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    \textbf{SEU122+1.p} MPTP bushy problem t2\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{t2\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU167+3.p} Basic properties of sets, theorem 119






$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ ( \mbox{cartesian\_product}_2(a,c)\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }\mbox{cartesian\_product}_2(c,a)\subseteq \mbox{cartesian\_product}_2(c,b) ) )\qquad\mbox{fof}(\mbox{t118\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( a\subseteq b\mbox{ and }c\subseteq d ) \ \Rightarrow \ \mbox{cartesian\_product}_2(a,c)\subseteq \mbox{cartesian\_product}_2(b,d) )\qquad\mbox{fof}(\mbox{t119\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SEU856$\wedge$5.p} TPS problem GAZING-THM46






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o}{:}\;( s = t\iff ( \lambda \mbox{xz}{:}\;a{:}\;( ( t@ \mbox{xz}\mbox{ and }\neg\,s@ \mbox{xz} )\mbox{ or }( s@ \mbox{xz}\mbox{ and }\neg\,t@ \mbox{xz} ) ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM46\_pme},\mbox{conjecture})$

\medskip\textbf{SEU990$\wedge$5.p} TPS problem from GRAPHS-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\exists \mbox{xc}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) \ \Rightarrow \ ( x@ ( \mbox{xc}@ x ) ) ) \ \Rightarrow \ ( \exists \mbox{xr}{:}\;a\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;\exists \mbox{xy}{:}\;b{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xy}{:}\;b{:}\;\exists \mbox{xx}{:}\;a{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}_1{:}\;b,\mbox{xy}_2{:}\;b{:}\;( ( \mbox{xr}@ \mbox{xx}@ \mbox{xy}_1\mbox{ and }\mbox{xr}@ \mbox{xx}@ \mbox{xy}_2 ) \ \Rightarrow \ \mbox{xy}_1 = \mbox{xy}_2 ) )\mbox{ or }\exists \mbox{xs}{:}\;b\to a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b{:}\;\exists \mbox{xy}{:}\;a{:}\;( \mbox{xs}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xy}{:}\;a{:}\;\exists \mbox{xx}{:}\;b{:}\;( \mbox{xs}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}_1{:}\;a,\mbox{xy}_2{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xx}@ \mbox{xy}_1\mbox{ and }\mbox{xs}@ \mbox{xx}@ \mbox{xy}_2 ) \ \Rightarrow \ \mbox{xy}_1 = \mbox{xy}_2 ) ) )\qquad\mbox{thf}(\mbox{cTHM610\_pme},\mbox{conjecture})$

\medskip\textbf{SEU819$\wedge$2.p} Ordinals

(! X:i.(! x:i.in x X $\to$ ordinal x) $\to$ transitiveset (setunion X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{nonempty} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\ne \mbox{emptyset} )\qquad\mbox{thf}(\mbox{nonempty},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{transitiveset} = ( \lambda a{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ ( \subseteq @ x@ a ) ) )\qquad\mbox{thf}(\mbox{transitiveset},\mbox{definition})$

$\mbox{setunionTransitive}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionTransitive\_type},\mbox{type})$

$\mbox{setunionTransitive} = ( \forall x{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{transitiveset}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{transitiveset}@ ( \mbox{setunion}@ x ) ) ) )\qquad\mbox{thf}(\mbox{setunionTransitive},\mbox{definition})$

$\mbox{stricttotalorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{stricttotalorderedByIn\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\mbox{in}@ x@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{in}@ x@ y\mbox{ or }\mbox{in}@ y@ x ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \neg\,\mbox{in}@ x@ x ) ) )\qquad\mbox{thf}(\mbox{stricttotalorderedByIn},\mbox{definition})$

$\mbox{wellorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{wellorderedByIn\_type},\mbox{type})$

$\mbox{wellorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{stricttotalorderedByIn}@ a\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{nonempty}@ x ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ x ) \ \Rightarrow \ ( \mbox{xx} = y\mbox{ or }\mbox{in}@ \mbox{xx}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{wellorderedByIn},\mbox{definition})$

$\mbox{ordinal}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ordinal\_type},\mbox{type})$

$\mbox{ordinal} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{transitiveset}@ \mbox{xx}\mbox{ and }\mbox{wellorderedByIn}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{ordinal},\mbox{definition})$

$\mbox{setunionTransitive} \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{ordinal}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{transitiveset}@ ( \mbox{setunion}@ x ) ) )\qquad\mbox{thf}(\mbox{setunionOrdinalLem}_1,\mbox{conjecture})$

\medskip\textbf{SEU503$\wedge$2.p} Preliminary Notions - Basic Laws of Logic

(! x:i.~(in x emptyset))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{in}@ \mbox{xx}@ \mbox{emptyset}\qquad\mbox{thf}(\mbox{notinemptyset},\mbox{conjecture})$

\medskip\textbf{SEU904$\wedge$5.p} TPS problem THM126

The composition of homomorphisms of binary operators is a
 homomorphism. Suggested by [BL+86].




$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xh}_1{:}\;g\to b,\mbox{xh}_2{:}\;b\to a,\mbox{xs}_1{:}\;g\to \mbox{\$o},\mbox{xf}_1{:}\;g\to g\to g,\mbox{xs}_2{:}\;b\to \mbox{\$o},\mbox{xf}_2{:}\;b\to b\to b,\mbox{xs}_3{:}\;a\to \mbox{\$o},\mbox{xf}_3{:}\;a\to a\to a{:}\;( ( \forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx}\mbox{ and }\mbox{xs}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xf}_2@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{xf}_2@ ( \mbox{xh}_1@ \mbox{xx} )@ ( \mbox{xh}_1@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx}\mbox{ and }\mbox{xs}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xf}_2@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx}\mbox{ and }\mbox{xs}_3@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_2@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx}\mbox{ and }\mbox{xs}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_2@ ( \mbox{xf}_2@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ \mbox{xx} )@ ( \mbox{xh}_2@ \mbox{xy} ) ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx}\mbox{ and }\mbox{xs}_3@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_2@ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xx} ) )@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM126\_pme},\mbox{conjecture})$

\medskip\textbf{SEU550$\wedge$2.p} A simple congruence property of exu

(forall phi:i$>$o.forall psi:i$>$o.(forall x:i.forall y:i.x = y $\to$
 (phi x $<$$\to$ psi y)) $\to$ (exu (lambda x:i.phi x) $<$$\to$
 exu (lambda x:i.psi x)))




$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu\_def},\mbox{definition})$

$\forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xpsi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} )\iff ( \mbox{xpsi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) )\iff ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xpsi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{exu\_\_Cong},\mbox{conjecture})$

\medskip\textbf{SEU874$\wedge$5.p} TPS problem from SET-TOPOLOGY-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cX}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cX},\mbox{type})$

$\mbox{cX} = ( \lambda \mbox{xx}{:}\;a{:}\;\exists s{:}\;a\to \mbox{\$o}{:}\;( \forall x_0{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x_0@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}_0{:}\;a\to \mbox{\$o}{:}\;( ( x_0@ \mbox{xx}_0 ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( s@ \mbox{xt} ) \ \Rightarrow \ ( x_0@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}_0@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x_0@ s ) )\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( s@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{cX}@ \mbox{xx}_0 ) )\mbox{ and }s@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cDOMLEMMA3\_pme},\mbox{conjecture})$

\medskip\textbf{SEU219+1.p} MPTP bushy problem t55\_funct\_1






$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{one\_to\_one}(a) \ \Rightarrow \ \mbox{function\_inverse}(a) = \mbox{relation\_inverse}(a) ) )\qquad\mbox{fof}(\mbox{d9\_funct}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(\mbox{function\_inverse}(a))\mbox{ and }\mbox{function}(\mbox{function\_inverse}(a)) ) )\qquad\mbox{fof}(\mbox{dt\_k2\_funct}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_inverse}(a)) )\qquad\mbox{fof}(\mbox{dt\_k4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) \ \Rightarrow \ ( \mbox{relation}(\mbox{relation\_inverse}(a))\mbox{ and }\mbox{function}(\mbox{relation\_inverse}(a)) ) )\qquad\mbox{fof}(\mbox{fc3\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation\_inverse}(\mbox{relation\_inverse}(a)) = a )\qquad\mbox{fof}(\mbox{involutiveness\_k4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{relation\_rng}(a) = \mbox{relation\_dom}(\mbox{relation\_inverse}(a))\mbox{ and }\mbox{relation\_dom}(a) = \mbox{relation\_rng}(\mbox{relation\_inverse}(a)) ) )\qquad\mbox{fof}(\mbox{t37\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{one\_to\_one}(a) \ \Rightarrow \ ( \mbox{relation\_rng}(a) = \mbox{relation\_dom}(\mbox{function\_inverse}(a))\mbox{ and }\mbox{relation\_dom}(a) = \mbox{relation\_rng}(\mbox{function\_inverse}(a)) ) ) )\qquad\mbox{fof}(\mbox{t55\_funct}_1,\mbox{conjecture})$

\medskip\textbf{SEU932$\wedge$5.p} TPS problem THM141

If some function which commutes with f has a unique fixed point,
 then f has a fixed point.




$\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \exists \mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ ( \mbox{xg}@ \mbox{xx} ) ) ) = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xg}@ ( \mbox{xf}@ \mbox{xx} ) ) )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xg}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xg}@ \mbox{xz} ) = \mbox{xz} \ \Rightarrow \ \mbox{xz} = \mbox{xx} ) ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM141\_pme},\mbox{conjecture})$

\medskip\textbf{SEU651$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.! z:i.! u:i.setadjoin (setadjoin x emptyset)
 (setadjoin (setadjoin x (setadjoin y emptyset)) emptyset) =
 setadjoin (setadjoin z emptyset) (setadjoin (setadjoin z
 (setadjoin u emptyset)) emptyset) $\to$ z = u $\to$ y = u)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{secondinupair}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{secondinupair\_type},\mbox{type})$

$\mbox{secondinupair} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{secondinupair},\mbox{definition})$

$\mbox{setukpairinjR}_{12}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjR12\_type},\mbox{type})$

$\mbox{setukpairinjR}_{12} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{setukpairinjR}_{12},\mbox{definition})$

$\mbox{uniqinunit} \ \Rightarrow \ ( \mbox{secondinupair} \ \Rightarrow \ ( \mbox{setukpairinjR}_{12} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xu}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) \ \Rightarrow \ ( \mbox{xz} = \mbox{xu} \ \Rightarrow \ \mbox{xy} = \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{setukpairinjR}_1,\mbox{conjecture})$

\medskip\textbf{SEU793$\wedge$2.p} More about Functions - Images of Functions

(! A:i.! f:i$>$i.? B:i.! x:i.in x B $<$$\to$ (? y:i.in y A \& x = f y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu},\mbox{definition})$

$\mbox{replAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{replAx\_type},\mbox{type})$

$\mbox{replAx} = ( \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{exu}@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) ) ) \ \Rightarrow \ \exists b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b )\iff \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xphi}@ \mbox{xy}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{replAx},\mbox{definition})$

$\mbox{replAx} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b )\iff \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xx} = ( \mbox{xf}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{image1Ex},\mbox{conjecture})$

\medskip\textbf{SEU733$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x (setminus A X) $\to$
 in x (setminus A (binintersect X Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{complementTnotintersectT}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementTnotintersectT\_type},\mbox{type})$

$\mbox{complementTnotintersectT} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{complementTnotintersectT},\mbox{definition})$

$\mbox{setminusI} \ \Rightarrow \ ( \mbox{complementTnotintersectT} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{complementImpComplementIntersect},\mbox{conjecture})$

\medskip\textbf{SEU565$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.! x:i.subset A B $\to$ ~(in x B) $\to$ ~(in x A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ a ) )\qquad\mbox{thf}(\mbox{subsetE}_2,\mbox{conjecture})$

\medskip\textbf{SEU818$\wedge$2.p} Ordinals

(! X:i.ordinal X $\to$ (! A:i.in A X $\to$ subset A X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{nonempty} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\ne \mbox{emptyset} )\qquad\mbox{thf}(\mbox{nonempty},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{transitiveset} = ( \lambda a{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ ( \subseteq @ x@ a ) ) )\qquad\mbox{thf}(\mbox{transitiveset},\mbox{definition})$

$\mbox{stricttotalorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{stricttotalorderedByIn\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\mbox{in}@ x@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{in}@ x@ y\mbox{ or }\mbox{in}@ y@ x ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \neg\,\mbox{in}@ x@ x ) ) )\qquad\mbox{thf}(\mbox{stricttotalorderedByIn},\mbox{definition})$

$\mbox{wellorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{wellorderedByIn\_type},\mbox{type})$

$\mbox{wellorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{stricttotalorderedByIn}@ a\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{nonempty}@ x ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ x ) \ \Rightarrow \ ( \mbox{xx} = y\mbox{ or }\mbox{in}@ \mbox{xx}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{wellorderedByIn},\mbox{definition})$

$\mbox{ordinal}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ordinal\_type},\mbox{type})$

$\mbox{ordinal} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{transitiveset}@ \mbox{xx}\mbox{ and }\mbox{wellorderedByIn}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{ordinal},\mbox{definition})$

$\mbox{ordinalTransSet}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ordinalTransSet\_type},\mbox{type})$

$\mbox{ordinalTransSet} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{ordinalTransSet},\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{ordinalTransSet} \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ ( \subseteq @ a@ x ) ) ) )\qquad\mbox{thf}(\mbox{ordinalTransSet}_1,\mbox{conjecture})$

\medskip\textbf{SEU504$\wedge$2.p} Preliminary Notions - Basic Laws of Logic

(! phi:i$>$o.exu ($\wedge$ x:i.phi x) $\to$ (? x:i.phi x))




$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu},\mbox{definition})$

$\mbox{exuE}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{exuE1\_type},\mbox{type})$

$\mbox{exuE}_1 = ( \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{exuE}_1,\mbox{definition})$

$\mbox{exuE}_1 \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{exuE3e},\mbox{conjecture})$

\medskip\textbf{SEU158+1.p} MPTP bushy problem t37\_zfmisc\_1






$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq b\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{t37\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq b\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l2\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU529$\wedge$2.p} Preliminary Notions - Equality Laws

(! x:i.! y:i.x = y $\to$ in x (setadjoin y emptyset))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{eqinunit},\mbox{conjecture})$

\medskip\textbf{SEU505$\wedge$2.p} Preliminary Notions - Basic Laws of Logic

(! A:i.! B:i.(! x:i.in x A $\to$ in x B) $\to$ (! x:i.in x B $\to$ in x A)
 $\to$ A = B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setextAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextAx\_type},\mbox{type})$

$\mbox{setextAx} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ a = b ) )\qquad\mbox{thf}(\mbox{setextAx},\mbox{definition})$

$\mbox{setextAx} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) )\qquad\mbox{thf}(\mbox{setext},\mbox{conjecture})$

\medskip\textbf{SEU127+1.p} MPTP bushy problem t17\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b)\subseteq a\qquad\mbox{fof}(\mbox{t17\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU257+1.p} MPTP bushy problem t32\_wellord1






$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{well\_ordering}(a)\iff ( \mbox{reflexive}(a)\mbox{ and }\mbox{transitive}(a)\mbox{ and }\mbox{antisymmetric}(a)\mbox{ and }\mbox{connected}(a)\mbox{ and }\mbox{well\_founded\_relation}(a) ) ) )\qquad\mbox{fof}(\mbox{d4\_wellord}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;\mbox{relation\_restriction}(a,b) = \mbox{set\_intersection}_2(a,\mbox{cartesian\_product}_2(b,b)) )\qquad\mbox{fof}(\mbox{d6\_wellord}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k2\_wellord}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( \mbox{reflexive}(b) \ \Rightarrow \ \mbox{reflexive}(\mbox{relation\_restriction}(b,a)) ) )\qquad\mbox{fof}(\mbox{t22\_wellord}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( \mbox{connected}(b) \ \Rightarrow \ \mbox{connected}(\mbox{relation\_restriction}(b,a)) ) )\qquad\mbox{fof}(\mbox{t23\_wellord}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( \mbox{transitive}(b) \ \Rightarrow \ \mbox{transitive}(\mbox{relation\_restriction}(b,a)) ) )\qquad\mbox{fof}(\mbox{t24\_wellord}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( \mbox{antisymmetric}(b) \ \Rightarrow \ \mbox{antisymmetric}(\mbox{relation\_restriction}(b,a)) ) )\qquad\mbox{fof}(\mbox{t25\_wellord}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( \mbox{well\_founded\_relation}(b) \ \Rightarrow \ \mbox{well\_founded\_relation}(\mbox{relation\_restriction}(b,a)) ) )\qquad\mbox{fof}(\mbox{t31\_wellord}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( \mbox{well\_ordering}(b) \ \Rightarrow \ \mbox{well\_ordering}(\mbox{relation\_restriction}(b,a)) ) )\qquad\mbox{fof}(\mbox{t32\_wellord}_1,\mbox{conjecture})$

\medskip\textbf{SEU729$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$
 subset X (setminus A (setminus A X)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{complementT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementT\_lem\_type},\mbox{type})$

$\mbox{complementT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{complementT\_lem},\mbox{definition})$

$\mbox{subsetTI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetTI\_type},\mbox{type})$

$\mbox{subsetTI} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \subseteq @ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{subsetTI},\mbox{definition})$

$\mbox{doubleComplementI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{doubleComplementI1\_type},\mbox{type})$

$\mbox{doubleComplementI}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{setminus}@ a@ x ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{doubleComplementI}_1,\mbox{definition})$

$\mbox{complementT\_lem} \ \Rightarrow \ ( \mbox{subsetTI} \ \Rightarrow \ ( \mbox{doubleComplementI}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \subseteq @ x@ ( \mbox{setminus}@ a@ ( \mbox{setminus}@ a@ x ) ) ) ) ) )\qquad\mbox{thf}(\mbox{doubleComplementSub}_1,\mbox{conjecture})$

\medskip\textbf{SEU957$\wedge$5.p} TPS problem from FUNCTION-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\exists \mbox{xc}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) \ \Rightarrow \ ( x@ ( \mbox{xc}@ x ) ) ) \ \Rightarrow \ ( \exists \mbox{xg}{:}\;a\to b{:}\;\forall y{:}\;b{:}\;\exists x{:}\;a{:}\;( \mbox{xg}@ x ) = y \ \Rightarrow \ \exists \mbox{xf}{:}\;b\to a{:}\;\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xf}@ \mbox{xx} ) = ( \mbox{xf}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cTHM607\_pme},\mbox{conjecture})$

\medskip\textbf{SEU782$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$
 breln1 A (binunion R S)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionEcases}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionEcases\_type},\mbox{type})$

$\mbox{binunionEcases} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{binunionEcases},\mbox{definition})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ ( \mbox{binunionEcases} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \mbox{breln}_1@ a@ ( \mbox{binunion}@ r@ s ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionprop},\mbox{conjecture})$

\medskip\textbf{SEU776$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! x:i.in x A $\to$ (! y:i.in y A $\to$
 in (kpair x y) R $\to$ in (kpair y x) (breln1invset A R))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dpsetconstrI\_type},\mbox{type})$

$\mbox{dpsetconstrI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xz}@ \mbox{xu} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrI},\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{breln1invset}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1invset\_type},\mbox{type})$

$\mbox{breln1invset} = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xy}@ \mbox{xx} )@ r ) ) )\qquad\mbox{thf}(\mbox{breln1invset},\mbox{definition})$

$\mbox{dpsetconstrI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xy}@ \mbox{xx} )@ ( \mbox{breln1invset}@ a@ r ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1invI},\mbox{conjecture})$

\medskip\textbf{SEU773$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ in (kpair x y) R $\to$ in (kpair x y) S)) $\to$
 subset R S))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{subbreln}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subbreln\_type},\mbox{type})$

$\mbox{subbreln} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ s ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) \ \Rightarrow \ ( \subseteq @ r@ s ) ) ) ) )\qquad\mbox{thf}(\mbox{subbreln},\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{subbreln} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) \ \Rightarrow \ ( \subseteq @ r@ s ) ) ) )\qquad\mbox{thf}(\mbox{subbreln}_1,\mbox{conjecture})$

\medskip\textbf{SEU861$\wedge$5.p} TPS problem THM531E

Subset of a finite set is finite.




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cB}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cC}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$( \forall p{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \forall e{:}\;a\to \mbox{\$o}{:}\;( \neg\,\exists \mbox{xt}{:}\;a{:}\;( e@ \mbox{xt} ) \ \Rightarrow \ ( p@ e ) )\mbox{ and }\forall y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a,z{:}\;a\to \mbox{\$o}{:}\;( ( p@ y\mbox{ and }\forall \mbox{xx}_{27}{:}\;a{:}\;( ( z@ \mbox{xx}_{27} ) \ \Rightarrow \ ( y@ \mbox{xx}_{27}\mbox{ or }\mbox{xx}_{27} = \mbox{xx} ) ) ) \ \Rightarrow \ ( p@ z ) ) ) \ \Rightarrow \ ( p@ \mbox{cC} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cB}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cC}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall p{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \forall e{:}\;a\to \mbox{\$o}{:}\;( \neg\,\exists \mbox{xt}{:}\;a{:}\;( e@ \mbox{xt} ) \ \Rightarrow \ ( p@ e ) )\mbox{ and }\forall y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a,z{:}\;a\to \mbox{\$o}{:}\;( ( p@ y\mbox{ and }\forall \mbox{xx}_{28}{:}\;a{:}\;( ( z@ \mbox{xx}_{28} ) \ \Rightarrow \ ( y@ \mbox{xx}_{28}\mbox{ or }\mbox{xx}_{28} = \mbox{xx} ) ) ) \ \Rightarrow \ ( p@ z ) ) ) \ \Rightarrow \ ( p@ \mbox{cB} ) )\qquad\mbox{thf}(\mbox{cTHM531E\_pme},\mbox{conjecture})$

\medskip\textbf{SEU659$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! A:i.! B:i.! x:i.! y:i.in (kpair x y) (cartprod A B) $\to$ in x A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprodmempair}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodmempair1\_type},\mbox{type})$

$\mbox{cartprodmempair}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprodmempair}_1,\mbox{definition})$

$\mbox{setukpairinjL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjL\_type},\mbox{type})$

$\mbox{setukpairinjL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) )\qquad\mbox{thf}(\mbox{setukpairinjL},\mbox{definition})$

$\mbox{cartprodmempair}_1 \ \Rightarrow \ ( \mbox{setukpairinjL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{cartprodpairmemEL},\mbox{conjecture})$

\medskip\textbf{SEU734$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 subset (setminus A X) (setminus A (binintersect X Y))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{binintersectT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectT\_lem\_type},\mbox{type})$

$\mbox{binintersectT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{binintersect}@ x@ y )@ ( \mbox{powerset}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectT\_lem},\mbox{definition})$

$\mbox{complementT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementT\_lem\_type},\mbox{type})$

$\mbox{complementT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{complementT\_lem},\mbox{definition})$

$\mbox{subsetTI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetTI\_type},\mbox{type})$

$\mbox{subsetTI} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \subseteq @ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{subsetTI},\mbox{definition})$

$\mbox{complementImpComplementIntersect}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementImpComplementIntersect\_type},\mbox{type})$

$\mbox{complementImpComplementIntersect} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{complementImpComplementIntersect},\mbox{definition})$

$\mbox{binintersectT\_lem} \ \Rightarrow \ ( \mbox{complementT\_lem} \ \Rightarrow \ ( \mbox{subsetTI} \ \Rightarrow \ ( \mbox{complementImpComplementIntersect} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \subseteq @ ( \mbox{setminus}@ a@ x )@ ( \mbox{setminus}@ a@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{complementSubsetComplementIntersect},\mbox{conjecture})$

\medskip\textbf{SEU561$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.A = B $\to$ subset A B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\subseteq  = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{subset},\mbox{definition})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ ( \subseteq @ a@ b ) )\qquad\mbox{thf}(\mbox{eqimpsubset}_1,\mbox{conjecture})$

\medskip\textbf{SEU746$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! phi:o.! x:i.in x A $\to$ in x (binunion X Y) $\to$
 (in x X $\to$ phi) $\to$ (in x Y $\to$ phi) $\to$ phi)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionE\_type},\mbox{type})$

$\mbox{binunionE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a\mbox{ or }\mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binunionE},\mbox{definition})$

$\mbox{binunionE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ y ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionTE},\mbox{conjecture})$

\medskip\textbf{SEU758$\wedge$2.p} Typed Set Theory - First Wizard of Oz Examples - WoZ1 Lemmas

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x (binintersect X Y) $\to$ in x A)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE\_type},\mbox{type})$

$\mbox{powersetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetE},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectEL\_type},\mbox{type})$

$\mbox{binintersectEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{definition})$

$\mbox{powersetE} \ \Rightarrow \ ( \mbox{binintersectEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{woz13rule}_0,\mbox{conjecture})$

\medskip\textbf{SEU499$\wedge$2.p} Foundation - Axioms - Logical Axioms

(!phi:i$>$o.exu($\wedge$x:i.phi x)$\to$(?x:i.phi x \& (!y:i.phi y $\to$ x = y)))




$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu},\mbox{definition})$

$\forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exuE}_1,\mbox{conjecture})$

\medskip\textbf{SEU973$\wedge$5.p} TPS problem from COINDUCTIVE-PU-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xt} = \mbox{cZ}\iff \mbox{xu} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ \mbox{xt} = \mbox{xu} ) ) ) \ \Rightarrow \ \forall \mbox{xv}{:}\;a{:}\;( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{cZ} ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xv} ) ) \ \Rightarrow \ \exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ \mbox{xv}@ ( \mbox{cP}@ \mbox{xv}@ \mbox{cZ} ) )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xu} = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_NAT\_LESS\_SUCC\_pme},\mbox{conjecture})$

\medskip\textbf{SEU463$\wedge$1.p} The transitive closure of a binary relation is transitive, part 3






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i},s{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{trans}@ s\mbox{ and }\mbox{subrel}@ r@ s\mbox{ and }\mbox{tc}@ r@ x@ y\mbox{ and }\mbox{tc}@ r@ y@ z ) \ \Rightarrow \ ( s@ x@ z ) )\qquad\mbox{thf}(\mbox{transitive\_closure\_is\_transitive}_3,\mbox{conjecture})$

\medskip\textbf{SEU937$\wedge$5.p} TPS problem THM48

The composition of injective functions is injective.




$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$c{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\forall f{:}\;b\to a,g{:}\;c\to b{:}\;( ( \forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;c,\mbox{xy}{:}\;c{:}\;( ( g@ \mbox{xx} ) = ( g@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;c,\mbox{xy}{:}\;c{:}\;( ( f@ ( g@ \mbox{xx} ) ) = ( f@ ( g@ \mbox{xy} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cTHM48\_pme},\mbox{conjecture})$

\medskip\textbf{SEU602$\wedge$2.p} Preliminary Notions - Operations on Sets - Set Difference

(! A:i.! B:i.! x:i.in x (setminus A B) $\to$ in x A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminus} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminus},\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) )\qquad\mbox{thf}(\mbox{setminusEL},\mbox{conjecture})$

\medskip\textbf{SEU604$\wedge$2.p} Preliminary Notions - Operations on Sets - Set Difference

(! A:i.! B:i.subset A B $\to$ setminus A B = emptyset)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{subsetemptysetimpeq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetemptysetimpeq\_type},\mbox{type})$

$\mbox{subsetemptysetimpeq} = ( \forall a{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ \mbox{emptyset} ) \ \Rightarrow \ a = \mbox{emptyset} ) )\qquad\mbox{thf}(\mbox{subsetemptysetimpeq},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusEL\_type},\mbox{type})$

$\mbox{setminusEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{setminusEL},\mbox{definition})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ ( \mbox{subsetemptysetimpeq} \ \Rightarrow \ ( \mbox{setminusEL} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( \mbox{setminus}@ a@ b ) = \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusSubset}_2,\mbox{conjecture})$

\medskip\textbf{SEU646$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! A:i.! B:i.! u:i.in u (cartprod A B) $\to$ in (kfst u) A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprodmempair}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodmempair1\_type},\mbox{type})$

$\mbox{cartprodmempair}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprodmempair}_1,\mbox{definition})$

$\mbox{kfst}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kfst\_type},\mbox{type})$

$\mbox{kfstpairEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{kfstpairEq\_type},\mbox{type})$

$\mbox{kfstpairEq} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{kfst}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{kfstpairEq},\mbox{definition})$

$\mbox{cartprodmempair}_1 \ \Rightarrow \ ( \mbox{kfstpairEq} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kfst}@ \mbox{xu} )@ a ) ) )\qquad\mbox{thf}(\mbox{cartprodfstin},\mbox{conjecture})$

\medskip\textbf{SEU775$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ breln1 A (breln1invset A R))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{setOfPairsIsBReln}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setOfPairsIsBReln1\_type},\mbox{type})$

$\mbox{setOfPairsIsBReln}_1 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{breln}_1@ a@ ( \mbox{dpsetconstr}@ a@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{setOfPairsIsBReln}_1,\mbox{definition})$

$\mbox{breln1invset}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1invset\_type},\mbox{type})$

$\mbox{breln1invset} = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xy}@ \mbox{xx} )@ r ) ) )\qquad\mbox{thf}(\mbox{breln1invset},\mbox{definition})$

$\mbox{setOfPairsIsBReln}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ ( \mbox{breln}_1@ a@ ( \mbox{breln1invset}@ a@ r ) ) )\qquad\mbox{thf}(\mbox{breln1invprop},\mbox{conjecture})$

\medskip\textbf{SEU941$\wedge$5.p} TPS problem FN-THM-1






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$c{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\forall f{:}\;a\to b,g{:}\;b\to c{:}\;( ( \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( g@ \mbox{xx} ) = ( g@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( g@ ( f@ \mbox{xx} ) ) = ( g@ ( f@ \mbox{xy} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cFN\_THM\_1\_pme},\mbox{conjecture})$

\medskip\textbf{SEU272+1.p} MPTP bushy problem s1\_xboole\_0\_\_e3\_38\_1\_\_ordinal1






$\forall a,b{:}\;( \mbox{ordinal}(b) \ \Rightarrow \ \exists c{:}\;\forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,\mbox{succ}(b))\mbox{ and }\exists e{:}\;( \mbox{ordinal}(e)\mbox{ and }d = e\mbox{ and }\mbox{in}(e,a) ) ) ) )\qquad\mbox{fof}(\mbox{s1\_xboole\_0\_\_e8\_6\_\_wellord2\_}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) \ \Rightarrow \ \mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{cc2\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) ) )\qquad\mbox{fof}(\mbox{cc3\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc3\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{succ}(a))\qquad\mbox{fof}(\mbox{fc1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) )\qquad\mbox{fof}(\mbox{cc1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ ( \neg\,\mbox{empty}(\mbox{succ}(a))\mbox{ and }\mbox{epsilon\_transitive}(\mbox{succ}(a))\mbox{ and }\mbox{epsilon\_connected}(\mbox{succ}(a))\mbox{ and }\mbox{ordinal}(\mbox{succ}(a)) ) )\qquad\mbox{fof}(\mbox{fc3\_ordinal}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{ordinal}(b) \ \Rightarrow \ ( \forall c,d,e{:}\;( ( c = d\mbox{ and }\exists f{:}\;( \mbox{ordinal}(f)\mbox{ and }d = f\mbox{ and }\mbox{in}(f,a) )\mbox{ and }c = e\mbox{ and }\exists g{:}\;( \mbox{ordinal}(g)\mbox{ and }e = g\mbox{ and }\mbox{in}(g,a) ) ) \ \Rightarrow \ d = e ) \ \Rightarrow \ \exists c{:}\;\forall d{:}\;( \mbox{in}(d,c)\iff \exists e{:}\;( \mbox{in}(e,\mbox{succ}(b))\mbox{ and }e = d\mbox{ and }\exists h{:}\;( \mbox{ordinal}(h)\mbox{ and }d = h\mbox{ and }\mbox{in}(h,a) ) ) ) ) )\qquad\mbox{fof}(\mbox{s1\_tarski\_\_e8\_6\_\_wellord2\_}_1,\mbox{axiom})$

\medskip\textbf{SEU805$\wedge$2.p} The Foundation Axiom

(! A:i.nonempty A $\to$ (? X:i.in X A \& binintersect X A = emptyset))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{foundationAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{foundationAx\_type},\mbox{type})$

$\mbox{foundationAx} = ( \forall a{:}\;\mbox{\$i}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \exists b{:}\;\mbox{\$i}{:}\;( \mbox{in}@ b@ a\mbox{ and }\neg\,\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b\mbox{ and }\mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{foundationAx},\mbox{definition})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{nonempty} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\ne \mbox{emptyset} )\qquad\mbox{thf}(\mbox{nonempty},\mbox{definition})$

$\mbox{nonemptyE}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{nonemptyE1\_type},\mbox{type})$

$\mbox{nonemptyE}_1 = ( \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{nonempty}@ a ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{nonemptyE}_1,\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{disjointsetsI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{disjointsetsI1\_type},\mbox{type})$

$\mbox{disjointsetsI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \neg\,\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{binintersect}@ a@ b ) = \mbox{emptyset} ) )\qquad\mbox{thf}(\mbox{disjointsetsI}_1,\mbox{definition})$

$\mbox{foundationAx} \ \Rightarrow \ ( \mbox{nonemptyE}_1 \ \Rightarrow \ ( \mbox{disjointsetsI}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{nonempty}@ a ) \ \Rightarrow \ \exists x{:}\;\mbox{\$i}{:}\;( \mbox{in}@ x@ a\mbox{ and }( \mbox{binintersect}@ x@ a ) = \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{foundation}_2,\mbox{conjecture})$

\medskip\textbf{SEU831$\wedge$5.p} TPS problem GAZING-THM32






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( s@ \mbox{xz}\mbox{ or }s@ \mbox{xz} ) ) = s\qquad\mbox{thf}(\mbox{cGAZING\_THM32\_pme},\mbox{conjecture})$

\medskip\textbf{SEU894$\wedge$5.p} TPS problem THM15-0






$\forall f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists g{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \forall p{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( p@ f\mbox{ and }\forall h{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( p@ h ) \ \Rightarrow \ ( p@ \lambda t{:}\;\mbox{\$i}{:}\;( f@ ( h@ t ) ) ) ) ) \ \Rightarrow \ ( p@ g ) )\mbox{ and }\exists x{:}\;\mbox{\$i}{:}\;( ( g@ x ) = x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( g@ y ) = y \ \Rightarrow \ x = y ) ) ) \ \Rightarrow \ \exists y{:}\;\mbox{\$i}{:}\;( f@ y ) = y )\qquad\mbox{thf}(\mbox{cTHM15\_0\_pme},\mbox{conjecture})$

\medskip\textbf{SEU284+1.p} MPTP bushy problem s3\_funct\_1\_\_e16\_22\_\_wellord2






$\forall a{:}\;\exists b{:}\;( \mbox{relation}(b)\mbox{ and }\mbox{function}(b)\mbox{ and }\mbox{relation\_dom}(b) = a\mbox{ and }\forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{apply}(b,c) = \mbox{singleton}(c) ) )\qquad\mbox{fof}(\mbox{s3\_funct\_1\_\_e16\_22\_\_wellord}_2,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_funct}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \forall b,c,d{:}\;( ( \mbox{in}(b,a)\mbox{ and }c = \mbox{singleton}(b)\mbox{ and }d = \mbox{singleton}(b) ) \ \Rightarrow \ c = d )\mbox{ and }\forall b{:}\;\neg\,\mbox{in}(b,a)\mbox{ and }\forall c{:}\;c\ne \mbox{singleton}(b) ) \ \Rightarrow \ \exists b{:}\;( \mbox{relation}(b)\mbox{ and }\mbox{function}(b)\mbox{ and }\mbox{relation\_dom}(b) = a\mbox{ and }\forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{apply}(b,c) = \mbox{singleton}(c) ) ) )\qquad\mbox{fof}(\mbox{s2\_funct\_1\_\_e16\_22\_\_wellord2\_}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU991$\wedge$5.p} TPS problem from GRAPHS-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\exists s{:}\;a\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;\exists \mbox{xy}{:}\;b{:}\;( s@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}_1{:}\;b,\mbox{xy}_2{:}\;b{:}\;( ( s@ \mbox{xx}@ \mbox{xy}_1\mbox{ and }s@ \mbox{xx}@ \mbox{xy}_2 ) \ \Rightarrow \ \mbox{xy}_1 = \mbox{xy}_2 )\mbox{ and }\forall \mbox{xx}_1{:}\;a,\mbox{xx}_2{:}\;a,\mbox{xy}{:}\;b{:}\;( ( s@ \mbox{xx}_1@ \mbox{xy}\mbox{ and }s@ \mbox{xx}_2@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}_1 = \mbox{xx}_2 ) )\mbox{ or }\exists r{:}\;b\to a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b{:}\;\exists \mbox{xy}{:}\;a{:}\;( r@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}_1{:}\;a,\mbox{xy}_2{:}\;a{:}\;( ( r@ \mbox{xx}@ \mbox{xy}_1\mbox{ and }r@ \mbox{xx}@ \mbox{xy}_2 ) \ \Rightarrow \ \mbox{xy}_1 = \mbox{xy}_2 )\mbox{ and }\forall \mbox{xx}_1{:}\;b,\mbox{xx}_2{:}\;b,\mbox{xy}{:}\;a{:}\;( ( r@ \mbox{xx}_1@ \mbox{xy}\mbox{ and }r@ \mbox{xx}_2@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}_1 = \mbox{xx}_2 ) )\qquad\mbox{thf}(\mbox{cTHM612\_pme},\mbox{conjecture})$

\medskip\textbf{SEU668$\wedge$2.p} Ordered Pairs - Sets of Pairs

(! A:i.! B:i.! phi:i$>$(i$>$o).! x:i.in x A $\to$ (! y:i.in y B $\to$
 in (kpair x y) (dpsetconstr A B ($\wedge$ z,u:i.phi z u)) $\to$ phi x y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{setukpairinjL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjL\_type},\mbox{type})$

$\mbox{setukpairinjL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) )\qquad\mbox{thf}(\mbox{setukpairinjL},\mbox{definition})$

$\mbox{setukpairinjR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjR\_type},\mbox{type})$

$\mbox{setukpairinjR} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ \mbox{xy} = \mbox{xu} ) )\qquad\mbox{thf}(\mbox{setukpairinjR},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstr} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{dsetconstr}@ ( \mbox{cartprod}@ a@ b )@ \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xphi}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstr},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{setukpairinjL} \ \Rightarrow \ ( \mbox{setukpairinjR} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xz}@ \mbox{xu} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrERa},\mbox{conjecture})$

\medskip\textbf{SEU603$\wedge$2.p} Preliminary Notions - Operations on Sets - Set Difference

(! A:i.! B:i.! x:i.in x (setminus A B) $\to$ ~(in x B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminus} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminus},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b )\qquad\mbox{thf}(\mbox{setminusER},\mbox{conjecture})$

\medskip\textbf{SEU612$\wedge$2.p} Preliminary Notions - Operations on Sets - Symmetric Difference

(! A:i.! B:i.! x:i.in x A $\to$ ~(in x B) $\to$ in x (symdiff A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIL\_type},\mbox{type})$

$\mbox{binunionIL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIL},\mbox{definition})$

$\mbox{symdiff}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{symdiff\_type},\mbox{type})$

$\mbox{symdiff} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{binunion}@ a@ b )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{in}@ \mbox{xx}@ a\mbox{ or }\neg\,\mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{symdiff},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{binunionIL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{symdiff}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{symdiffI}_1,\mbox{conjecture})$

\medskip\textbf{SEU643$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! u:i.iskpair u $\to$ singleton (dsetconstr (setunion u) ($\wedge$ x:i.in
 (setadjoin x emptyset) u)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{iskpair}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{iskpair\_type},\mbox{type})$

$\mbox{iskpair} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setunion}@ a )\mbox{ and }a = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{iskpair},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{ex1I}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ex1I\_type},\mbox{type})$

$\mbox{ex1I} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xy} = \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{ex}_1@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ex1I},\mbox{definition})$

$\mbox{setukpairinjL}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjL1\_type},\mbox{type})$

$\mbox{setukpairinjL}_1 = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) )\qquad\mbox{thf}(\mbox{setukpairinjL}_1,\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{ex1I} \ \Rightarrow \ ( \mbox{setukpairinjL}_1 \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{iskpair}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ ( \mbox{setunion}@ \mbox{xu} )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ \mbox{xu} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{kfstsingleton},\mbox{conjecture})$

\medskip\textbf{SEU779$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ (! z:i.in z A $\to$ in (kpair x z) R $\to$
 in (kpair z y) S $\to$ in (kpair x y) (breln1compset A R S))))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dpsetconstrI\_type},\mbox{type})$

$\mbox{dpsetconstrI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xz}@ \mbox{xu} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrI},\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{breln1compset}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1compset\_type},\mbox{type})$

$\mbox{breln1compset} = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i},s{:}\;\mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xz}@ a\mbox{ and }\mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xz} )@ r\mbox{ and }\mbox{in}@ ( \mbox{kpair}@ \mbox{xz}@ \mbox{xy} )@ s ) ) )\qquad\mbox{thf}(\mbox{breln1compset},\mbox{definition})$

$\mbox{dpsetconstrI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xz}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xz} )@ r ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xz}@ \mbox{xy} )@ s ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{breln1compset}@ a@ r@ s ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1compI},\mbox{conjecture})$

\medskip\textbf{SEU582$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.! x:i.in x A $\to$ in x (binunion A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{setunionI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionI\_type},\mbox{type})$

$\mbox{setunionI} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{setunionI},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunion} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{binunion},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{setunionI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIL},\mbox{conjecture})$

\medskip\textbf{SEU511$\wedge$2.p} Preliminary Notions - Basic Laws of Logic

(! A:i.(? x:i.in x A) $\to$ nonempty A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{nonempty} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\ne \mbox{emptyset} )\qquad\mbox{thf}(\mbox{nonempty},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{nonempty}@ a ) )\qquad\mbox{thf}(\mbox{nonemptyI}_1,\mbox{conjecture})$

\medskip\textbf{SEU760$\wedge$2.p} Typed Set Theory - First Wizard of Oz Examples - WoZ1 Lemmas

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! Z:i.in Z (powerset A) $\to$ subset Y Z $\to$
 subset (binintersect X Y) Z)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectER\_type},\mbox{type})$

$\mbox{binintersectER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersectER},\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ ( \mbox{binintersectER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ y@ z ) \ \Rightarrow \ ( \subseteq @ ( \mbox{binintersect}@ x@ y )@ z ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz13rule}_2,\mbox{conjecture})$

\medskip\textbf{SEU126+2.p} MPTP chainy problem t12\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_union}_2(a,b) = b )\qquad\mbox{fof}(\mbox{t12\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{lemma})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{t2\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t3\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\neg\,\exists c{:}\;\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t4\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;a\subseteq \mbox{set\_union}_2(a,b)\qquad\mbox{fof}(\mbox{t7\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }c\subseteq b ) \ \Rightarrow \ \mbox{set\_union}_2(a,c)\subseteq b )\qquad\mbox{fof}(\mbox{t8\_xboole}_1,\mbox{lemma})$

\medskip\textbf{SEU552$\wedge$2.p} A simple congruence property of setadjoin

(forall x:i.forall y:i.x = y $\to$ (forall z:i.forall u:i.z = u $\to$
 setadjoin x z = setadjoin y u))




$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ \forall \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xz} = \mbox{xu} \ \Rightarrow \ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xz} ) = ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{setadjoin\_\_Cong},\mbox{conjecture})$

\medskip\textbf{SEU852$\wedge$5.p} TPS problem GAZING-THM36






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o},u{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( t@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx} ) )\iff \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx}\mbox{ and }u@ \mbox{xx}\mbox{ and }\neg\,t@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM36\_pme},\mbox{conjecture})$

\medskip\textbf{SEU173+1.p} MPTP bushy problem l71\_subset\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{powerset}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c\subseteq a ) )\qquad\mbox{fof}(\mbox{d1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \neg\,\mbox{empty}(a) \ \Rightarrow \ ( \mbox{element}(b,a)\iff \mbox{in}(b,a) ) )\mbox{ and }( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{element}(b,a)\iff \mbox{empty}(b) ) ) )\qquad\mbox{fof}(\mbox{d2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) \ \Rightarrow \ \mbox{element}(a,\mbox{powerset}(b)) )\qquad\mbox{fof}(\mbox{l71\_subset}_1,\mbox{conjecture})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU217+3.p} Functions and their basic properties, theorem 35






$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;\mbox{relation}(\mbox{identity\_relation}(a))\qquad\mbox{fof}(\mbox{dt\_k6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(\mbox{identity\_relation}(a))\mbox{ and }\mbox{function}(\mbox{identity\_relation}(a)) )\qquad\mbox{fof}(\mbox{fc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(b,a) \ \Rightarrow \ \mbox{apply}(\mbox{identity\_relation}(a),b) = b )\qquad\mbox{fof}(\mbox{t35\_funct}_1,\mbox{conjecture})$

$\forall a,b{:}\;( ( \mbox{relation}(b)\mbox{ and }\mbox{function}(b) ) \ \Rightarrow \ ( b = \mbox{identity\_relation}(a)\iff ( \mbox{relation\_dom}(b) = a\mbox{ and }\forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{apply}(b,c) = c ) ) ) )\qquad\mbox{fof}(\mbox{t34\_funct}_1,\mbox{axiom})$

\medskip\textbf{SEU662$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! A:i.! B:i.! x:i.in x A $\to$ (! y:i.in y B $\to$
 kfst (kpair x y) = x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kfst}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kfst\_type},\mbox{type})$

$\mbox{kfstpairEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{kfstpairEq\_type},\mbox{type})$

$\mbox{kfstpairEq} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{kfst}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{kfstpairEq},\mbox{definition})$

$\mbox{cartprodmempaircEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodmempaircEq\_type},\mbox{type})$

$\mbox{cartprodmempaircEq} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cartprodmempaircEq},\mbox{definition})$

$\mbox{kfstpairEq} \ \Rightarrow \ ( \mbox{cartprodmempaircEq} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{kfst}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cartprodfstpairEq},\mbox{conjecture})$

\medskip\textbf{SEU817$\wedge$2.p} Ordinals

(! X:i.ordinal X $\to$ (! x:i.! A:i.in A X $\to$ in x A $\to$ in x X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{nonempty} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\ne \mbox{emptyset} )\qquad\mbox{thf}(\mbox{nonempty},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{transitiveset} = ( \lambda a{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ ( \subseteq @ x@ a ) ) )\qquad\mbox{thf}(\mbox{transitiveset},\mbox{definition})$

$\mbox{stricttotalorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{stricttotalorderedByIn\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\mbox{in}@ x@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{in}@ x@ y\mbox{ or }\mbox{in}@ y@ x ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \neg\,\mbox{in}@ x@ x ) ) )\qquad\mbox{thf}(\mbox{stricttotalorderedByIn},\mbox{definition})$

$\mbox{wellorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{wellorderedByIn\_type},\mbox{type})$

$\mbox{wellorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{stricttotalorderedByIn}@ a\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{nonempty}@ x ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ x ) \ \Rightarrow \ ( \mbox{xx} = y\mbox{ or }\mbox{in}@ \mbox{xx}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{wellorderedByIn},\mbox{definition})$

$\mbox{ordinal}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ordinal\_type},\mbox{type})$

$\mbox{ordinal} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{transitiveset}@ \mbox{xx}\mbox{ and }\mbox{wellorderedByIn}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{ordinal},\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) )\qquad\mbox{thf}(\mbox{ordinalTransSet},\mbox{conjecture})$

\medskip\textbf{SEU540$\wedge$2.p} Preliminary Notions - Dependent Connective Laws

(! phi:o.phi $\to$ in emptyset (prop2set phi))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{prop2set}{:}\;\mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{prop2set\_type},\mbox{type})$

$\mbox{prop2set} = ( \lambda \mbox{xphi}{:}\;\mbox{\$o}{:}\;( \mbox{dsetconstr}@ ( \mbox{powerset}@ \mbox{emptyset} )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{prop2set},\mbox{definition})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{powersetI} \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( \mbox{xphi} \ \Rightarrow \ ( \mbox{in}@ \mbox{emptyset}@ ( \mbox{prop2set}@ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{prop2setI},\mbox{conjecture})$

\medskip\textbf{SEU672$\wedge$2.p} Functions

(! A:i.! B:i.! f:i.func A B f $\to$ (! x:i.in x A $\to$ singleton
 (dsetconstr B ($\wedge$ y:i.in (kpair x y) f))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) ) ) ) )\qquad\mbox{thf}(\mbox{funcImageSingleton},\mbox{conjecture})$

\medskip\textbf{SEU816$\wedge$2.p} Ordinals

(! X:i.ordinal X $\to$ (! Y:i.ordinal Y $\to$
 transitiveset (binintersect X Y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{nonempty} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\ne \mbox{emptyset} )\qquad\mbox{thf}(\mbox{nonempty},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{transitiveset} = ( \lambda a{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ ( \subseteq @ x@ a ) ) )\qquad\mbox{thf}(\mbox{transitiveset},\mbox{definition})$

$\mbox{binintTransitive}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintTransitive\_type},\mbox{type})$

$\mbox{binintTransitive} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{transitiveset}@ x ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{transitiveset}@ y ) \ \Rightarrow \ ( \mbox{transitiveset}@ ( \mbox{binintersect}@ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{binintTransitive},\mbox{definition})$

$\mbox{stricttotalorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{stricttotalorderedByIn\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\mbox{in}@ x@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{in}@ x@ y\mbox{ or }\mbox{in}@ y@ x ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \neg\,\mbox{in}@ x@ x ) ) )\qquad\mbox{thf}(\mbox{stricttotalorderedByIn},\mbox{definition})$

$\mbox{wellorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{wellorderedByIn\_type},\mbox{type})$

$\mbox{wellorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{stricttotalorderedByIn}@ a\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{nonempty}@ x ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ x ) \ \Rightarrow \ ( \mbox{xx} = y\mbox{ or }\mbox{in}@ \mbox{xx}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{wellorderedByIn},\mbox{definition})$

$\mbox{ordinal}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ordinal\_type},\mbox{type})$

$\mbox{ordinal} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{transitiveset}@ \mbox{xx}\mbox{ and }\mbox{wellorderedByIn}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{ordinal},\mbox{definition})$

$\mbox{binintTransitive} \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ y ) \ \Rightarrow \ ( \mbox{transitiveset}@ ( \mbox{binintersect}@ x@ y ) ) ) )\qquad\mbox{thf}(\mbox{ordinalMinLem}_1,\mbox{conjecture})$

\medskip\textbf{SEU129+1.p} MPTP bushy problem t26\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_intersection}_2(a,c)\subseteq \mbox{set\_intersection}_2(b,c) )\qquad\mbox{fof}(\mbox{t26\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU254+1.p} MPTP bushy problem t24\_wellord1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;\mbox{relation\_restriction}(a,b) = \mbox{set\_intersection}_2(a,\mbox{cartesian\_product}_2(b,b)) )\qquad\mbox{fof}(\mbox{d6\_wellord}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k2\_wellord}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{transitive}(a)\iff \forall b,c,d{:}\;( ( \mbox{in}(\mbox{ordered\_pair}(b,c),a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(b,d),a) ) ) )\qquad\mbox{fof}(\mbox{l2\_wellord}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{t106\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_restriction}(c,b))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(a,\mbox{cartesian\_product}_2(b,b)) ) ) )\qquad\mbox{fof}(\mbox{t16\_wellord}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( \mbox{transitive}(b) \ \Rightarrow \ \mbox{transitive}(\mbox{relation\_restriction}(b,a)) ) )\qquad\mbox{fof}(\mbox{t24\_wellord}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU203+1.p} MPTP bushy problem t143\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b,c{:}\;( c = \mbox{relation\_image}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e{:}\;( \mbox{in}(\mbox{ordered\_pair}(e,d),a)\mbox{ and }\mbox{in}(e,b) ) ) ) )\qquad\mbox{fof}(\mbox{d13\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k9\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_image}(c,b))\iff \exists d{:}\;( \mbox{in}(d,\mbox{relation\_dom}(c))\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,a),c)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{t143\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU548$\wedge$2.p} A simple congruence property of in

(forall A:i.forall B:i.A = B $\to$ (forall x:i.forall y:i.x = y $\to$
 (in x A $<$$\to$ in y B)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{inCongP}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{inCongP\_type},\mbox{type})$

$\mbox{inCongP} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{inCongP},\mbox{definition})$

$\mbox{inCongP} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{conjecture})$

\medskip\textbf{SEU923$\wedge$5.p} TPS problem THM54






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cF}{:}\;b\to a\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{cF}@ \mbox{xx} ) = ( \mbox{cF}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{cF}@ \mbox{xx} ) = ( \mbox{cF}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM54\_pme},\mbox{conjecture})$

\medskip\textbf{SEU688$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i.in f (funcSet A B) $\to$ (! x:i.in x A $\to$
 (! y:i.in y B $\to$ in (kpair x y) f $\to$ ap A B f x = y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{infuncsetfunc}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{infuncsetfunc\_type},\mbox{type})$

$\mbox{infuncsetfunc} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ ( \mbox{func}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{infuncsetfunc},\mbox{definition})$

$\mbox{funcGraphProp}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcGraphProp2\_type},\mbox{type})$

$\mbox{funcGraphProp}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{funcGraphProp}_2,\mbox{definition})$

$\mbox{infuncsetfunc} \ \Rightarrow \ ( \mbox{funcGraphProp}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{funcGraphProp}_4,\mbox{conjecture})$

\medskip\textbf{SEU192+1.p} MPTP bushy problem t86\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_dom\_restriction}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff ( \mbox{in}(d,b)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,e),a) ) ) ) ) )\qquad\mbox{fof}(\mbox{d11\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_dom\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k7\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_dom}(\mbox{relation\_dom\_restriction}(c,b)))\iff ( \mbox{in}(a,b)\mbox{ and }\mbox{in}(a,\mbox{relation\_dom}(c)) ) ) )\qquad\mbox{fof}(\mbox{t86\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU752$\wedge$2.p} Typed Set Theory - Laws for Typed Sets - DeMorgan Laws

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x (binunion (setminus A X) (setminus A Y)) $\to$
 in x (setminus A (binintersect X Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionE\_type},\mbox{type})$

$\mbox{binunionE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a\mbox{ or }\mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binunionE},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{binintersectTELcontra}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectTELcontra\_type},\mbox{type})$

$\mbox{binintersectTELcontra} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ x \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectTELcontra},\mbox{definition})$

$\mbox{binintersectTERcontra}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectTERcontra\_type},\mbox{type})$

$\mbox{binintersectTERcontra} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ y \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectTERcontra},\mbox{definition})$

$\mbox{binunionE} \ \Rightarrow \ ( \mbox{setminusI} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ ( \mbox{binintersectTELcontra} \ \Rightarrow \ ( \mbox{binintersectTERcontra} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{setminus}@ a@ y ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{demorgan1b},\mbox{conjecture})$

\medskip\textbf{SEU846$\wedge$5.p} TPS problem GAZING-THM11






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o},u{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( t@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( u@ \mbox{xx}\mbox{ and }\neg\,t@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx}\mbox{ and }\neg\,s@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM11\_pme},\mbox{conjecture})$

\medskip\textbf{SEU959$\wedge$5.p} TPS problem from FUNCTION-THMS






$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xx}@ ( \mbox{xh}@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx} ) = ( g@ \mbox{xy} ) \ \Rightarrow \ ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ ( g@ \mbox{xx} ) ) ) = f )\qquad\mbox{thf}(\mbox{cTHM588AC2\_pme},\mbox{conjecture})$

\medskip\textbf{SEU534$\wedge$2.p} Preliminary Notions - Bounded Quantifier Laws

(! A:i.! phi:i$>$o.(? x:i.in x A \& phi x) $\to$ dsetconstr A
 ($\wedge$ x:i.phi x) = emptyset $\to$ false)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{emptysetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) = \mbox{emptyset} \ \Rightarrow \ \mbox{\$false} ) ) )\qquad\mbox{thf}(\mbox{emptyE}_1,\mbox{conjecture})$

\medskip\textbf{SEU538$\wedge$2.p} Preliminary Notions - Bounded Quantifier Laws

(! A:i.! phi:i$>$o.~(? x:i.in x A \& phi x) $\to$ (! x:i.in x A $\to$
 ~(phi x)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in},\mbox{type})$

$\forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \neg\,\mbox{xphi}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{quantDeMorgan}_3,\mbox{conjecture})$

\medskip\textbf{SEU693$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i.in f (funcSet A B) $\to$ (! x:i.in x A $\to$
 ap A B f x = ap A B f x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in},\mbox{type})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap},\mbox{type})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{ap2apEq}_1,\mbox{conjecture})$

\medskip\textbf{SEU208+1.p} MPTP bushy problem t166\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b,c{:}\;( c = \mbox{relation\_inverse\_image}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),a)\mbox{ and }\mbox{in}(e,b) ) ) ) )\qquad\mbox{fof}(\mbox{d14\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) ) )\qquad\mbox{fof}(\mbox{d5\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k10\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{fc6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_rng}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_rng}(a)) ) )\qquad\mbox{fof}(\mbox{fc8\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_inverse\_image}(c,b))\iff \exists d{:}\;( \mbox{in}(d,\mbox{relation\_rng}(c))\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(a,d),c)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{t166\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU171+1.p} MPTP bushy problem t50\_subset\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( ( \neg\,\mbox{empty}(a) \ \Rightarrow \ ( \mbox{element}(b,a)\iff \mbox{in}(b,a) ) )\mbox{ and }( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{element}(b,a)\iff \mbox{empty}(b) ) ) )\qquad\mbox{fof}(\mbox{d2\_subset}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{subset\_complement}(a,b) = \mbox{set\_difference}(a,b) )\qquad\mbox{fof}(\mbox{d5\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{element}(\mbox{subset\_complement}(a,b),\mbox{powerset}(a)) )\qquad\mbox{fof}(\mbox{dt\_k3\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{subset\_complement}(a,\mbox{subset\_complement}(a,b)) = b )\qquad\mbox{fof}(\mbox{involutiveness\_k3\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( a\ne \mbox{empty\_set} \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \forall c{:}\;( \mbox{element}(c,a) \ \Rightarrow \ ( \neg\,\mbox{in}(c,b) \ \Rightarrow \ \mbox{in}(c,\mbox{subset\_complement}(a,b)) ) ) ) )\qquad\mbox{fof}(\mbox{t50\_subset}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU486$\wedge$1.p} Confluence implies local confluence






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{confl}@ r ) \ \Rightarrow \ ( \mbox{lconfl}@ r ) )\qquad\mbox{thf}(\mbox{confluence\_implies\_local\_confluence},\mbox{conjecture})$

\medskip\textbf{SEU654$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.! z:i.! u:i.kpair x y = kpair z u $\to$ y = u)




$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{setukpairinjR}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjR2\_type},\mbox{type})$

$\mbox{setukpairinjR}_2 = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xu}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xy} = \mbox{xu} ) )\qquad\mbox{thf}(\mbox{setukpairinjR}_2,\mbox{definition})$

$\mbox{setukpairinjR}_2 \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ \mbox{xy} = \mbox{xu} )\qquad\mbox{thf}(\mbox{setukpairinjR},\mbox{conjecture})$

\medskip\textbf{SEU781$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ in (kpair x y) (breln1compset A R S) $\to$
 (! phi:o.(! z:i.in z A $\to$ in (kpair x z) R $\to$ in (kpair z y) S $\to$
 phi) $\to$ phi)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{breln1compset}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1compset\_type},\mbox{type})$

$\mbox{breln1compE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1compE\_type},\mbox{type})$

$\mbox{breln1compE} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{breln1compset}@ a@ r@ s ) ) \ \Rightarrow \ \exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xz}@ a\mbox{ and }\mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xz} )@ r\mbox{ and }\mbox{in}@ ( \mbox{kpair}@ \mbox{xz}@ \mbox{xy} )@ s ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1compE},\mbox{definition})$

$\mbox{breln1compE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{breln1compset}@ a@ r@ s ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( \forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xz}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xz} )@ r ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xz}@ \mbox{xy} )@ s ) \ \Rightarrow \ \mbox{xphi} ) ) ) \ \Rightarrow \ \mbox{xphi} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1compEex},\mbox{conjecture})$

\medskip\textbf{SEU707$\wedge$2.p} Conditionals

(! A:i.! phi:o.! x:i.in x A $\to$ (! y:i.in y A $\to$ phi $\to$
 if A phi x y = x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{iftrueProp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{iftrueProp1\_type},\mbox{type})$

$\mbox{iftrueProp}_1 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{xphi} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iftrueProp}_1,\mbox{definition})$

$\mbox{ifSingleton}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ifSingleton\_type},\mbox{type})$

$\mbox{ifSingleton} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ifSingleton},\mbox{definition})$

$\mbox{if}{:}\;\mbox{\$i}\to \mbox{\$o}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{if\_type},\mbox{type})$

$\mbox{if} = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{if},\mbox{definition})$

$\mbox{theeq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{theeq\_type},\mbox{type})$

$\mbox{theeq} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{singleton}@ x ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{setunion}@ x ) = \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{theeq},\mbox{definition})$

$\mbox{iftrueProp}_1 \ \Rightarrow \ ( \mbox{ifSingleton} \ \Rightarrow \ ( \mbox{theeq} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{xphi} \ \Rightarrow \ ( \mbox{if}@ a@ \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) = \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{iftrue},\mbox{conjecture})$

\medskip\textbf{SEU452$\wedge$1.p} Hofman's Marktoberdorf exercise

The equivalence of two characterizations of the smallest
 "quasi-PER" containing a given binary relation R, one the obvious
 inductive characterization.




$r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(r,\mbox{type})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall s{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( r@ x@ y ) \ \Rightarrow \ ( s@ x@ y ) )\mbox{ and }\forall w{:}\;\mbox{\$i},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( ( s@ x@ y\mbox{ and }s@ z@ y\mbox{ and }s@ z@ w ) \ \Rightarrow \ ( s@ x@ w ) ) ) \ \Rightarrow \ ( s@ a@ b ) )\iff \forall p{:}\;\mbox{\$i}\to \mbox{\$o},q{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( r@ x@ y ) \ \Rightarrow \ ( ( p@ x )\iff ( q@ y ) ) ) \ \Rightarrow \ ( ( p@ a )\iff ( q@ b ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SEU629$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! B:i.! x:i.in x A $\to$ (! y:i.in y B $\to$ in (kpair x y)
 (powerset (powerset (binunion A B)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{ubforcartprodlem}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ubforcartprodlem2\_type},\mbox{type})$

$\mbox{ubforcartprodlem}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) )@ ( \mbox{powerset}@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ubforcartprodlem}_2,\mbox{definition})$

$\mbox{ubforcartprodlem}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{powerset}@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ubforcartprodlem}_3,\mbox{conjecture})$

\medskip\textbf{SEU626$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! B:i.! x:i.in x A $\to$ (! y:i.in y B $\to$ in (setadjoin x
 (setadjoin y emptyset)) (powerset (binunion A B))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{powersetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI1\_type},\mbox{type})$

$\mbox{powersetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ b@ a ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI}_1,\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{upairsubunion}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{upairsubunion\_type},\mbox{type})$

$\mbox{upairsubunion} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \subseteq @ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ ( \mbox{binunion}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{upairsubunion},\mbox{definition})$

$\mbox{powersetI}_1 \ \Rightarrow \ ( \mbox{upairsubunion} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{upairinpowunion},\mbox{conjecture})$

\medskip\textbf{SEU594$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.binintersect A B = B $\to$ subset B A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersect} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersect},\mbox{definition})$

$\mbox{binintersectEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectEL\_type},\mbox{type})$

$\mbox{binintersectEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{definition})$

$\mbox{in\_\_Cong} \ \Rightarrow \ ( \mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{binintersectEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{binintersect}@ a@ b ) = b \ \Rightarrow \ ( \subseteq @ b@ a ) ) ) )\qquad\mbox{thf}(\mbox{binintersectSubset}_3,\mbox{conjecture})$

\medskip\textbf{SEU152+1.p} MPTP bushy problem l23\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_union}_2(\mbox{singleton}(a),b) = b )\qquad\mbox{fof}(\mbox{l23\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq b\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l2\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_union}_2(a,b) = b )\qquad\mbox{fof}(\mbox{t12\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SEU759$\wedge$2.p} Typed Set Theory - First Wizard of Oz Examples - WoZ1 Lemmas

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! Z:i.in Z (powerset A) $\to$ subset X Z $\to$
 subset (binintersect X Y) Z)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectEL\_type},\mbox{type})$

$\mbox{binintersectEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ ( \mbox{binintersectEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ x@ z ) \ \Rightarrow \ ( \subseteq @ ( \mbox{binintersect}@ x@ y )@ z ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz13rule}_1,\mbox{conjecture})$

\medskip\textbf{SEU125+2.p} MPTP chainy problem t8\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{lemma})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{t2\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t3\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\neg\,\exists c{:}\;\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t4\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;a\subseteq \mbox{set\_union}_2(a,b)\qquad\mbox{fof}(\mbox{t7\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }c\subseteq b ) \ \Rightarrow \ \mbox{set\_union}_2(a,c)\subseteq b )\qquad\mbox{fof}(\mbox{t8\_xboole}_1,\mbox{conjecture})$

\medskip\textbf{SEU958$\wedge$5.p} TPS problem from FUNCTION-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\exists \mbox{xc}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) \ \Rightarrow \ ( x@ ( \mbox{xc}@ x ) ) ) \ \Rightarrow \ ( \exists \mbox{xg}{:}\;a\to b{:}\;\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \mbox{xg}@ \mbox{xx} ) = ( \mbox{xg}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\mbox{ or }\exists \mbox{xf}{:}\;b\to a{:}\;\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xf}@ \mbox{xx} ) = ( \mbox{xf}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cTHM611\_pme},\mbox{conjecture})$

\medskip\textbf{SEU183+1.p} MPTP bushy problem t45\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) ) )\qquad\mbox{fof}(\mbox{d5\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \forall c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_composition}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff \exists f{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,f),a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(f,e),b) ) ) ) ) ) )\qquad\mbox{fof}(\mbox{d8\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_composition}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k5\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_rng}(\mbox{relation\_composition}(a,b))\subseteq \mbox{relation\_rng}(b) ) )\qquad\mbox{fof}(\mbox{t45\_relat}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU933$\wedge$5.p} TPS problem THM196B

It is not true that if [k COMPOSE j] is an iterate of j, then k 
 must be an iterate of j, provided  we have distinct elements a 
 and b.




$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$a\ne b \ \Rightarrow \ \neg\,\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xj}\mbox{ and }\forall \mbox{xj}_2{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj}_2 ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xj}@ ( \mbox{xj}_2@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xj}\mbox{ and }\forall \mbox{xj}_3{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj}_3 ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xj}@ ( \mbox{xj}_3@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xk} ) ) )\qquad\mbox{thf}(\mbox{cTHM196B\_pme},\mbox{conjecture})$

\medskip\textbf{SEU263+1.p} MPTP bushy problem t14\_relset\_1






$\forall a,b,c{:}\;( \mbox{element}(c,\mbox{powerset}(\mbox{cartesian\_product}_2(a,b))) \ \Rightarrow \ \mbox{relation}(c) )\qquad\mbox{fof}(\mbox{cc1\_relset}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation\_of}_2(c,a,b)\iff c\subseteq \mbox{cartesian\_product}_2(a,b) )\qquad\mbox{fof}(\mbox{d1\_relset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_relset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation\_of2\_as\_subset}(c,a,b) \ \Rightarrow \ \mbox{element}(c,\mbox{powerset}(\mbox{cartesian\_product}_2(a,b))) )\qquad\mbox{fof}(\mbox{dt\_m2\_relset}_1,\mbox{axiom})$

$\forall a,b{:}\;\exists c{:}\;\mbox{relation\_of}_2(c,a,b)\qquad\mbox{fof}(\mbox{existence\_m1\_relset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\exists c{:}\;\mbox{relation\_of2\_as\_subset}(c,a,b)\qquad\mbox{fof}(\mbox{existence\_m2\_relset}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation\_of2\_as\_subset}(c,a,b)\iff \mbox{relation\_of}_2(c,a,b) )\qquad\mbox{fof}(\mbox{redefinition\_m2\_relset}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( a\subseteq b\mbox{ and }c\subseteq d ) \ \Rightarrow \ \mbox{cartesian\_product}_2(a,c)\subseteq \mbox{cartesian\_product}_2(b,d) )\qquad\mbox{fof}(\mbox{t119\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation\_of2\_as\_subset}(c,a,b) \ \Rightarrow \ ( \mbox{relation\_dom}(c)\subseteq a\mbox{ and }\mbox{relation\_rng}(c)\subseteq b ) )\qquad\mbox{fof}(\mbox{t12\_relset}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{relation\_of2\_as\_subset}(d,c,a) \ \Rightarrow \ ( \mbox{relation\_rng}(d)\subseteq b \ \Rightarrow \ \mbox{relation\_of2\_as\_subset}(d,c,b) ) )\qquad\mbox{fof}(\mbox{t14\_relset}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ a\subseteq \mbox{cartesian\_product}_2(\mbox{relation\_dom}(a),\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{t21\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

\medskip\textbf{SEU741$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! Z:i.in Z (powerset A) $\to$ (! x:i.in x A $\to$
 in x (binintersect X Y) $\to$ in x (binunion Y Z)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIL\_type},\mbox{type})$

$\mbox{binunionIL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIL},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectER\_type},\mbox{type})$

$\mbox{binintersectER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersectER},\mbox{definition})$

$\mbox{binunionIL} \ \Rightarrow \ ( \mbox{binintersectER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ y@ z ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{inIntersectImpInUnion}_2,\mbox{conjecture})$

\medskip\textbf{SEU649$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.x = y $\to$ setadjoin x (setadjoin y emptyset) =
 setadjoin x emptyset)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{eqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{eqinunit\_type},\mbox{type})$

$\mbox{eqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{eqinunit},\mbox{definition})$

$\mbox{upairset2E}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{upairset2E\_type},\mbox{type})$

$\mbox{upairset2E} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) \ \Rightarrow \ ( \mbox{xz} = \mbox{xx}\mbox{ or }\mbox{xz} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{upairset2E},\mbox{definition})$

$\mbox{setext} \ \Rightarrow \ ( \mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ ( \mbox{eqinunit} \ \Rightarrow \ ( \mbox{upairset2E} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{setukpairinjR}_{11},\mbox{conjecture})$

\medskip\textbf{SEU275+1.p} MPTP bushy problem t7\_wellord2






$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) )\qquad\mbox{fof}(\mbox{cc1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) \ \Rightarrow \ \mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{cc2\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{well\_ordering}(a)\iff ( \mbox{reflexive}(a)\mbox{ and }\mbox{transitive}(a)\mbox{ and }\mbox{antisymmetric}(a)\mbox{ and }\mbox{connected}(a)\mbox{ and }\mbox{well\_founded\_relation}(a) ) ) )\qquad\mbox{fof}(\mbox{d4\_wellord}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{relation}(\mbox{inclusion\_relation}(a))\qquad\mbox{fof}(\mbox{dt\_k1\_wellord}_2,\mbox{axiom})$

$\exists a{:}\;( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{reflexive}(\mbox{inclusion\_relation}(a))\qquad\mbox{fof}(\mbox{t2\_wellord}_2,\mbox{axiom})$

$\forall a{:}\;\mbox{transitive}(\mbox{inclusion\_relation}(a))\qquad\mbox{fof}(\mbox{t3\_wellord}_2,\mbox{axiom})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ \mbox{connected}(\mbox{inclusion\_relation}(a)) )\qquad\mbox{fof}(\mbox{t4\_wellord}_2,\mbox{axiom})$

$\forall a{:}\;\mbox{antisymmetric}(\mbox{inclusion\_relation}(a))\qquad\mbox{fof}(\mbox{t5\_wellord}_2,\mbox{axiom})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ \mbox{well\_founded\_relation}(\mbox{inclusion\_relation}(a)) )\qquad\mbox{fof}(\mbox{t6\_wellord}_2,\mbox{axiom})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ \mbox{well\_ordering}(\mbox{inclusion\_relation}(a)) )\qquad\mbox{fof}(\mbox{t7\_wellord}_2,\mbox{conjecture})$

\medskip\textbf{SEU635$\wedge$2.p} Ordered Pairs - Singletons

(! A:i.subset A (setunion (setadjoin A emptyset)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{setunionI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionI\_type},\mbox{type})$

$\mbox{setunionI} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{setunionI},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{setunionI} \ \Rightarrow \ ( \mbox{subsetI}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( \subseteq @ a@ ( \mbox{setunion}@ ( \mbox{setadjoin}@ a@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{setunionsingleton}_2,\mbox{conjecture})$

\medskip\textbf{SEU164+1.p} MPTP bushy problem t99\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{powerset}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c\subseteq a ) )\qquad\mbox{fof}(\mbox{d1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq b\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l2\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{union}(\mbox{powerset}(a)) = a\qquad\mbox{fof}(\mbox{t99\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU809$\wedge$2.p} Omega and Peano

(! x:i.in x omega $\to$ in (setadjoin x x) omega)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{omega}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{omega\_type},\mbox{type})$

$\mbox{omegaSAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{omegaSAx\_type},\mbox{type})$

$\mbox{omegaSAx} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{omega} ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xx} )@ \mbox{omega} ) ) )\qquad\mbox{thf}(\mbox{omegaSAx},\mbox{definition})$

$\mbox{omegaSAx} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{omega} ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xx} )@ \mbox{omega} ) )\qquad\mbox{thf}(\mbox{omegaSclos},\mbox{conjecture})$

\medskip\textbf{SEU660$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! A:i.! B:i.! x:i.! y:i.in (kpair x y) (cartprod A B) $\to$ in y B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprodmempair}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodmempair1\_type},\mbox{type})$

$\mbox{cartprodmempair}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprodmempair}_1,\mbox{definition})$

$\mbox{setukpairinjR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjR\_type},\mbox{type})$

$\mbox{setukpairinjR} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ \mbox{xy} = \mbox{xu} ) )\qquad\mbox{thf}(\mbox{setukpairinjR},\mbox{definition})$

$\mbox{cartprodmempair}_1 \ \Rightarrow \ ( \mbox{setukpairinjR} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ b ) ) )\qquad\mbox{thf}(\mbox{cartprodpairmemER},\mbox{conjecture})$

\medskip\textbf{SEU462$\wedge$1.p} The transitive closure of a binary relation is transitive, part 2






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},s{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{trans}@ s\mbox{ and }\mbox{subrel}@ r@ s\mbox{ and }\mbox{tc}@ r@ x@ y ) \ \Rightarrow \ ( s@ x@ y ) )\qquad\mbox{thf}(\mbox{transitive\_closure\_is\_transitive}_2,\mbox{conjecture})$

\medskip\textbf{SEU774$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ in (kpair x y) R $\to$ in (kpair x y) S)) $\to$
 (! x:i.in x A $\to$ (! y:i.in y A $\to$ in (kpair x y) S $\to$
 in (kpair x y) R)) $\to$ R = S))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{eqbreln}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{eqbreln\_type},\mbox{type})$

$\mbox{eqbreln} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ s ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) \ \Rightarrow \ r = s ) ) ) ) )\qquad\mbox{thf}(\mbox{eqbreln},\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{eqbreln} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) \ \Rightarrow \ r = s ) ) ) )\qquad\mbox{thf}(\mbox{eqbreln}_1,\mbox{conjecture})$

\medskip\textbf{SEU717$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x X $\to$ in x Y) $\to$ in X (powerset Y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{powersetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI1\_type},\mbox{type})$

$\mbox{powersetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ b@ a ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI}_1,\mbox{definition})$

$\mbox{subsetTI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetTI\_type},\mbox{type})$

$\mbox{subsetTI} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \subseteq @ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{subsetTI},\mbox{definition})$

$\mbox{powersetI}_1 \ \Rightarrow \ ( \mbox{subsetTI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \mbox{in}@ x@ ( \mbox{powerset}@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{powersetTI}_1,\mbox{conjecture})$

\medskip\textbf{SEU893$\wedge$5.p} TPS problem THM34A






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cF}{:}\;b\to a\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cS}{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cR}{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\forall \mbox{xx}{:}\;a{:}\;( \exists \mbox{xt}{:}\;b{:}\;( ( \mbox{cR}@ \mbox{xt}\mbox{ or }\mbox{cS}@ \mbox{xt} )\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) ) \ \Rightarrow \ ( \exists \mbox{xt}{:}\;b{:}\;( \mbox{cR}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) )\mbox{ or }\exists \mbox{xt}{:}\;b{:}\;( \mbox{cS}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM34A\_pme},\mbox{conjecture})$

\medskip\textbf{SEU896$\wedge$5.p} TPS problem THM500






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$y{:}\;b\qquad\mbox{thf}(y,\mbox{type})$

$g{:}\;b\to a\qquad\mbox{thf}(g,\mbox{type})$

$x{:}\;b\qquad\mbox{thf}(x,\mbox{type})$

$( \lambda \mbox{xz}{:}\;a{:}\;\exists \mbox{xt}{:}\;b{:}\;( ( \mbox{xt} = x\mbox{ or }\mbox{xt} = y )\mbox{ and }\mbox{xz} = ( g@ \mbox{xt} ) ) ) = ( \lambda \mbox{xv}{:}\;a{:}\;( \mbox{xv} = ( g@ x )\mbox{ or }\mbox{xv} = ( g@ y ) ) )\qquad\mbox{thf}(\mbox{cTHM500\_pme},\mbox{conjecture})$

\medskip\textbf{SEU619$\wedge$2.p} Ordered Pairs - Kuratowski Pairs

(! x:i.! y:i.iskpair (setadjoin (setadjoin x emptyset) (setadjoin
 (setadjoin x (setadjoin y emptyset)) emptyset)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{iskpair}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{iskpair\_type},\mbox{type})$

$\mbox{iskpair} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setunion}@ a )\mbox{ and }a = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{iskpair},\mbox{definition})$

$\mbox{setukpairIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairIL\_type},\mbox{type})$

$\mbox{setukpairIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{setukpairIL},\mbox{definition})$

$\mbox{setukpairIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairIR\_type},\mbox{type})$

$\mbox{setukpairIR} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setunion}@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{setukpairIR},\mbox{definition})$

$\mbox{setukpairIL} \ \Rightarrow \ ( \mbox{setukpairIR} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{iskpair}@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{kpairiskpair},\mbox{conjecture})$

\medskip\textbf{SEU477$\wedge$1.p} Another definition of terminating

The definition of terminating is the same as saying there is no
 non-empty subset A in which every element has an R successor
 (in A).




$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{term} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\exists a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i}{:}\;( a@ x )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( a@ x ) \ \Rightarrow \ \exists y{:}\;\mbox{\$i}{:}\;( a@ y\mbox{ and }r@ x@ y ) ) ) )\qquad\mbox{thf}(\mbox{alternative\_formulation\_of\_terminating},\mbox{conjecture})$

\medskip\textbf{SEU735$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 in (setminus A X) (powerset (setminus A (binintersect X Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{powersetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI1\_type},\mbox{type})$

$\mbox{powersetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ b@ a ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI}_1,\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{complementSubsetComplementIntersect}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementSubsetComplementIntersect\_type},\mbox{type})$

$\mbox{complementSubsetComplementIntersect} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \subseteq @ ( \mbox{setminus}@ a@ x )@ ( \mbox{setminus}@ a@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{complementSubsetComplementIntersect},\mbox{definition})$

$\mbox{powersetI}_1 \ \Rightarrow \ ( \mbox{complementSubsetComplementIntersect} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{powerset}@ ( \mbox{setminus}@ a@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{complementInPowersetComplementIntersect},\mbox{conjecture})$

\medskip\textbf{SEU641$\wedge$2.p} Ordered Pairs - Singletons

(! x:i.! y:i.setadjoin x emptyset = setadjoin y emptyset $\to$ x = y)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) = ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{singletonsuniq},\mbox{conjecture})$

\medskip\textbf{SEU143+1.p} MPTP bushy problem l1\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{singleton}(a)\ne \mbox{empty\_set}\qquad\mbox{fof}(\mbox{l1\_zfmisc}_1,\mbox{conjecture})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

\medskip\textbf{SEU187+1.p} MPTP bushy problem t60\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) ) )\qquad\mbox{fof}(\mbox{d5\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{fc6\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\mbox{relation\_dom}(\mbox{empty\_set}) = \mbox{empty\_set}\mbox{ and }\mbox{relation\_rng}(\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t60\_relat}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU262+1.p} MPTP bushy problem t12\_relset\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{element}(c,\mbox{powerset}(\mbox{cartesian\_product}_2(a,b))) \ \Rightarrow \ \mbox{relation}(c) )\qquad\mbox{fof}(\mbox{cc1\_relset}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) ) )\qquad\mbox{fof}(\mbox{d5\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_relset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation\_of2\_as\_subset}(c,a,b) \ \Rightarrow \ \mbox{element}(c,\mbox{powerset}(\mbox{cartesian\_product}_2(a,b))) )\qquad\mbox{fof}(\mbox{dt\_m2\_relset}_1,\mbox{axiom})$

$\forall a,b{:}\;\exists c{:}\;\mbox{relation\_of}_2(c,a,b)\qquad\mbox{fof}(\mbox{existence\_m1\_relset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\exists c{:}\;\mbox{relation\_of2\_as\_subset}(c,a,b)\qquad\mbox{fof}(\mbox{existence\_m2\_relset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation\_of2\_as\_subset}(c,a,b)\iff \mbox{relation\_of}_2(c,a,b) )\qquad\mbox{fof}(\mbox{redefinition\_m2\_relset}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{t106\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation\_of2\_as\_subset}(c,a,b) \ \Rightarrow \ ( \mbox{relation\_dom}(c)\subseteq a\mbox{ and }\mbox{relation\_rng}(c)\subseteq b ) )\qquad\mbox{fof}(\mbox{t12\_relset}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU518$\wedge$2.p} Preliminary Notions - Power Sets and Unions

(! A:i.! B:i.(! x:i.in x B $\to$ in x A) $\to$ in B (powerset A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetAx\_type},\mbox{type})$

$\mbox{powersetAx} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) )\iff \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetAx},\mbox{definition})$

$\mbox{powersetAx} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{conjecture})$

\medskip\textbf{SEU877$\wedge$5.p} TPS problem from SET-TOPOLOGY-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cA}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\mbox{cB}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cB}@ \mbox{xx} ) ) \ \Rightarrow \ ( \lambda u{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( u@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cA}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( u@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xe}{:}\;a\to \mbox{\$o}{:}\;( \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}_0{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx}_0 ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}_0@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xe} ) )\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xx}_0 ) )\mbox{ and }\forall \mbox{xy}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xy}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xy}@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ ( u@ \mbox{xy} ) ) ) ) ) ) = ( \lambda u{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;\exists v{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( v@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cB}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( v@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xe}{:}\;a\to \mbox{\$o}{:}\;( \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}_0{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx}_0 ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}_0@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xe} ) )\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xx}_0 ) )\mbox{ and }\forall \mbox{xy}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cB}@ \mbox{xy}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xy}@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ ( v@ \mbox{xy} ) ) ) )\mbox{ and }u = ( \lambda \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( v@ \mbox{xx}\mbox{ and }\mbox{cA}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cDOMTHM3\_pme},\mbox{conjecture})$

\medskip\textbf{SEU901$\wedge$5.p} TPS problem THM131D






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xh}_1{:}\;g\to b,\mbox{xh}_2{:}\;b\to a,\mbox{xs}_1{:}\;g\to \mbox{\$o},\mbox{xf}_1{:}\;g\to g,\mbox{xs}_2{:}\;b\to \mbox{\$o},\mbox{xf}_2{:}\;b\to b,\mbox{xs}_3{:}\;a\to \mbox{\$o},\mbox{xf}_3{:}\;a\to a{:}\;( ( \forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_1@ ( \mbox{xf}_1@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xf}_2@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx} ) ) = ( \mbox{xf}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xf}_2@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_2@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xh}_2@ ( \mbox{xf}_2@ \mbox{xx} ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xh}_2@ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx} ) ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM131D\_pme},\mbox{conjecture})$

\medskip\textbf{SEU736$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 subset X (setminus A Y) $\to$ (! x:i.in x A $\to$ in x Y $\to$
 in x (setminus A X))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{contrasubsetT}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{contrasubsetT\_type},\mbox{type})$

$\mbox{contrasubsetT} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \subseteq @ x@ ( \mbox{setminus}@ a@ y ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ y ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ x ) ) ) ) ) )\qquad\mbox{thf}(\mbox{contrasubsetT},\mbox{definition})$

$\mbox{setminusI} \ \Rightarrow \ ( \mbox{contrasubsetT} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ x@ ( \mbox{setminus}@ a@ y ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{contraSubsetComplement},\mbox{conjecture})$

\medskip\textbf{SEU929$\wedge$5.p} TPS problem THM170






$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ f\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( f@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ g ) ) \ \Rightarrow \ ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( f@ ( g@ \mbox{xx} ) ) ) = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( g@ ( f@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cTHM170\_pme},\mbox{conjecture})$

\medskip\textbf{SEU570$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.! C:i.subset A B $\to$ subset B C $\to$ subset A C)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ c ) \ \Rightarrow \ ( \subseteq @ a@ c ) ) ) )\qquad\mbox{thf}(\mbox{subsetTrans},\mbox{conjecture})$

\medskip\textbf{SEU496$\wedge$1.p} Transitive closure and strict (partial) orders

The transitive closure of a strict (partial) order is a strict
 (partial) order.




$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{so}@ r ) \ \Rightarrow \ ( \mbox{so}@ ( \mbox{tc}@ r ) ) )\qquad\mbox{thf}(\mbox{transitive\_closure\_of\_strict\_order\_is\_strict\_order},\mbox{conjecture})$

\medskip\textbf{SEU899$\wedge$5.p} TPS problem THM34






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cF}{:}\;b\to a\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cS}{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cR}{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\forall \mbox{xx}{:}\;a{:}\;( \exists \mbox{xt}{:}\;b{:}\;( ( \mbox{cR}@ \mbox{xt}\mbox{ or }\mbox{cS}@ \mbox{xt} )\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) )\iff ( \exists \mbox{xt}{:}\;b{:}\;( \mbox{cR}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) )\mbox{ or }\exists \mbox{xt}{:}\;b{:}\;( \mbox{cS}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM34\_pme},\mbox{conjecture})$

\medskip\textbf{SEU839$\wedge$5.p} TPS problem GAZING-THM26






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( s@ \mbox{xz}\mbox{ or }t@ \mbox{xz} ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( t@ \mbox{xz}\mbox{ or }s@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM26\_pme},\mbox{conjecture})$

\medskip\textbf{SEU138+1.p} MPTP bushy problem t48\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_difference}(a,\mbox{set\_difference}(a,b)) = \mbox{set\_intersection}_2(a,b)\qquad\mbox{fof}(\mbox{t48\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU935$\wedge$5.p} TPS problem FN-THM-2






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$c{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\forall f{:}\;a\to b,g{:}\;b\to c{:}\;( ( \forall y{:}\;b{:}\;\exists x{:}\;a{:}\;( f@ x ) = y\mbox{ and }\forall y{:}\;c{:}\;\exists x{:}\;b{:}\;( g@ x ) = y ) \ \Rightarrow \ \forall y{:}\;c{:}\;\exists x{:}\;a{:}\;( g@ ( f@ x ) ) = y )\qquad\mbox{thf}(\mbox{cFN\_THM\_2\_pme},\mbox{conjecture})$

\medskip\textbf{SEU572$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! x:i.! B:i.subset A B $\to$ subset A (setadjoin x B))




$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetTrans}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetTrans\_type},\mbox{type})$

$\mbox{subsetTrans} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ c ) \ \Rightarrow \ ( \subseteq @ a@ c ) ) ) )\qquad\mbox{thf}(\mbox{subsetTrans},\mbox{definition})$

$\mbox{setadjoinSub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinSub\_type},\mbox{type})$

$\mbox{setadjoinSub} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i}{:}\;( \subseteq @ a@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{setadjoinSub},\mbox{definition})$

$\mbox{subsetTrans} \ \Rightarrow \ ( \mbox{setadjoinSub} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( \subseteq @ a@ ( \mbox{setadjoin}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{setadjoinSub}_2,\mbox{conjecture})$

\medskip\textbf{SEU684$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i.func A B f $\to$ (! x:i.in x A $\to$
 in (kpair x (ap A B f x)) f))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{theprop}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{theprop\_type},\mbox{type})$

$\mbox{theprop} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{singleton}@ x ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setunion}@ x )@ x ) ) )\qquad\mbox{thf}(\mbox{theprop},\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{funcImageSingleton}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcImageSingleton\_type},\mbox{type})$

$\mbox{funcImageSingleton} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{funcImageSingleton},\mbox{definition})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{ap} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{dsetconstr}@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{ap},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{theprop} \ \Rightarrow \ ( \mbox{funcImageSingleton} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) )@ \mbox{xf} ) ) ) ) )\qquad\mbox{thf}(\mbox{funcGraphProp}_1,\mbox{conjecture})$

\medskip\textbf{SEU211+1.p} MPTP bushy problem t178\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b,c{:}\;( c = \mbox{relation\_inverse\_image}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),a)\mbox{ and }\mbox{in}(e,b) ) ) ) )\qquad\mbox{fof}(\mbox{d14\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k10\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( a\subseteq b \ \Rightarrow \ \mbox{relation\_inverse\_image}(c,a)\subseteq \mbox{relation\_inverse\_image}(c,b) ) )\qquad\mbox{fof}(\mbox{t178\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU150+1.p} MPTP bushy problem t9\_zfmisc\_1






$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t8\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{t9\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU692$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i.in f (funcSet A B) $\to$ (! g:i.in g
 (funcSet A B) $\to$ (! x:i.in x A $\to$ ap A B f x = ap A B g x) $\to$
 f = g))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{infuncsetfunc}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{infuncsetfunc\_type},\mbox{type})$

$\mbox{infuncsetfunc} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ ( \mbox{func}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{infuncsetfunc},\mbox{definition})$

$\mbox{funcext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcext\_type},\mbox{type})$

$\mbox{funcext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xg}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xg} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = ( \mbox{ap}@ a@ b@ \mbox{xg}@ \mbox{xx} ) ) \ \Rightarrow \ \mbox{xf} = \mbox{xg} ) ) ) )\qquad\mbox{thf}(\mbox{funcext},\mbox{definition})$

$\mbox{infuncsetfunc} \ \Rightarrow \ ( \mbox{funcext} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ \forall \mbox{xg}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xg}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = ( \mbox{ap}@ a@ b@ \mbox{xg}@ \mbox{xx} ) ) \ \Rightarrow \ \mbox{xf} = \mbox{xg} ) ) ) )\qquad\mbox{thf}(\mbox{funcext}_2,\mbox{conjecture})$

\medskip\textbf{SEU491$\wedge$1.p} Terminating relations and confluence and local confluence

For a terminating relation confluence and local confluence are
 the same.




$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{term}@ r ) \ \Rightarrow \ ( ( \mbox{confl}@ r )\iff ( \mbox{lconfl}@ r ) ) )\qquad\mbox{thf}(\mbox{termination\_makes\_confluence\_equal\_to\_local\_confluence},\mbox{conjecture})$

\medskip\textbf{SEU533$\wedge$2.p} Preliminary Notions - Equality Laws - Kuratowski Pairs

(! x:i.! y:i.in y (setadjoin x (setadjoin y emptyset)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{setadjoinIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIR\_type},\mbox{type})$

$\mbox{setadjoinIR} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{setadjoinIR},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{setadjoinIR} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{upairsetIR},\mbox{conjecture})$

\medskip\textbf{SEU596$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.~(? x:i.in x A \& in x B) $\to$
 binintersect A B = emptyset)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptyI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptyI\_type},\mbox{type})$

$\mbox{emptyI} = ( \forall a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{in}@ \mbox{xx}@ a \ \Rightarrow \ a = \mbox{emptyset} ) )\qquad\mbox{thf}(\mbox{emptyI},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectEL\_type},\mbox{type})$

$\mbox{binintersectEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{definition})$

$\mbox{binintersectER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectER\_type},\mbox{type})$

$\mbox{binintersectER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersectER},\mbox{definition})$

$\mbox{emptyI} \ \Rightarrow \ ( \mbox{binintersectEL} \ \Rightarrow \ ( \mbox{binintersectER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \neg\,\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{binintersect}@ a@ b ) = \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{disjointsetsI}_1,\mbox{conjecture})$

\medskip\textbf{SEU599$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.binintersect A B = A $\to$ subset A B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectER\_type},\mbox{type})$

$\mbox{binintersectER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersectER},\mbox{definition})$

$\mbox{in\_\_Cong} \ \Rightarrow \ ( \mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{binintersectER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{binintersect}@ a@ b ) = a \ \Rightarrow \ ( \subseteq @ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binintersectSubset}_1,\mbox{conjecture})$

\medskip\textbf{SEU430+1.p} First and Second Order Cutting of Binary Relations T30






$\forall a,b{:}\;( \mbox{m1\_subset}_1(b,\mbox{k1\_zfmisc}_1(\mbox{k1\_zfmisc}_1(a))) \ \Rightarrow \ ( \mbox{k5\_setfam}_1(a,b) = \mbox{k1\_xboole}_0\iff \forall c{:}\;( \mbox{r2\_hidden}(c,b) \ \Rightarrow \ c = \mbox{k1\_xboole}_0 ) ) )\qquad\mbox{fof}(\mbox{t30\_relset}_2,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{r2\_hidden}(a,b) \ \Rightarrow \ \neg\,\mbox{r2\_hidden}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{v1\_xboole}_0(a) \ \Rightarrow \ \mbox{v1\_relat}_1(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{k1\_xboole}_0\iff \forall b{:}\;\neg\,\mbox{r2\_hidden}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{k3\_tarski}(a)\iff \forall c{:}\;( \mbox{r2\_hidden}(c,b)\iff \exists d{:}\;( \mbox{r2\_hidden}(c,d)\mbox{ and }\mbox{r2\_hidden}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{m1\_subset}_1(b,\mbox{k1\_zfmisc}_1(\mbox{k1\_zfmisc}_1(a))) \ \Rightarrow \ \mbox{m1\_subset}_1(\mbox{k5\_setfam}_1(a,b),\mbox{k1\_zfmisc}_1(a)) )\qquad\mbox{fof}(\mbox{dt\_k5\_setfam}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{m1\_subset}_1(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{v1\_xboole}_0(\mbox{k1\_xboole}_0)\mbox{ and }\mbox{v1\_relat}_1(\mbox{k1\_xboole}_0)\mbox{ and }\mbox{v3\_relat}_1(\mbox{k1\_xboole}_0)\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{v1\_xboole}_0(\mbox{k1\_zfmisc}_1(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{v1\_xboole}_0(\mbox{k1\_xboole}_0)\mbox{ and }\mbox{v1\_relat}_1(\mbox{k1\_xboole}_0)\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{v1\_xboole}_0(a)\mbox{ and }\mbox{v1\_relat}_1(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{v1\_xboole}_0(a) \ \Rightarrow \ \exists b{:}\;( \mbox{m1\_subset}_1(b,\mbox{k1\_zfmisc}_1(a))\mbox{ and }\neg\,\mbox{v1\_xboole}_0(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{v1\_xboole}_0(a)\mbox{ and }\mbox{v1\_relat}_1(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{m1\_subset}_1(b,\mbox{k1\_zfmisc}_1(a))\mbox{ and }\mbox{v1\_xboole}_0(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{v1\_relat}_1(a)\mbox{ and }\mbox{v3\_relat}_1(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{m1\_subset}_1(b,\mbox{k1\_zfmisc}_1(\mbox{k1\_zfmisc}_1(a))) \ \Rightarrow \ \mbox{k5\_setfam}_1(a,b) = \mbox{k3\_tarski}(b) )\qquad\mbox{fof}(\mbox{redefinition\_k5\_setfam}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{r1\_tarski}(a,a)\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{r2\_hidden}(a,b) \ \Rightarrow \ \mbox{m1\_subset}_1(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{m1\_subset}_1(a,b) \ \Rightarrow \ ( \mbox{v1\_xboole}_0(b)\mbox{ or }\mbox{r2\_hidden}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{m1\_subset}_1(a,\mbox{k1\_zfmisc}_1(b))\iff \mbox{r1\_tarski}(a,b) )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{r2\_hidden}(a,b)\mbox{ and }\mbox{m1\_subset}_1(b,\mbox{k1\_zfmisc}_1(c)) ) \ \Rightarrow \ \mbox{m1\_subset}_1(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{r2\_hidden}(a,b)\mbox{ and }\mbox{m1\_subset}_1(b,\mbox{k1\_zfmisc}_1(c))\mbox{ and }\mbox{v1\_xboole}_0(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{v1\_xboole}_0(a) \ \Rightarrow \ a = \mbox{k1\_xboole}_0 )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{r2\_hidden}(a,b)\mbox{ and }\mbox{v1\_xboole}_0(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{v1\_xboole}_0(a)\mbox{ and }a\ne b\mbox{ and }\mbox{v1\_xboole}_0(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU158+3.p} Basic properties of sets, theorem 37






$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq b\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{t37\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq b\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l2\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU531$\wedge$2.p} Preliminary Notions - Equality Laws

(! x:i.! y:i.! z:i.in z (setadjoin x (setadjoin y emptyset)) $\to$
 z = x | z = y)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinE\_type},\mbox{type})$

$\mbox{setadjoinE} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( \mbox{xy} = \mbox{xx} \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{setadjoinE},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{setadjoinE} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) \ \Rightarrow \ ( \mbox{xz} = \mbox{xx}\mbox{ or }\mbox{xz} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{upairsetE},\mbox{conjecture})$

\medskip\textbf{SEU712$\wedge$2.p} Typed Set Theory - Types of Set Operators

(! A:i.! X:i.in X (powerset A) $\to$
 in (powerset X) (powerset (powerset A)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\mbox{powersetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE\_type},\mbox{type})$

$\mbox{powersetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetE},\mbox{definition})$

$\mbox{powersetI} \ \Rightarrow \ ( \mbox{powersetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{powerset}@ x )@ ( \mbox{powerset}@ ( \mbox{powerset}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{powersetT\_lem},\mbox{conjecture})$

\medskip\textbf{SEU836$\wedge$5.p} TPS problem GAZING-THM19






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx}\mbox{ or }s@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM19\_pme},\mbox{conjecture})$

\medskip\textbf{SEU792$\wedge$2.p} Binary Relations on a Set - Second Wizard of Oz Examples

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$
 (! T:i.breln1 A T $\to$ breln1compset A (binunion R S) T =
 binunion (breln1invset A (breln1compset A (breln1invset A T)
 (breln1invset A S))) (breln1invset A (breln1compset A
 (breln1invset A T) (breln1invset A R))))))




$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln1invset}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1invset\_type},\mbox{type})$

$\mbox{breln1invprop}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1invprop\_type},\mbox{type})$

$\mbox{breln1invprop} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ ( \mbox{breln}_1@ a@ ( \mbox{breln1invset}@ a@ r ) ) ) )\qquad\mbox{thf}(\mbox{breln1invprop},\mbox{definition})$

$\mbox{breln1compset}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1compset\_type},\mbox{type})$

$\mbox{breln1compprop}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1compprop\_type},\mbox{type})$

$\mbox{breln1compprop} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \mbox{breln}_1@ a@ ( \mbox{breln1compset}@ a@ r@ s ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1compprop},\mbox{definition})$

$\mbox{breln1unionCommutes}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1unionCommutes\_type},\mbox{type})$

$\mbox{breln1unionCommutes} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \mbox{binunion}@ r@ s ) = ( \mbox{binunion}@ s@ r ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionCommutes},\mbox{definition})$

$\mbox{woz2Ex}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{woz2Ex\_type},\mbox{type})$

$\mbox{woz2Ex} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ r = ( \mbox{breln1invset}@ a@ ( \mbox{breln1invset}@ a@ r ) ) ) )\qquad\mbox{thf}(\mbox{woz2Ex},\mbox{definition})$

$\mbox{woz2W}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{woz2W\_type},\mbox{type})$

$\mbox{woz2W} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \mbox{breln1invset}@ a@ ( \mbox{breln1compset}@ a@ r@ s ) ) = ( \mbox{breln1compset}@ a@ ( \mbox{breln1invset}@ a@ s )@ ( \mbox{breln1invset}@ a@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{woz2W},\mbox{definition})$

$\mbox{woz2A}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{woz2A\_type},\mbox{type})$

$\mbox{woz2A} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall t{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ t ) \ \Rightarrow \ ( \mbox{breln1compset}@ a@ ( \mbox{binunion}@ r@ s )@ t ) = ( \mbox{binunion}@ ( \mbox{breln1compset}@ a@ r@ t )@ ( \mbox{breln1compset}@ a@ s@ t ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz2A},\mbox{definition})$

$\mbox{breln1invprop} \ \Rightarrow \ ( \mbox{breln1compprop} \ \Rightarrow \ ( \mbox{breln1unionCommutes} \ \Rightarrow \ ( \mbox{woz2Ex} \ \Rightarrow \ ( \mbox{woz2W} \ \Rightarrow \ ( \mbox{woz2A} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall t{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ t ) \ \Rightarrow \ ( \mbox{breln1compset}@ a@ ( \mbox{binunion}@ r@ s )@ t ) = ( \mbox{binunion}@ ( \mbox{breln1invset}@ a@ ( \mbox{breln1compset}@ a@ ( \mbox{breln1invset}@ a@ t )@ ( \mbox{breln1invset}@ a@ s ) ) )@ ( \mbox{breln1invset}@ a@ ( \mbox{breln1compset}@ a@ ( \mbox{breln1invset}@ a@ t )@ ( \mbox{breln1invset}@ a@ r ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz2B},\mbox{conjecture})$

\medskip\textbf{SEU701$\wedge$2.p} Conditionals

(! A:i.! phi:o.! x:i.in x A $\to$ (! y:i.in y A $\to$ ~phi $\to$
 dsetconstr A ($\wedge$ z:i.phi \& z = x | ~phi \& z = y) =
 setadjoin y emptyset))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{eqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{eqinunit\_type},\mbox{type})$

$\mbox{eqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{eqinunit},\mbox{definition})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\mbox{iffalseProp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{iffalseProp1\_type},\mbox{type})$

$\mbox{iffalseProp}_1 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \neg\,\mbox{xphi} \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iffalseProp}_1,\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{setext} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ ( \mbox{eqinunit} \ \Rightarrow \ ( \mbox{in\_\_Cong} \ \Rightarrow \ ( \mbox{iffalseProp}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \neg\,\mbox{xphi} \ \Rightarrow \ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) = ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iffalseProp}_2,\mbox{conjecture})$

\medskip\textbf{SEU992$\wedge$5.p} TPS problem from GRAPHS-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\exists \mbox{xc}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) \ \Rightarrow \ ( x@ ( \mbox{xc}@ x ) ) ) \ \Rightarrow \ ( \exists r{:}\;a\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;\exists \mbox{xy}{:}\;b{:}\;( r@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xy}{:}\;b{:}\;\exists \mbox{xx}{:}\;a{:}\;( r@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}_1{:}\;b,\mbox{xy}_2{:}\;b{:}\;( ( r@ \mbox{xx}@ \mbox{xy}_1\mbox{ and }r@ \mbox{xx}@ \mbox{xy}_2 ) \ \Rightarrow \ \mbox{xy}_1 = \mbox{xy}_2 ) ) \ \Rightarrow \ \exists s{:}\;b\to a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b{:}\;\exists \mbox{xy}{:}\;a{:}\;( s@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}_1{:}\;a,\mbox{xy}_2{:}\;a{:}\;( ( s@ \mbox{xx}@ \mbox{xy}_1\mbox{ and }s@ \mbox{xx}@ \mbox{xy}_2 ) \ \Rightarrow \ \mbox{xy}_1 = \mbox{xy}_2 )\mbox{ and }\forall \mbox{xx}_1{:}\;b,\mbox{xx}_2{:}\;b,\mbox{xy}{:}\;a{:}\;( ( s@ \mbox{xx}_1@ \mbox{xy}\mbox{ and }s@ \mbox{xx}_2@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}_1 = \mbox{xx}_2 ) ) )\qquad\mbox{thf}(\mbox{cTHM608\_pme},\mbox{conjecture})$

\medskip\textbf{SEU170+1.p} MPTP bushy problem t43\_subset\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{subset\_complement}(a,b) = \mbox{set\_difference}(a,b) )\qquad\mbox{fof}(\mbox{d5\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{element}(\mbox{subset\_complement}(a,b),\mbox{powerset}(a)) )\qquad\mbox{fof}(\mbox{dt\_k3\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{subset\_complement}(a,\mbox{subset\_complement}(a,b)) = b )\qquad\mbox{fof}(\mbox{involutiveness\_k3\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \forall c{:}\;( \mbox{in}(c,b) \ \Rightarrow \ \mbox{in}(c,a) ) )\qquad\mbox{fof}(\mbox{l3\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \forall c{:}\;( \mbox{element}(c,\mbox{powerset}(a)) \ \Rightarrow \ ( \mbox{disjoint}(b,c)\iff b\subseteq \mbox{subset\_complement}(a,c) ) ) )\qquad\mbox{fof}(\mbox{t43\_subset}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU865$\wedge$5.p} TPS problem from FINITE-SET-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cB}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cC}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$( \forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{cC} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cB}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cC}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{cB} ) )\qquad\mbox{thf}(\mbox{cTHM531\_pme},\mbox{conjecture})$

\medskip\textbf{SEU699$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i.in f (funcSet A B) $\to$
 lam A B ($\wedge$ x:i.ap A B f x) = f)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{ap2p}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ap2p\_type},\mbox{type})$

$\mbox{ap2p} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} )@ b ) ) ) )\qquad\mbox{thf}(\mbox{ap2p},\mbox{definition})$

$\mbox{lam}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{lam\_type},\mbox{type})$

$\mbox{lam2p}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{lam2p\_type},\mbox{type})$

$\mbox{lam2p} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{lam}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) )@ ( \mbox{funcSet}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{lam2p},\mbox{definition})$

$\mbox{funcext}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcext2\_type},\mbox{type})$

$\mbox{funcext}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ \forall \mbox{xg}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xg}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = ( \mbox{ap}@ a@ b@ \mbox{xg}@ \mbox{xx} ) ) \ \Rightarrow \ \mbox{xf} = \mbox{xg} ) ) ) )\qquad\mbox{thf}(\mbox{funcext}_2,\mbox{definition})$

$\mbox{beta}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{beta2\_type},\mbox{type})$

$\mbox{beta}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ ( \mbox{lam}@ a@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) )@ \mbox{xx} ) = ( \mbox{xf}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{beta}_2,\mbox{definition})$

$\mbox{ap2p} \ \Rightarrow \ ( \mbox{lam2p} \ \Rightarrow \ ( \mbox{funcext}_2 \ \Rightarrow \ ( \mbox{beta}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ ( \mbox{lam}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) ) = \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{eta}_2,\mbox{conjecture})$

\medskip\textbf{SEU961$\wedge$5.p} TPS problem from FUNCTION-THMS






$( \forall \mbox{xa}{:}\;\mbox{\$i},\mbox{xb}{:}\;\mbox{\$i}{:}\;\mbox{xa} = \mbox{xb}\mbox{ and }\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = ( \mbox{xg}@ \mbox{xx} ) \ \Rightarrow \ \mbox{xf} = \mbox{xg} ) ) \ \Rightarrow \ \forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xj}\mbox{ and }\forall \mbox{xj}_0{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj}_0 ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xj}@ ( \mbox{xj}_0@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xj}\mbox{ and }\forall \mbox{xj}_1{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj}_1 ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xj}@ ( \mbox{xj}_1@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xk} ) ) )\qquad\mbox{thf}(\mbox{cTHM196C\_pme},\mbox{conjecture})$

\medskip\textbf{SEU132+1.p} MPTP bushy problem t33\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_difference}(a,c)\subseteq \mbox{set\_difference}(b,c) )\qquad\mbox{fof}(\mbox{t33\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU166+1.p} MPTP bushy problem t118\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ ( \mbox{cartesian\_product}_2(a,c)\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }\mbox{cartesian\_product}_2(c,a)\subseteq \mbox{cartesian\_product}_2(c,b) ) )\qquad\mbox{fof}(\mbox{t118\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU737$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 subset X (setminus A Y) $\to$ subset Y (setminus A X)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{complementT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementT\_lem\_type},\mbox{type})$

$\mbox{complementT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{complementT\_lem},\mbox{definition})$

$\mbox{subsetTI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetTI\_type},\mbox{type})$

$\mbox{subsetTI} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \subseteq @ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{subsetTI},\mbox{definition})$

$\mbox{contraSubsetComplement}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{contraSubsetComplement\_type},\mbox{type})$

$\mbox{contraSubsetComplement} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ x@ ( \mbox{setminus}@ a@ y ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{contraSubsetComplement},\mbox{definition})$

$\mbox{complementT\_lem} \ \Rightarrow \ ( \mbox{subsetTI} \ \Rightarrow \ ( \mbox{contraSubsetComplement} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ x@ ( \mbox{setminus}@ a@ y ) ) \ \Rightarrow \ ( \subseteq @ y@ ( \mbox{setminus}@ a@ x ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{complementTcontraSubset},\mbox{conjecture})$

\medskip\textbf{SEU622$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! x:i.in x A $\to$ in (setadjoin x emptyset) (powerset A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{powersetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI1\_type},\mbox{type})$

$\mbox{powersetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ b@ a ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI}_1,\mbox{definition})$

$\mbox{singletonsubset}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{singletonsubset\_type},\mbox{type})$

$\mbox{singletonsubset} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \subseteq @ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ a ) ) )\qquad\mbox{thf}(\mbox{singletonsubset},\mbox{definition})$

$\mbox{powersetI}_1 \ \Rightarrow \ ( \mbox{singletonsubset} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{singletoninpowerset},\mbox{conjecture})$

\medskip\textbf{SEU245+1.p} MPTP bushy problem t16\_wellord1






$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k2\_wellord}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;\mbox{relation\_restriction}(a,b) = \mbox{set\_intersection}_2(a,\mbox{cartesian\_product}_2(b,b)) )\qquad\mbox{fof}(\mbox{d6\_wellord}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_restriction}(c,b))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(a,\mbox{cartesian\_product}_2(b,b)) ) ) )\qquad\mbox{fof}(\mbox{t16\_wellord}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

\medskip\textbf{SEU785$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ in (kpair x y) R | in (kpair x y) S $\to$
 in (kpair x y) (binunion R S)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln1unionIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1unionIL\_type},\mbox{type})$

$\mbox{breln1unionIL} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionIL},\mbox{definition})$

$\mbox{breln1unionIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1unionIR\_type},\mbox{type})$

$\mbox{breln1unionIR} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionIR},\mbox{definition})$

$\mbox{breln1unionIL} \ \Rightarrow \ ( \mbox{breln1unionIR} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r\mbox{ or }\mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionI},\mbox{conjecture})$

\medskip\textbf{SEU696$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i.func A B f $\to$ lam A B ($\wedge$ x:i.ap A B f x) = f)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{app}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{app\_type},\mbox{type})$

$\mbox{app} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} )@ b ) ) ) )\qquad\mbox{thf}(\mbox{app},\mbox{definition})$

$\mbox{lam}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{lam\_type},\mbox{type})$

$\mbox{lam} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{lam},\mbox{definition})$

$\mbox{lamp}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{lamp\_type},\mbox{type})$

$\mbox{lamp} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ ( \mbox{func}@ a@ b@ ( \mbox{lam}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{lamp},\mbox{definition})$

$\mbox{funcext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcext\_type},\mbox{type})$

$\mbox{funcext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xg}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xg} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = ( \mbox{ap}@ a@ b@ \mbox{xg}@ \mbox{xx} ) ) \ \Rightarrow \ \mbox{xf} = \mbox{xg} ) ) ) )\qquad\mbox{thf}(\mbox{funcext},\mbox{definition})$

$\mbox{beta}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{beta1\_type},\mbox{type})$

$\mbox{beta}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ ( \mbox{lam}@ a@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) )@ \mbox{xx} ) = ( \mbox{xf}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{beta}_1,\mbox{definition})$

$\mbox{app} \ \Rightarrow \ ( \mbox{lamp} \ \Rightarrow \ ( \mbox{funcext} \ \Rightarrow \ ( \mbox{beta}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ ( \mbox{lam}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) ) = \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{eta}_1,\mbox{conjecture})$

\medskip\textbf{SEU167+1.p} MPTP bushy problem t119\_zfmisc\_1






$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ ( \mbox{cartesian\_product}_2(a,c)\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }\mbox{cartesian\_product}_2(c,a)\subseteq \mbox{cartesian\_product}_2(c,b) ) )\qquad\mbox{fof}(\mbox{t118\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( a\subseteq b\mbox{ and }c\subseteq d ) \ \Rightarrow \ \mbox{cartesian\_product}_2(a,c)\subseteq \mbox{cartesian\_product}_2(b,d) )\qquad\mbox{fof}(\mbox{t119\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SEU623$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! B:i.! x:i.in x A $\to$ in (setadjoin x emptyset) (powerset
 (binunion A B)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{powersetsubset}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetsubset\_type},\mbox{type})$

$\mbox{powersetsubset} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( \subseteq @ ( \mbox{powerset}@ a )@ ( \mbox{powerset}@ b ) ) ) )\qquad\mbox{thf}(\mbox{powersetsubset},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionLsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionLsub\_type},\mbox{type})$

$\mbox{binunionLsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \subseteq @ a@ ( \mbox{binunion}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{binunionLsub},\mbox{definition})$

$\mbox{singletoninpowerset}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{singletoninpowerset\_type},\mbox{type})$

$\mbox{singletoninpowerset} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{singletoninpowerset},\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ ( \mbox{powersetsubset} \ \Rightarrow \ ( \mbox{binunionLsub} \ \Rightarrow \ ( \mbox{singletoninpowerset} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{singletoninpowunion},\mbox{conjecture})$

\medskip\textbf{SEU718$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in X (powerset Y) $\to$ in x X $\to$ in x Y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{powersetE}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE1\_type},\mbox{type})$

$\mbox{powersetE}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \subseteq @ b@ a ) ) )\qquad\mbox{thf}(\mbox{powersetE}_1,\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ ( \mbox{powersetE}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ x@ ( \mbox{powerset}@ y ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{powersetTE}_1,\mbox{conjecture})$

\medskip\textbf{SEU513$\wedge$2.p} Preliminary Notions - Adjoining Elements to Sets

in emptyset (setadjoin emptyset emptyset)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{in}@ \mbox{emptyset}@ ( \mbox{setadjoin}@ \mbox{emptyset}@ \mbox{emptyset} ) )\qquad\mbox{thf}(\mbox{emptyinunitempty},\mbox{conjecture})$

\medskip\textbf{SEU126+1.p} MPTP bushy problem t12\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_union}_2(a,b) = b )\qquad\mbox{fof}(\mbox{t12\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU205+1.p} MPTP bushy problem t145\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b,c{:}\;( c = \mbox{relation\_image}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e{:}\;( \mbox{in}(\mbox{ordered\_pair}(e,d),a)\mbox{ and }\mbox{in}(e,b) ) ) ) )\qquad\mbox{fof}(\mbox{d13\_relat}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k9\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{set\_intersection}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc1\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_image}(c,b))\iff \exists d{:}\;( \mbox{in}(d,\mbox{relation\_dom}(c))\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,a),c)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{t143\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_image}(b,a) = \mbox{relation\_image}(b,\mbox{set\_intersection}_2(\mbox{relation\_dom}(b),a)) )\qquad\mbox{fof}(\mbox{t145\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU866$\wedge$5.p} TPS problem from PIGEON-HOLE






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xp}{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xp} ) ) \ \Rightarrow \ \neg\,\exists \mbox{xq}{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xq}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) )\mbox{ and }\exists \mbox{xx}{:}\;a{:}\;( \neg\,\mbox{xq}@ \mbox{xx}\mbox{ and }\mbox{xp}@ \mbox{xx} )\mbox{ and }\exists \mbox{xf}{:}\;a\to a{:}\;\forall \mbox{xy}{:}\;a{:}\;( ( \mbox{xp}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;a{:}\;( \mbox{xq}@ \mbox{xx}\mbox{ and }\mbox{xy} = ( \mbox{xf}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM161\_pme},\mbox{conjecture})$

\medskip\textbf{SEU732$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x (setminus A X) $\to$
 ~(in x (binintersect X Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{binintersectTELcontra}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectTELcontra\_type},\mbox{type})$

$\mbox{binintersectTELcontra} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ x \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectTELcontra},\mbox{definition})$

$\mbox{setminusER} \ \Rightarrow \ ( \mbox{binintersectTELcontra} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{complementTnotintersectT},\mbox{conjecture})$

\medskip\textbf{SEU940$\wedge$5.p} TPS problem THM112A






$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists m{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( m@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\mbox{ and }\forall g{:}\;\mbox{\$i}\to \mbox{\$i},h{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( m@ g\mbox{ and }m@ h ) \ \Rightarrow \ ( m@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( g@ ( h@ \mbox{xx} ) )\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( p@ y ) \ \Rightarrow \ ( p@ ( g@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM112A\_pme},\mbox{conjecture})$

\medskip\textbf{SEU644$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! X:i.singleton X $\to$ in (setunion X) X)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{eqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{eqinunit\_type},\mbox{type})$

$\mbox{eqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{eqinunit},\mbox{definition})$

$\mbox{setunionsingleton}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionsingleton\_type},\mbox{type})$

$\mbox{setunionsingleton} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{setunionsingleton},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{eqinunit} \ \Rightarrow \ ( \mbox{setunionsingleton} \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{singleton}@ x ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setunion}@ x )@ x ) ) )\qquad\mbox{thf}(\mbox{theprop},\mbox{conjecture})$

\medskip\textbf{SEU469$\wedge$1.p} Transitive reflexive closure is transitive and reflexive

The transitive reflexive closure of a binary relation is
 transitive and reflexive.




$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{trans}@ ( \mbox{trc}@ r )\mbox{ and }\mbox{refl}@ ( \mbox{trc}@ r ) )\qquad\mbox{thf}(\mbox{transitive\_reflexive\_closure\_is\_transitive\_reflexive},\mbox{conjecture})$

\medskip\textbf{SEU975$\wedge$5.p} TPS problem from COINDUCTIVE-PU-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xt} = \mbox{cZ}\iff \mbox{xu} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ \mbox{xt} = \mbox{xu} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xu} = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) ) \ \Rightarrow \ \exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xx} )@ ( \mbox{cL}@ \mbox{xy} ) )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xu} = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_LEM2B\_pme},\mbox{conjecture})$

\medskip\textbf{SEU789$\wedge$2.p} Binary Relations on a Set - Second Wizard of Oz Examples

(! A:i.! R:i.breln1 A R $\to$ R = breln1invset A (breln1invset A R))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{setextsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextsub\_type},\mbox{type})$

$\mbox{setextsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ a ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setextsub},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{subbreln}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subbreln1\_type},\mbox{type})$

$\mbox{subbreln}_1 = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) \ \Rightarrow \ ( \subseteq @ r@ s ) ) ) ) )\qquad\mbox{thf}(\mbox{subbreln}_1,\mbox{definition})$

$\mbox{breln1invset}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1invset\_type},\mbox{type})$

$\mbox{breln1invprop}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1invprop\_type},\mbox{type})$

$\mbox{breln1invprop} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ ( \mbox{breln}_1@ a@ ( \mbox{breln1invset}@ a@ r ) ) ) )\qquad\mbox{thf}(\mbox{breln1invprop},\mbox{definition})$

$\mbox{breln1invI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1invI\_type},\mbox{type})$

$\mbox{breln1invI} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xy}@ \mbox{xx} )@ ( \mbox{breln1invset}@ a@ r ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1invI},\mbox{definition})$

$\mbox{breln1invE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1invE\_type},\mbox{type})$

$\mbox{breln1invE} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xy}@ \mbox{xx} )@ ( \mbox{breln1invset}@ a@ r ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1invE},\mbox{definition})$

$\mbox{setextsub} \ \Rightarrow \ ( \mbox{subbreln}_1 \ \Rightarrow \ ( \mbox{breln1invprop} \ \Rightarrow \ ( \mbox{breln1invI} \ \Rightarrow \ ( \mbox{breln1invE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ r = ( \mbox{breln1invset}@ a@ ( \mbox{breln1invset}@ a@ r ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz2Ex},\mbox{conjecture})$

\medskip\textbf{SEU994$\wedge$5.p} TPS problem from LATTICES






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cR}{:}\;a\to a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\forall \mbox{xs}{:}\;a\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xz}{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xz}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xj}{:}\;a{:}\;( \forall \mbox{xz}{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xz}@ \mbox{xj} ) ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx}@ \mbox{xj} ) ) )\mbox{ and }\forall \mbox{xs}{:}\;a\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;a{:}\;( \forall \mbox{xz}{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xj}{:}\;a{:}\;( \forall \mbox{xz}{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xj}@ \mbox{xz} ) ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xj}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \mbox{cR}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{cR}@ \mbox{xy}@ \mbox{xx} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cCLATTICE\_pme},\mbox{conjecture})$

\medskip\textbf{SEU756$\wedge$2.p} Typed Set Theory - Laws for Typed Sets - DeMorgan Laws

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x (binintersect (setminus A X) (setminus A Y))
 $\to$ in x (setminus A (binunion X Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectEL\_type},\mbox{type})$

$\mbox{binintersectEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{definition})$

$\mbox{binintersectER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectER\_type},\mbox{type})$

$\mbox{binintersectER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersectER},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{demorgan2b}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{demorgan2b2\_type},\mbox{type})$

$\mbox{demorgan2b}_2 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ y ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{demorgan2b}_2,\mbox{definition})$

$\mbox{binintersectEL} \ \Rightarrow \ ( \mbox{binintersectER} \ \Rightarrow \ ( \mbox{demorgan2b}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{setminus}@ a@ y ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{demorgan2b},\mbox{conjecture})$

\medskip\textbf{SEU747$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ ~(in x X) $\to$ ~(in x Y) $\to$
 ~(in x (binunion X Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionE\_type},\mbox{type})$

$\mbox{binunionE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a\mbox{ or }\mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binunionE},\mbox{definition})$

$\mbox{binunionE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ x \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ y \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionTEcontra},\mbox{conjecture})$

\medskip\textbf{SEU320+1.p} MPTP bushy problem t30\_tops\_1






$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{element}(\mbox{subset\_complement}(a,b),\mbox{powerset}(a)) )\qquad\mbox{fof}(\mbox{dt\_k3\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{top\_str}(a) \ \Rightarrow \ \mbox{one\_sorted\_str}(a) )\qquad\mbox{fof}(\mbox{dt\_l1\_pre\_topc},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_l1\_struct}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_u1\_struct}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{top\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_pre\_topc},\mbox{axiom})$

$\exists a{:}\;\mbox{one\_sorted\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_struct}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{subset\_complement}(a,\mbox{subset\_complement}(a,b)) = b )\qquad\mbox{fof}(\mbox{involutiveness\_k3\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{top\_str}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) \ \Rightarrow \ ( \mbox{closed\_subset}(b,a)\iff \mbox{open\_subset}(\mbox{subset\_complement}(\mbox{the\_carrier}(a),b),a) ) ) )\qquad\mbox{fof}(\mbox{t29\_tops}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{top\_str}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) \ \Rightarrow \ ( \mbox{open\_subset}(b,a)\iff \mbox{closed\_subset}(\mbox{subset\_complement}(\mbox{the\_carrier}(a),b),a) ) ) )\qquad\mbox{fof}(\mbox{t30\_tops}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

\medskip\textbf{SEU702$\wedge$2.p} Conditionals

(! A:i.! phi:o.! x:i.in x A $\to$ (! y:i.in y A $\to$ phi $\to$
 in x (dsetconstr A ($\wedge$ z:i.phi \& z = x | ~phi \& z = y))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{xphi} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iftrueProp}_1,\mbox{conjecture})$

\medskip\textbf{SEU640$\wedge$2.p} Ordered Pairs - Singletons

(! A:i.! phi:i$>$o.(! x:i.in x A $\to$ (! y:i.in y A $\to$ phi x $\to$
 phi y $\to$ x = y)) $\to$ (? x:i.in x A \& phi x) $\to$ ex1 A ($\wedge$ x:i.phi x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{singletonprop}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{singletonprop\_type},\mbox{type})$

$\mbox{singletonprop} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) ) ) \ \Rightarrow \ ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{singletonprop},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{singletonprop} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) ) ) \ \Rightarrow \ ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{ex}_1@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex1I}_2,\mbox{conjecture})$

\medskip\textbf{SEU618$\wedge$2.p} Ordered Pairs - Kuratowski Pairs

(! x:i.! y:i.in y (setunion (setadjoin (setadjoin x emptyset)
 (setadjoin (setadjoin x (setadjoin y emptyset)) emptyset))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{setunionI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionI\_type},\mbox{type})$

$\mbox{setunionI} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{setunionI},\mbox{definition})$

$\mbox{secondinupair}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{secondinupair\_type},\mbox{type})$

$\mbox{secondinupair} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{secondinupair},\mbox{definition})$

$\mbox{setunionI} \ \Rightarrow \ ( \mbox{secondinupair} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setunion}@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{setukpairIR},\mbox{conjecture})$

\medskip\textbf{SEU169+1.p} MPTP bushy problem l3\_subset\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{powerset}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c\subseteq a ) )\qquad\mbox{fof}(\mbox{d1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \neg\,\mbox{empty}(a) \ \Rightarrow \ ( \mbox{element}(b,a)\iff \mbox{in}(b,a) ) )\mbox{ and }( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{element}(b,a)\iff \mbox{empty}(b) ) ) )\qquad\mbox{fof}(\mbox{d2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \forall c{:}\;( \mbox{in}(c,b) \ \Rightarrow \ \mbox{in}(c,a) ) )\qquad\mbox{fof}(\mbox{l3\_subset}_1,\mbox{conjecture})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU484$\wedge$1.p} A reflexive relation is non-terminating






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{refl}@ r ) \ \Rightarrow \ \neg\,\mbox{term}@ r )\qquad\mbox{thf}(\mbox{reflexive\_implies\_non\_terminating},\mbox{conjecture})$

\medskip\textbf{SEU631$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! B:i.! u:i.in u (cartprod A B) $\to$ (? x:i.in x A \&
 (? y:i.in y B \& u = kpair x y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprod} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{powerset}@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xz}@ b\mbox{ and }\mbox{xx} = ( \mbox{kpair}@ \mbox{xy}@ \mbox{xz} ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprod},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cartprodmempair}_1,\mbox{conjecture})$

\medskip\textbf{SEU527$\wedge$2.p} Preliminary Notions - Equality Laws

(! x:i.! y:i.in x (setadjoin y emptyset) $\to$ x = y)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{setadjoinE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinE\_type},\mbox{type})$

$\mbox{setadjoinE} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( \mbox{xy} = \mbox{xx} \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{setadjoinE},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ ( \mbox{setadjoinE} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{conjecture})$

\medskip\textbf{SEU694$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i.func A B f $\to$ (! x:i.in x A $\to$
 ap A B f x = ap A B f x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in},\mbox{type})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap},\mbox{type})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{ap2apEq}_2,\mbox{conjecture})$

\medskip\textbf{SEU706$\wedge$2.p} Conditionals

(! X:i.singleton X $\to$ (! x:i.in x X $\to$ setunion X = x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\mbox{setadjoin\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoin\_\_Cong\_type},\mbox{type})$

$\mbox{setadjoin\_\_Cong} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ \forall \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xz} = \mbox{xu} \ \Rightarrow \ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xz} ) = ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{setadjoin\_\_Cong},\mbox{definition})$

$\mbox{setunion\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunion\_\_Cong\_type},\mbox{type})$

$\mbox{setunion\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ ( \mbox{setunion}@ a ) = ( \mbox{setunion}@ b ) ) )\qquad\mbox{thf}(\mbox{setunion\_\_Cong},\mbox{definition})$

$\mbox{setunionsingleton}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionsingleton\_type},\mbox{type})$

$\mbox{setunionsingleton} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{setunionsingleton},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{uniqinunit} \ \Rightarrow \ ( \mbox{in\_\_Cong} \ \Rightarrow \ ( \mbox{setadjoin\_\_Cong} \ \Rightarrow \ ( \mbox{setunion\_\_Cong} \ \Rightarrow \ ( \mbox{setunionsingleton} \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{singleton}@ x ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{setunion}@ x ) = \mbox{xx} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{theeq},\mbox{conjecture})$

\medskip\textbf{SEU517$\wedge$2.p} Preliminary Notions - Power Sets and Unions

(! A:i.dsetconstr A ($\wedge$ x:i.true) = A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{setext} \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{\$true} ) = a ) )\qquad\mbox{thf}(\mbox{setoftrueEq},\mbox{conjecture})$

\medskip\textbf{SEU669$\wedge$2.p} Ordered Pairs - Sets of Pairs

(! A:i.! B:i.! phi:i$>$(i$>$o).! x:i.! y:i.in (kpair x y)
 (dpsetconstr A B ($\wedge$ z,u:i.phi z u)) $\to$ in x A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprodpairmemEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodpairmemEL\_type},\mbox{type})$

$\mbox{cartprodpairmemEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{cartprodpairmemEL},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstr} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{dsetconstr}@ ( \mbox{cartprod}@ a@ b )@ \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xphi}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstr},\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{cartprodpairmemEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xz}@ \mbox{xu} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrEL}_1,\mbox{conjecture})$

\medskip\textbf{SEU647$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.! z:i.! u:i.setadjoin (setadjoin x emptyset)
 (setadjoin (setadjoin x (setadjoin y emptyset)) emptyset) =
 setadjoin (setadjoin z emptyset) (setadjoin (setadjoin z
 (setadjoin u emptyset)) emptyset) $\to$ x = z)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{setukpairinjL}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjL1\_type},\mbox{type})$

$\mbox{setukpairinjL}_1 = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) )\qquad\mbox{thf}(\mbox{setukpairinjL}_1,\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{setukpairinjL}_1 \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xu}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) )\qquad\mbox{thf}(\mbox{setukpairinjL}_2,\mbox{conjecture})$

\medskip\textbf{SEU230+1.p} MPTP bushy problem t10\_ordinal1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{succ}(a) = \mbox{set\_union}_2(a,\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d1\_ordinal}_1,\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_ordinal}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{succ}(a))\qquad\mbox{fof}(\mbox{fc1\_ordinal}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{in}(a,\mbox{succ}(a))\qquad\mbox{fof}(\mbox{t10\_ordinal}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU714$\wedge$2.p} Typed Set Theory - Types of Set Operators

(! A:i.! X:i.in X (powerset A) $\to$ in (setminus A X) (powerset A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusEL\_type},\mbox{type})$

$\mbox{setminusEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{setminusEL},\mbox{definition})$

$\mbox{powersetI} \ \Rightarrow \ ( \mbox{setminusEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{complementT\_lem},\mbox{conjecture})$

\medskip\textbf{SEU611$\wedge$2.p} Preliminary Notions - Operations on Sets - Symmetric Difference

(! A:i.! B:i.! x:i.in x (symdiff A B) $\to$ (! phi:o.(in x A $\to$
 ~(in x B) $\to$ phi) $\to$ (~(in x A) $\to$ in x B $\to$ phi) $\to$ phi))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionE\_type},\mbox{type})$

$\mbox{binunionE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a\mbox{ or }\mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binunionE},\mbox{definition})$

$\mbox{symdiff}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{symdiff\_type},\mbox{type})$

$\mbox{symdiff} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{binunion}@ a@ b )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{in}@ \mbox{xx}@ a\mbox{ or }\neg\,\mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{symdiff},\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{binunionE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{symdiff}@ a@ b ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ \mbox{xphi} ) ) \ \Rightarrow \ ( ( \neg\,\mbox{in}@ \mbox{xx}@ a \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \mbox{xphi} ) ) \ \Rightarrow \ \mbox{xphi} ) ) ) ) )\qquad\mbox{thf}(\mbox{symdiffE},\mbox{conjecture})$

\medskip\textbf{SEU723$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ subset X (setminus A Y) $\to$ in x Y $\to$ ~(in x X))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \subseteq @ x@ ( \mbox{setminus}@ a@ y ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ y ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ x ) ) ) ) ) )\qquad\mbox{thf}(\mbox{contrasubsetT},\mbox{conjecture})$

\medskip\textbf{SEU459$\wedge$1.p} The symmetric closure operator is inflationary






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{infl}@ \mbox{sc}\qquad\mbox{thf}(\mbox{symmetric\_closure\_op\_is\_inflationary},\mbox{conjecture})$

\medskip\textbf{SEU512$\wedge$2.p} Preliminary Notions - Adjoining Elements to Sets

(! x:i.! y:i.in x (setadjoin x y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinAx\_type},\mbox{type})$

$\mbox{setadjoinAx} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) )\iff ( \mbox{xy} = \mbox{xx}\mbox{ or }\mbox{in}@ \mbox{xy}@ a ) ) )\qquad\mbox{thf}(\mbox{setadjoinAx},\mbox{definition})$

$\mbox{setadjoinAx} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{conjecture})$

\medskip\textbf{SEU163+3.p} Basic properties of sets, theorem 92






$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ a\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{t92\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ a\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{l50\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU356+1.p} MPTP bushy problem t15\_yellow\_0






$\forall a{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{ex\_sup\_of\_relstr\_set}(a,b)\iff \exists c{:}\;( \mbox{element}(c,\mbox{the\_carrier}(a))\mbox{ and }\mbox{relstr\_set\_smaller}(a,b,c)\mbox{ and }\forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_set\_smaller}(a,b,d) \ \Rightarrow \ \mbox{related}(a,c,d) ) )\mbox{ and }\forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( ( \mbox{relstr\_set\_smaller}(a,b,d)\mbox{ and }\forall e{:}\;( \mbox{element}(e,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_set\_smaller}(a,b,e) \ \Rightarrow \ \mbox{related}(a,d,e) ) ) ) \ \Rightarrow \ d = c ) ) ) ) )\qquad\mbox{fof}(\mbox{d7\_yellow}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \mbox{one\_sorted\_str}(a) )\qquad\mbox{fof}(\mbox{dt\_l1\_orders}_2,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_l1\_struct}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_u1\_struct}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{rel\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_orders}_2,\mbox{axiom})$

$\exists a{:}\;\mbox{one\_sorted\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_struct}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{antisymmetric\_relstr}(a)\mbox{ and }\mbox{rel\_str}(a) ) \ \Rightarrow \ \forall b{:}\;( \mbox{ex\_sup\_of\_relstr\_set}(a,b)\iff \exists c{:}\;( \mbox{element}(c,\mbox{the\_carrier}(a))\mbox{ and }\mbox{relstr\_set\_smaller}(a,b,c)\mbox{ and }\forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_set\_smaller}(a,b,d) \ \Rightarrow \ \mbox{related}(a,c,d) ) ) ) ) )\qquad\mbox{fof}(\mbox{t15\_yellow}_0,\mbox{conjecture})$

$\forall a{:}\;( ( \mbox{antisymmetric\_relstr}(a)\mbox{ and }\mbox{rel\_str}(a) ) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{the\_carrier}(a)) \ \Rightarrow \ \forall c{:}\;( \mbox{element}(c,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( ( \mbox{related}(a,b,c)\mbox{ and }\mbox{related}(a,c,b) ) \ \Rightarrow \ b = c ) ) ) )\qquad\mbox{fof}(\mbox{t25\_orders}_2,\mbox{axiom})$

\medskip\textbf{SEU845$\wedge$5.p} TPS problem GAZING-THM12






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx} ) ) \ \Rightarrow \ s = ( \lambda \mbox{xx}{:}\;a{:}\;( t@ \mbox{xx}\mbox{ and }\neg\,t@ \mbox{xx}\mbox{ and }\neg\,s@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM12\_pme},\mbox{conjecture})$

\medskip\textbf{SEU488$\wedge$1.p} Confluence implies semi confluence






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{confl}@ r ) \ \Rightarrow \ ( \mbox{sconfl}@ r ) )\qquad\mbox{thf}(\mbox{confluence\_implies\_semi\_confluence},\mbox{conjecture})$

\medskip\textbf{SEU162+3.p} Basic properties of sets, theorem 65






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{disjoint}(\mbox{singleton}(a),b)\mbox{ and }\mbox{in}(a,b)\qquad\mbox{fof}(\mbox{l25\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{in}(a,b) \ \Rightarrow \ \mbox{disjoint}(\mbox{singleton}(a),b) )\qquad\mbox{fof}(\mbox{l28\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{set\_difference}(a,\mbox{singleton}(b)) = a\iff \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{t65\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_difference}(a,b) = a )\qquad\mbox{fof}(\mbox{t83\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SEU730$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$
 subset (setminus A (setminus A X)) X)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{complementT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementT\_lem\_type},\mbox{type})$

$\mbox{complementT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{complementT\_lem},\mbox{definition})$

$\mbox{subsetTI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetTI\_type},\mbox{type})$

$\mbox{subsetTI} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \subseteq @ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{subsetTI},\mbox{definition})$

$\mbox{doubleComplementE}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{doubleComplementE1\_type},\mbox{type})$

$\mbox{doubleComplementE}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{setminus}@ a@ x ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{doubleComplementE}_1,\mbox{definition})$

$\mbox{complementT\_lem} \ \Rightarrow \ ( \mbox{subsetTI} \ \Rightarrow \ ( \mbox{doubleComplementE}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \subseteq @ ( \mbox{setminus}@ a@ ( \mbox{setminus}@ a@ x ) )@ x ) ) ) )\qquad\mbox{thf}(\mbox{doubleComplementSub}_2,\mbox{conjecture})$

\medskip\textbf{SEU545$\wedge$2.p} Preliminary Notions - Equivalence Laws

(! phi:i$>$o.(? x:i.phi x) $\to$ (! x:i.! y:i.phi x $\to$ phi y $\to$
 x = y) $\to$ exu ($\wedge$ x:i.phi x))




$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu},\mbox{definition})$

$\mbox{exuI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{exuI1\_type},\mbox{type})$

$\mbox{exuI}_1 = ( \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{exuI}_1,\mbox{definition})$

$\mbox{exuI}_1 \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{exuI}_3,\mbox{conjecture})$

\medskip\textbf{SEU199+1.p} MPTP bushy problem t117\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \forall c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_rng\_restriction}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff ( \mbox{in}(e,a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,e),b) ) ) ) ) )\qquad\mbox{fof}(\mbox{d12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( a\subseteq b\iff \forall c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(c,d),a) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(c,d),b) ) ) ) )\qquad\mbox{fof}(\mbox{d3\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_rng\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k8\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_rng\_restriction}(a,b)\subseteq b )\qquad\mbox{fof}(\mbox{t117\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU891$\wedge$5.p} TPS problem THM34B






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cF}{:}\;b\to a\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cS}{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cR}{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\forall \mbox{xx}{:}\;a{:}\;( ( \exists \mbox{xt}{:}\;b{:}\;( \mbox{cR}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) )\mbox{ or }\exists \mbox{xt}{:}\;b{:}\;( \mbox{cS}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) ) ) \ \Rightarrow \ \exists \mbox{xt}{:}\;b{:}\;( ( \mbox{cR}@ \mbox{xt}\mbox{ or }\mbox{cS}@ \mbox{xt} )\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cTHM34B\_pme},\mbox{conjecture})$

\medskip\textbf{SEU163+1.p} MPTP bushy problem t92\_zfmisc\_1






$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ a\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{t92\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ a\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{l50\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU829$\wedge$1.p} About powersets 2






$\mbox{subseteq}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subseteq\_type},\mbox{type})$

$\mbox{subseteq} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall u{:}\;\mbox{\$i}{:}\;( ( x@ u ) \ \Rightarrow \ ( y@ u ) ) )\qquad\mbox{thf}(\mbox{subseteq},\mbox{definition})$

$\mbox{powerset}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powerset} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{subseteq}@ y@ x ) )\qquad\mbox{thf}(\mbox{poserset},\mbox{definition})$

$\mbox{emptyset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptyset} = ( \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$false} )\qquad\mbox{thf}(\mbox{emptyset},\mbox{definition})$

$( \mbox{powerset}@ \mbox{emptyset} ) = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;x = \mbox{emptyset} )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SEU621$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! x:i.in x A $\to$ subset (setadjoin x emptyset) A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{uniqinunit} \ \Rightarrow \ ( \mbox{subsetI}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \subseteq @ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ a ) ) )\qquad\mbox{thf}(\mbox{singletonsubset},\mbox{conjecture})$

\medskip\textbf{SEU645$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.kfst (kpair x y) = x)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{iskpair}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{iskpair\_type},\mbox{type})$

$\mbox{iskpair} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setunion}@ a )\mbox{ and }a = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{iskpair},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{kpairp}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{kpairp\_type},\mbox{type})$

$\mbox{kpairp} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{iskpair}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{kpairp},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{setukpairinjL}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjL1\_type},\mbox{type})$

$\mbox{setukpairinjL}_1 = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) )\qquad\mbox{thf}(\mbox{setukpairinjL}_1,\mbox{definition})$

$\mbox{kfstsingleton}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{kfstsingleton\_type},\mbox{type})$

$\mbox{kfstsingleton} = ( \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{iskpair}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ ( \mbox{setunion}@ \mbox{xu} )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ \mbox{xu} ) ) ) ) )\qquad\mbox{thf}(\mbox{kfstsingleton},\mbox{definition})$

$\mbox{theprop}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{theprop\_type},\mbox{type})$

$\mbox{theprop} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{singleton}@ x ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setunion}@ x )@ x ) ) )\qquad\mbox{thf}(\mbox{theprop},\mbox{definition})$

$\mbox{kfst}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kfst\_type},\mbox{type})$

$\mbox{kfst} = ( \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{dsetconstr}@ ( \mbox{setunion}@ \mbox{xu} )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ \mbox{xu} ) ) ) )\qquad\mbox{thf}(\mbox{kfst},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{kpairp} \ \Rightarrow \ ( \mbox{setukpairinjL}_1 \ \Rightarrow \ ( \mbox{kfstsingleton} \ \Rightarrow \ ( \mbox{theprop} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{kfst}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{kfstpairEq},\mbox{conjecture})$

\medskip\textbf{SEU724$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ subset X Y $\to$ ~(in x Y) $\to$ ~(in x X))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \subseteq @ x@ y ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ y \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{contrasubsetT}_1,\mbox{conjecture})$

\medskip\textbf{SEU677$\wedge$2.p} Functions - Lambda Abstraction

(! A:i.! B:i.! f:i.func A B f $\to$ in f (funcSet A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{powersetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI1\_type},\mbox{type})$

$\mbox{powersetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ b@ a ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI}_1,\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{funcSet} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{powerset}@ ( \mbox{cartprod}@ a@ b ) )@ \lambda \mbox{xf}{:}\;\mbox{\$i}{:}\;( \mbox{func}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{funcSet},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{powersetI}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{funcinfuncset},\mbox{conjecture})$

\medskip\textbf{SEU872$\wedge$5.p} TPS problem from FINITE-SET-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cF}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cE}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$( \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{cE}@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{cE} ) )\mbox{ and }\forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{cF}@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{cF} ) ) ) \ \Rightarrow \ \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{cE}@ \mbox{xt}\mbox{ or }\mbox{cF}@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{cE}@ \mbox{xz}\mbox{ or }\mbox{cF}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cDOMLEMMA2\_pme},\mbox{conjecture})$

\medskip\textbf{SEU151+1.p} MPTP bushy problem t10\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c,d{:}\;\neg\,\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(c,d)\mbox{ and }a\ne c\mbox{ and }a\ne d\qquad\mbox{fof}(\mbox{t10\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU151+3.p} Basic properties of sets, theorem 10






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c,d{:}\;\neg\,\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(c,d)\mbox{ and }a\ne c\mbox{ and }a\ne d\qquad\mbox{fof}(\mbox{t10\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU595$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.! x:i.in x (binintersect A B) $\to$ in x B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersect} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersect},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{binintersectER},\mbox{conjecture})$

\medskip\textbf{SEU116+1.p} Boolean domains, theorem 30






$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(\mbox{powerset}(a))\mbox{ and }\mbox{cup\_closed}(\mbox{powerset}(a))\mbox{ and }\mbox{diff\_closed}(\mbox{powerset}(a))\mbox{ and }\mbox{preboolean}(\mbox{powerset}(a)) )\qquad\mbox{fof}(\mbox{fc1\_finsub}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{preboolean}(a) \ \Rightarrow \ ( \mbox{cup\_closed}(a)\mbox{ and }\mbox{diff\_closed}(a) ) )\qquad\mbox{fof}(\mbox{cc1\_finsub}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{cup\_closed}(a)\mbox{ and }\mbox{diff\_closed}(a) ) \ \Rightarrow \ \mbox{preboolean}(a) )\qquad\mbox{fof}(\mbox{cc2\_finsub}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{finite}(a) )\qquad\mbox{fof}(\mbox{cc1\_finset}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{finite}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{finite}(b) ) )\qquad\mbox{fof}(\mbox{cc2\_finset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{preboolean}(\mbox{finite\_subsets}(a))\qquad\mbox{fof}(\mbox{dt\_k5\_finsub}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(\mbox{finite\_subsets}(a))\mbox{ and }\mbox{cup\_closed}(\mbox{finite\_subsets}(a))\mbox{ and }\mbox{diff\_closed}(\mbox{finite\_subsets}(a))\mbox{ and }\mbox{preboolean}(\mbox{finite\_subsets}(a)) )\qquad\mbox{fof}(\mbox{fc2\_finsub}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{finite\_subsets}(a)) \ \Rightarrow \ \mbox{finite}(b) )\qquad\mbox{fof}(\mbox{cc3\_finsub}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{cup\_closed}(a)\mbox{ and }\mbox{cap\_closed}(a)\mbox{ and }\mbox{diff\_closed}(a)\mbox{ and }\mbox{preboolean}(a) )\qquad\mbox{fof}(\mbox{rc1\_finsub}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{finite}(a) )\qquad\mbox{fof}(\mbox{rc1\_finset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b)\mbox{ and }\mbox{relation}(b)\mbox{ and }\mbox{function}(b)\mbox{ and }\mbox{one\_to\_one}(b)\mbox{ and }\mbox{epsilon\_transitive}(b)\mbox{ and }\mbox{epsilon\_connected}(b)\mbox{ and }\mbox{ordinal}(b)\mbox{ and }\mbox{natural}(b)\mbox{ and }\mbox{finite}(b) )\qquad\mbox{fof}(\mbox{rc2\_finset}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b)\mbox{ and }\mbox{finite}(b) ) )\qquad\mbox{fof}(\mbox{rc3\_finset}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b)\mbox{ and }\mbox{finite}(b) ) )\qquad\mbox{fof}(\mbox{rc4\_finset}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{finite\_subsets}(a)) \ \Rightarrow \ \mbox{finite}(b) )\qquad\mbox{fof}(\mbox{t30\_finsub}_1,\mbox{conjecture})$

\medskip\textbf{SEU229+3.p} Ordinal numbers, theorem 3






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( d = \mbox{unordered\_triple}(a,b,c)\iff \forall e{:}\;( \mbox{in}(e,d)\iff \neg\,e\ne a\mbox{ and }e\ne b\mbox{ and }e\ne c ) )\qquad\mbox{fof}(\mbox{d1\_enumset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{in}(b,c)\mbox{ and }\mbox{in}(c,a)\qquad\mbox{fof}(\mbox{t3\_ordinal}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\forall d{:}\;\neg\,\mbox{in}(d,b)\mbox{ and }\mbox{in}(d,c)\qquad\mbox{fof}(\mbox{t7\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU928$\wedge$5.p} TPS problem THM48A






$\forall f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( f@ ( f@ \mbox{xx} ) ) = ( f@ ( f@ \mbox{xy} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cTHM48A\_pme},\mbox{conjecture})$

\medskip\textbf{SEU892$\wedge$5.p} TPS problem X6104






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\exists \mbox{xi}{:}\;( a\to a )\to ( a\to a )\to \mbox{\$o}{:}\;( \forall \mbox{xg}{:}\;a\to a{:}\;( \mbox{xi}@ \mbox{xg}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{xx}\mbox{ and }\mbox{xi}@ \mbox{xg}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xg}@ ( \mbox{xg}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xf}{:}\;a\to a,\mbox{xy}{:}\;a{:}\;( ( \mbox{xi}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{xy}@ \mbox{xf} ) \ \Rightarrow \ ( \mbox{xf}@ \mbox{xy} ) = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cX}_{6104},\mbox{conjecture})$

\medskip\textbf{SEU908$\wedge$5.p} TPS problem from MISTAKEN-LEASTCLOSEDUNDER






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$f{:}\;a\to a\qquad\mbox{thf}(f,\mbox{type})$

$\forall \mbox{xx}{:}\;a{:}\;( \forall \mbox{xp}{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xp}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xp}@ ( f@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xp}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xp}@ ( f@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ ( \mbox{xp}@ ( f@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM527\_pme},\mbox{conjecture})$

\medskip\textbf{SEU532$\wedge$2.p} Preliminary Notions - Equality Laws

(! x:i.! y:i.in x (setadjoin x (setadjoin y emptyset)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{upairsetIL},\mbox{conjecture})$

\medskip\textbf{SEU886$\wedge$5.p} TPS problem X5203






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$f{:}\;b\to a\qquad\mbox{thf}(f,\mbox{type})$

$y{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(y,\mbox{type})$

$x{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$\forall \mbox{xx}_0{:}\;a{:}\;( \exists \mbox{xt}{:}\;b{:}\;( x@ \mbox{xt}\mbox{ and }y@ \mbox{xt}\mbox{ and }\mbox{xx}_0 = ( f@ \mbox{xt} ) ) \ \Rightarrow \ ( \exists \mbox{xt}{:}\;b{:}\;( x@ \mbox{xt}\mbox{ and }\mbox{xx}_0 = ( f@ \mbox{xt} ) )\mbox{ and }\exists \mbox{xt}{:}\;b{:}\;( y@ \mbox{xt}\mbox{ and }\mbox{xx}_0 = ( f@ \mbox{xt} ) ) ) )\qquad\mbox{thf}(\mbox{cX5203\_pme},\mbox{conjecture})$

\medskip\textbf{SEU454$\wedge$1.p} The reflexive closure operator is idempotent






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{idem}@ \mbox{rc}\qquad\mbox{thf}(\mbox{reflexive\_closure\_op\_is\_idempotent},\mbox{conjecture})$

\medskip\textbf{SEU966$\wedge$5.p} TPS problem from FUNCTION-THMS






$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$h{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(h,\mbox{type})$

$( ( h@ a ) = a\mbox{ and }( h@ b )\ne a\mbox{ and }\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = ( \mbox{xg}@ \mbox{xx} ) \ \Rightarrow \ \mbox{xf} = \mbox{xg} ) ) \ \Rightarrow \ \neg\,\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}_4{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj}_4 ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xj}@ ( \mbox{xj}_4@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}_5{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj}_5 ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xj}@ ( \mbox{xj}_5@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xk} ) ) )\qquad\mbox{thf}(\mbox{cTHM196A\_pme},\mbox{conjecture})$

\medskip\textbf{SEU989$\wedge$5.p} TPS problem from GRAPHS-THMS






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\exists s{:}\;b\to a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;b{:}\;\exists \mbox{xy}{:}\;a{:}\;( s@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}_1{:}\;a,\mbox{xy}_2{:}\;a{:}\;( ( s@ \mbox{xx}@ \mbox{xy}_1\mbox{ and }s@ \mbox{xx}@ \mbox{xy}_2 ) \ \Rightarrow \ \mbox{xy}_1 = \mbox{xy}_2 )\mbox{ and }\forall \mbox{xx}_1{:}\;b,\mbox{xx}_2{:}\;b,\mbox{xy}{:}\;a{:}\;( ( s@ \mbox{xx}_1@ \mbox{xy}\mbox{ and }s@ \mbox{xx}_2@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx}_1 = \mbox{xx}_2 ) ) \ \Rightarrow \ \exists r{:}\;a\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;\exists \mbox{xy}{:}\;b{:}\;( r@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xy}{:}\;b{:}\;\exists \mbox{xx}{:}\;a{:}\;( r@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}_1{:}\;b,\mbox{xy}_2{:}\;b{:}\;( ( r@ \mbox{xx}@ \mbox{xy}_1\mbox{ and }r@ \mbox{xx}@ \mbox{xy}_2 ) \ \Rightarrow \ \mbox{xy}_1 = \mbox{xy}_2 ) )\qquad\mbox{thf}(\mbox{cTHM554\_pme},\mbox{conjecture})$

\medskip\textbf{SEU844$\wedge$5.p} TPS problem GAZING-THM8






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cS}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cT}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cT},\mbox{type})$

$\forall s_0{:}\;a\to \mbox{\$o},t_0{:}\;a\to \mbox{\$o}{:}\;s_0 = t_0 \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cS}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cT}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cT}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM8\_pme},\mbox{conjecture})$

\medskip\textbf{SEU998$\wedge$5.p} TPS problem 3-DIAMOND-THM






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{jOIN}{:}\;a\to a\to a,\mbox{mEET}{:}\;a\to a\to a{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{jOIN}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{jOIN}@ \mbox{xx}@ ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{mEET}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{mEET}@ \mbox{xx}@ ( \mbox{mEET}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{mEET}@ \mbox{xy}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{jOIN}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{xy}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{mEET}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{xy} ) \ \Rightarrow \ ( \exists \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xa}{:}\;a,\mbox{xb}{:}\;a,\mbox{xc}{:}\;a{:}\;( \mbox{xa}\ne \mbox{xb}\mbox{ and }\mbox{xa}\ne \mbox{xc}\mbox{ and }\mbox{xa}\ne \mbox{xx}\mbox{ and }\mbox{xa}\ne \mbox{xy}\mbox{ and }\mbox{xb}\ne \mbox{xc}\mbox{ and }\mbox{xb}\ne \mbox{xx}\mbox{ and }\mbox{xb}\ne \mbox{xy}\mbox{ and }\mbox{xc}\ne \mbox{xx}\mbox{ and }\mbox{xc}\ne \mbox{xy}\mbox{ and }\mbox{xx}\ne \mbox{xy}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xa} ) = \mbox{xa}\mbox{ and }( \mbox{jOIN}@ \mbox{xx}@ \mbox{xa} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xb} ) = \mbox{xb}\mbox{ and }( \mbox{jOIN}@ \mbox{xx}@ \mbox{xb} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xc} ) = \mbox{xc}\mbox{ and }( \mbox{jOIN}@ \mbox{xx}@ \mbox{xc} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xa}@ \mbox{xc} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xa}@ \mbox{xc} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xa}@ \mbox{xy} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xa}@ \mbox{xy} ) = \mbox{xa}\mbox{ and }( \mbox{mEET}@ \mbox{xb}@ \mbox{xc} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xb}@ \mbox{xc} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xb}@ \mbox{xy} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xb}@ \mbox{xy} ) = \mbox{xb}\mbox{ and }( \mbox{mEET}@ \mbox{xc}@ \mbox{xy} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xc}@ \mbox{xy} ) = \mbox{xc} ) \ \Rightarrow \ \neg\,\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xz} ) ) = ( \mbox{jOIN}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xz} ) ) ) )\qquad\mbox{thf}(\mbox{c3\_DIAMOND\_THM\_pme},\mbox{conjecture})$

\medskip\textbf{SEU986$\wedge$5.p} TPS problem from FINITE-SETS-RELNS-THMS






$\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xw}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xp} ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xw}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xq} ) ) \ \Rightarrow \ \forall \mbox{xw}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xq}@ \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xs}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xs}@ \mbox{xu}@ \mbox{xv}\mbox{ or }( \mbox{xu} = \mbox{xx}\mbox{ and }\mbox{xv} = \mbox{xy} ) ) ) ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx}\mbox{ and }\mbox{xq}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM162\_pme},\mbox{conjecture})$

\medskip\textbf{SEU625$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! B:i.! x:i.in x A $\to$ (! y:i.in y B $\to$ subset (setadjoin x
 (setadjoin y emptyset)) (binunion A B)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionLsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionLsub\_type},\mbox{type})$

$\mbox{binunionLsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \subseteq @ a@ ( \mbox{binunion}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{binunionLsub},\mbox{definition})$

$\mbox{binunionRsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionRsub\_type},\mbox{type})$

$\mbox{binunionRsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \subseteq @ b@ ( \mbox{binunion}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{binunionRsub},\mbox{definition})$

$\mbox{upairset2E}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{upairset2E\_type},\mbox{type})$

$\mbox{upairset2E} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) \ \Rightarrow \ ( \mbox{xz} = \mbox{xx}\mbox{ or }\mbox{xz} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{upairset2E},\mbox{definition})$

$\mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ ( \mbox{binunionLsub} \ \Rightarrow \ ( \mbox{binunionRsub} \ \Rightarrow \ ( \mbox{upairset2E} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \subseteq @ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ ( \mbox{binunion}@ a@ b ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{upairsubunion},\mbox{conjecture})$

\medskip\textbf{SEU196+1.p} MPTP bushy problem t99\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_dom\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k7\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{fc6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_rng}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_rng}(a)) ) )\qquad\mbox{fof}(\mbox{fc8\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( a\subseteq b \ \Rightarrow \ ( \mbox{relation\_dom}(a)\subseteq \mbox{relation\_dom}(b)\mbox{ and }\mbox{relation\_rng}(a)\subseteq \mbox{relation\_rng}(b) ) ) ) )\qquad\mbox{fof}(\mbox{t25\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_dom\_restriction}(b,a)\subseteq b )\qquad\mbox{fof}(\mbox{t88\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_rng}(\mbox{relation\_dom\_restriction}(b,a))\subseteq \mbox{relation\_rng}(b) )\qquad\mbox{fof}(\mbox{t99\_relat}_1,\mbox{conjecture})$

\medskip\textbf{SEU125+1.p} MPTP bushy problem t8\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }c\subseteq b ) \ \Rightarrow \ \mbox{set\_union}_2(a,c)\subseteq b )\qquad\mbox{fof}(\mbox{t8\_xboole}_1,\mbox{conjecture})$

\medskip\textbf{SEU927$\wedge$5.p} TPS problem THM92

Trivial theorem which gives nice simple example of an expansion
 proof.




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xf}{:}\;a\to a,\mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xf}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM92\_pme},\mbox{conjecture})$

\medskip\textbf{SEU564$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.! x:i.subset A B $\to$ in x A $\to$ in x B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\subseteq  = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{subset},\mbox{definition})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{conjecture})$

\medskip\textbf{SEU885$\wedge$5.p} TPS problem THM30A






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cF}{:}\;b\to a\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cV}{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cV},\mbox{type})$

$\mbox{cU}{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cU},\mbox{type})$

$\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{cU}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cV}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( \exists \mbox{xt}{:}\;b{:}\;( \mbox{cU}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) ) \ \Rightarrow \ \exists \mbox{xt}{:}\;b{:}\;( \mbox{cV}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cTHM30A\_pme},\mbox{conjecture})$

\medskip\textbf{SEU490$\wedge$1.p} Semi confluence implies Church-Rosser property






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{sconfl}@ r ) \ \Rightarrow \ ( \mbox{cr}@ r ) )\qquad\mbox{thf}(\mbox{semi\_confluence\_implies\_church\_rosser},\mbox{conjecture})$

\medskip\textbf{SEU122+2.p} MPTP chainy problem t2\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{lemma})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{t2\_xboole}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\neg\,\exists c{:}\;\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t4\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU950$\wedge$5.p} TPS problem THM573

Challenge from Dana Scott stemming from injective Cantor Theorem.




$h{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(h,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xy}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( h@ \mbox{xx} ) = ( h@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists x{:}\;\mbox{\$i}{:}\;( \mbox{xg}@ x ) = y\qquad\mbox{thf}(\mbox{cTHM573\_pme},\mbox{conjecture})$

\medskip\textbf{SEU917$\wedge$5.p} TPS problem THM8






$\exists f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM8\_pme},\mbox{conjecture})$

\medskip\textbf{SEU934$\wedge$5.p} TPS problem THM14






$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \lambda y{:}\;\mbox{\$i}{:}\;y = \mbox{xx} ) = ( \lambda y{:}\;\mbox{\$i}{:}\;y = \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM14\_pme},\mbox{conjecture})$

\medskip\textbf{SEU850$\wedge$5.p} TPS problem GAZING-THM9






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o},u{:}\;a\to \mbox{\$o}{:}\;( ( s = t\mbox{ and }t = u ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( t@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( u@ \mbox{xx} ) \ \Rightarrow \ ( s@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM9\_pme},\mbox{conjecture})$

\medskip\textbf{SEU480$\wedge$1.p} Termination implies the induction principle






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{term}@ r ) \ \Rightarrow \ ( \mbox{ind}@ r ) )\qquad\mbox{thf}(\mbox{termination\_implies\_induction},\mbox{conjecture})$

\medskip\textbf{SEU987$\wedge$5.p} TPS problem from FUNS-AND-RELNS-THMS






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$g{:}\;b\to b\qquad\mbox{thf}(g,\mbox{type})$

$f{:}\;a\to a\qquad\mbox{thf}(f,\mbox{type})$

$\mbox{cS}{:}\;a\to b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$( \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( g@ \mbox{xx} ) = ( g@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xu}{:}\;a\to b\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;b{:}\;( ( \mbox{cS}@ \mbox{xx}@ \mbox{xy} )\iff ( \mbox{xu}@ ( f@ \mbox{xx} )@ ( g@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cSV7\_pme},\mbox{conjecture})$

\medskip\textbf{SEU535$\wedge$2.p} Preliminary Notions - Bounded Quantifier Laws

(! phi:i$>$o.! x:i.in x emptyset $\to$ phi x)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{vacuousDall},\mbox{conjecture})$

\medskip\textbf{SEU551$\wedge$3.p} A simple congruence property of emptyset

Reflexivity.




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall a{:}\;a{:}\;a = a\qquad\mbox{thf}(\mbox{cER\_eq\_},\mbox{conjecture})$

\medskip\textbf{SEU510$\wedge$2.p} Preliminary Notions - Basic Laws of Logic

(! A:i.! phi:i$>$o.! x:i.in x A $\to$ phi x $\to$ nonempty (dsetconstr A
 ($\wedge$ y:i.phi y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{nonempty} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\ne \mbox{emptyset} )\qquad\mbox{thf}(\mbox{nonempty},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{emptysetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{nonempty}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{nonemptyI},\mbox{conjecture})$

\medskip\textbf{SEU665$\wedge$2.p} Ordered Pairs - Sets of Pairs

(! A:i.! B:i.! phi:i$>$(i$>$o).! x:i.in x A $\to$ (! y:i.in y B $\to$
 phi x y $\to$ in (kpair x y) (dpsetconstr A B ($\wedge$ z,u:i.phi z u))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprodpairin}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodpairin\_type},\mbox{type})$

$\mbox{cartprodpairin} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{cartprod}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprodpairin},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstr} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{dsetconstr}@ ( \mbox{cartprod}@ a@ b )@ \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xphi}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstr},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{setext} \ \Rightarrow \ ( \mbox{cartprodpairin} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xz}@ \mbox{xu} ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrI},\mbox{conjecture})$

\medskip\textbf{SEU406+1.p} The Operation of Addition of Relational Structures T01






$\forall a,b,c,d{:}\;\neg\,\mbox{r2\_hidden}(a,\mbox{k2\_xboole}_0(c,d))\mbox{ and }\mbox{r2\_hidden}(b,\mbox{k2\_xboole}_0(c,d))\mbox{ and }\neg\,\mbox{r2\_hidden}(a,\mbox{k4\_xboole}_0(c,d))\mbox{ and }\mbox{r2\_hidden}(b,\mbox{k4\_xboole}_0(c,d))\mbox{ and }\neg\,\mbox{r2\_hidden}(a,d)\mbox{ and }\mbox{r2\_hidden}(b,d)\mbox{ and }\neg\,\mbox{r2\_hidden}(a,\mbox{k4\_xboole}_0(c,d))\mbox{ and }\mbox{r2\_hidden}(b,d)\mbox{ and }\neg\,\mbox{r2\_hidden}(a,d)\mbox{ and }\mbox{r2\_hidden}(b,\mbox{k4\_xboole}_0(c,d))\qquad\mbox{fof}(\mbox{t1\_latsum}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{r2\_hidden}(a,b) \ \Rightarrow \ \neg\,\mbox{r2\_hidden}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{k2\_xboole}_0(a,b) = \mbox{k2\_xboole}_0(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{k2\_xboole}_0(a,b)\iff \forall d{:}\;( \mbox{r2\_hidden}(d,c)\iff ( \mbox{r2\_hidden}(d,a)\mbox{ or }\mbox{r2\_hidden}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{m1\_subset}_1(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{v1\_xboole}_0(\mbox{k1\_xboole}_0)\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{v1\_xboole}_0(a) \ \Rightarrow \ \neg\,\mbox{v1\_xboole}_0(\mbox{k2\_xboole}_0(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{v1\_xboole}_0(a) \ \Rightarrow \ \neg\,\mbox{v1\_xboole}_0(\mbox{k2\_xboole}_0(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{k2\_xboole}_0(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{v1\_xboole}_0(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{v1\_xboole}_0(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{k2\_xboole}_0(a,\mbox{k1\_xboole}_0) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{r2\_hidden}(a,b) \ \Rightarrow \ \mbox{m1\_subset}_1(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{m1\_subset}_1(a,b) \ \Rightarrow \ ( \mbox{v1\_xboole}_0(b)\mbox{ or }\mbox{r2\_hidden}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;\mbox{k2\_xboole}_0(a,\mbox{k4\_xboole}_0(b,a)) = \mbox{k2\_xboole}_0(a,b)\qquad\mbox{fof}(\mbox{t39\_xboole}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{k4\_xboole}_0(a,\mbox{k1\_xboole}_0) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{k4\_xboole}_0(\mbox{k1\_xboole}_0,a) = \mbox{k1\_xboole}_0\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{v1\_xboole}_0(a) \ \Rightarrow \ a = \mbox{k1\_xboole}_0 )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{r2\_hidden}(a,b)\mbox{ and }\mbox{v1\_xboole}_0(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{v1\_xboole}_0(a)\mbox{ and }a\ne b\mbox{ and }\mbox{v1\_xboole}_0(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU120+1.p} MPTP bushy problem t4\_xboole\_0






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\neg\,\exists c{:}\;\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t4\_xboole}_0,\mbox{conjecture})$

\medskip\textbf{SEU509$\wedge$2.p} Preliminary Notions - Basic Laws of Logic

(! A:i.nonempty A $\to$ (? x:i.in x A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{nonempty} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\ne \mbox{emptyset} )\qquad\mbox{thf}(\mbox{nonempty},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ ( \mbox{setext} \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{nonempty}@ a ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{nonemptyE}_1,\mbox{conjecture})$

\medskip\textbf{SEU884$\wedge$5.p} TPS problem THM30B






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cS}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cR}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\forall \mbox{xx}{:}\;a{:}\;( \exists \mbox{xt}{:}\;a{:}\;( \mbox{cR}@ \mbox{xt}\mbox{ and }\mbox{xx} = \mbox{xt} ) \ \Rightarrow \ \exists \mbox{xt}{:}\;a{:}\;( \mbox{cS}@ \mbox{xt}\mbox{ and }\mbox{xx} = \mbox{xt} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cR}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTHM30B\_pme},\mbox{conjecture})$

\medskip\textbf{SEU243+1.p} MPTP bushy problem t5\_wellord1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{well\_founded\_relation}(a)\iff \forall b{:}\;\neg\,b\subseteq \mbox{relation\_field}(a)\mbox{ and }b\ne \mbox{empty\_set}\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\mbox{disjoint}(\mbox{fiber}(a,c),b) ) )\qquad\mbox{fof}(\mbox{d2\_wellord}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{is\_well\_founded\_in}(a,b)\iff \forall c{:}\;\neg\,c\subseteq b\mbox{ and }c\ne \mbox{empty\_set}\mbox{ and }\forall d{:}\;\neg\,\mbox{in}(d,c)\mbox{ and }\mbox{disjoint}(\mbox{fiber}(a,d),c) ) )\qquad\mbox{fof}(\mbox{d3\_wellord}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation\_field}(a) = \mbox{set\_union}_2(\mbox{relation\_dom}(a),\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{d6\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_wellord}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{well\_founded\_relation}(a)\iff \mbox{is\_well\_founded\_in}(a,\mbox{relation\_field}(a)) ) )\qquad\mbox{fof}(\mbox{t5\_wellord}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU457$\wedge$1.p} The symmetric closure of a binary relation is symmetric






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{symm}@ ( \mbox{sc}@ r ) )\qquad\mbox{thf}(\mbox{symmetric\_closure\_is\_symmetric},\mbox{conjecture})$

\medskip\textbf{SEU810$\wedge$2.p} Omega and Peano

(! x:i.in x omega $\to$ ~(omegaS x = emptyset))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{omega}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{omega\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\mbox{omegaS}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{omegaS\_type},\mbox{type})$

$\mbox{omegaS} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{omegaS},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ ( \mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{in\_\_Cong} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{omega} ) \ \Rightarrow \ ( \mbox{omegaS}@ \mbox{xx} )\ne \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{peano0notS},\mbox{conjecture})$

\medskip\textbf{SEU528$\wedge$2.p} Preliminary Notions - Equality Laws

(! x:i.! y:i.~(x = y) $\to$ ~(in y (setadjoin x emptyset)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{uniqinunit} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx}\ne \mbox{xy} \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) )\qquad\mbox{thf}(\mbox{notinsingleton},\mbox{conjecture})$

\medskip\textbf{SEU120+2.p} MPTP chainy problem t4\_xboole\_0






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\neg\,\exists c{:}\;\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t4\_xboole}_0,\mbox{conjecture})$

\medskip\textbf{SEU164+3.p} Basic properties of sets, theorem 99






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{powerset}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c\subseteq a ) )\qquad\mbox{fof}(\mbox{d1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq b\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l2\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{union}(\mbox{powerset}(a)) = a\qquad\mbox{fof}(\mbox{t99\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU568$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.! x:i.in x A $\to$ ~(in x B) $\to$ ~(A = B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{notsubsetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{notsubsetI\_type},\mbox{type})$

$\mbox{notsubsetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ \neg\,\subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{notsubsetI},\mbox{definition})$

$\mbox{notequalI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{notequalI1\_type},\mbox{type})$

$\mbox{notequalI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \neg\,\subseteq @ a@ b \ \Rightarrow \ a\ne b ) )\qquad\mbox{thf}(\mbox{notequalI}_1,\mbox{definition})$

$\mbox{notsubsetI} \ \Rightarrow \ ( \mbox{notequalI}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ a\ne b ) ) )\qquad\mbox{thf}(\mbox{notequalI}_2,\mbox{conjecture})$

\medskip\textbf{SEU230+3.p} Ordinal numbers, theorem  Ordinal numbers, theorem 10






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{succ}(a) = \mbox{set\_union}_2(a,\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d1\_ordinal}_1,\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{succ}(a))\qquad\mbox{fof}(\mbox{fc1\_ordinal}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_non\_empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc5\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{in}(a,\mbox{succ}(a))\qquad\mbox{fof}(\mbox{t10\_ordinal}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU798$\wedge$2.p} More about Functions - Injective Functions

(! A:i.! B:i.! f:i.in f (funcSet A B) $\to$ injective A B f $\to$
 in f (injFuncSet A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{injective}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{injective\_type},\mbox{type})$

$\mbox{injFuncSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{injFuncSet\_type},\mbox{type})$

$\mbox{injFuncSet} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{funcSet}@ a@ b )@ \lambda \mbox{xf}{:}\;\mbox{\$i}{:}\;( \mbox{injective}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{injFuncSet},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ ( ( \mbox{injective}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xf}@ ( \mbox{injFuncSet}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{injFuncInInjFuncSet},\mbox{conjecture})$

\medskip\textbf{SEU634$\wedge$2.p} Ordered Pairs - Singletons

(! A:i.subset (setunion (setadjoin A emptyset)) A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{setunionE}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionE2\_type},\mbox{type})$

$\mbox{setunionE}_2 = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$i}{:}\;( \mbox{in}@ x@ a\mbox{ and }\mbox{in}@ \mbox{xx}@ x ) ) )\qquad\mbox{thf}(\mbox{setunionE}_2,\mbox{definition})$

$\mbox{uniqinunit} \ \Rightarrow \ ( \mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{setunionE}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( \subseteq @ ( \mbox{setunion}@ ( \mbox{setadjoin}@ a@ \mbox{emptyset} ) )@ a ) ) )\qquad\mbox{thf}(\mbox{setunionsingleton}_1,\mbox{conjecture})$

\medskip\textbf{SEU492$\wedge$1.p} Alternative definition of a strict (partial) order: requiring

Alternative definition of a strict (partial) order: requiring
 irreflexibility instead of asymmetry.




$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{so} = ( \lambda r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{irrefl}@ r\mbox{ and }\mbox{trans}@ r ) )\qquad\mbox{thf}(\mbox{alternative\_definition\_of\_strict\_order},\mbox{conjecture})$

\medskip\textbf{SEU828$\wedge$1.p} About powersets 1






$\mbox{seteq}{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to ( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{seteq\_type},\mbox{type})$

$\mbox{seteq} = ( \lambda x{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o},y{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;\forall u{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( x@ u )\iff ( y@ u ) ) )\qquad\mbox{thf}(\mbox{seteq},\mbox{definition})$

$\mbox{subseteq}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subseteq\_type},\mbox{type})$

$\mbox{subseteq} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall u{:}\;\mbox{\$i}{:}\;( ( x@ u ) \ \Rightarrow \ ( y@ u ) ) )\qquad\mbox{thf}(\mbox{subseteq},\mbox{definition})$

$\mbox{powerset}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powerset} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{subseteq}@ y@ x ) )\qquad\mbox{thf}(\mbox{poserset},\mbox{definition})$

$\mbox{emptyset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptyset} = ( \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$false} )\qquad\mbox{thf}(\mbox{emptyset},\mbox{definition})$

$\mbox{seteq}@ ( \mbox{powerset}@ \mbox{emptyset} )@ \lambda x{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;x = \mbox{emptyset}\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SEU142+1.p} MPTP bushy problem t69\_enumset1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{unordered\_pair}(a,a) = \mbox{singleton}(a)\qquad\mbox{fof}(\mbox{t69\_enumset}_1,\mbox{conjecture})$

\medskip\textbf{SEU930$\wedge$5.p} TPS problem THM171A

If g commutes with f, any unique fixed point of gi is a fixed 
 point of f.




$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( f@ ( g@ \mbox{xx} ) ) ) = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( g@ ( f@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( ( g@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xz} ) = \mbox{xz} \ \Rightarrow \ \mbox{xz} = \mbox{xx} ) ) \ \Rightarrow \ ( f@ \mbox{xx} ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{cTHM171A\_pme},\mbox{conjecture})$

\medskip\textbf{SEU323+1.p} MPTP bushy problem t51\_tops\_1






$\forall a,b{:}\;( ( \mbox{topological\_space}(a)\mbox{ and }\mbox{top\_str}(a)\mbox{ and }\mbox{closed\_subset}(b,a)\mbox{ and }\mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) ) \ \Rightarrow \ \mbox{open\_subset}(\mbox{subset\_complement}(\mbox{the\_carrier}(a),b),a) )\qquad\mbox{fof}(\mbox{fc3\_tops}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{topological\_space}(a)\mbox{ and }\mbox{top\_str}(a) ) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a)))\mbox{ and }\mbox{closed\_subset}(b,a) ) )\qquad\mbox{fof}(\mbox{rc6\_pre\_topc},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{subset\_complement}(a,\mbox{subset\_complement}(a,b)) = b )\qquad\mbox{fof}(\mbox{involutiveness\_k3\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\exists a{:}\;\mbox{one\_sorted\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_struct}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{element}(\mbox{subset\_complement}(a,b),\mbox{powerset}(a)) )\qquad\mbox{fof}(\mbox{dt\_k3\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{top\_str}(a)\mbox{ and }\mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) ) \ \Rightarrow \ \mbox{element}(\mbox{topstr\_closure}(a,b),\mbox{powerset}(\mbox{the\_carrier}(a))) )\qquad\mbox{fof}(\mbox{dt\_k6\_pre\_topc},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_l1\_struct}_0,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{topological\_space}(a)\mbox{ and }\mbox{top\_str}(a)\mbox{ and }\mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) ) \ \Rightarrow \ \mbox{closed\_subset}(\mbox{topstr\_closure}(a,b),a) )\qquad\mbox{fof}(\mbox{fc2\_tops}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{topological\_space}(a)\mbox{ and }\mbox{top\_str}(a)\mbox{ and }\mbox{open\_subset}(b,a)\mbox{ and }\mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) ) \ \Rightarrow \ \mbox{closed\_subset}(\mbox{subset\_complement}(\mbox{the\_carrier}(a),b),a) )\qquad\mbox{fof}(\mbox{fc4\_tops}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{top\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_pre\_topc},\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{top\_str}(a)\mbox{ and }\mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) ) \ \Rightarrow \ \mbox{element}(\mbox{interior}(a,b),\mbox{powerset}(\mbox{the\_carrier}(a))) )\qquad\mbox{fof}(\mbox{dt\_k1\_tops}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{top\_str}(a) \ \Rightarrow \ \mbox{one\_sorted\_str}(a) )\qquad\mbox{fof}(\mbox{dt\_l1\_pre\_topc},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_u1\_struct}_0,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{topological\_space}(a)\mbox{ and }\mbox{top\_str}(a) ) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a)))\mbox{ and }\mbox{open\_subset}(b,a) ) )\qquad\mbox{fof}(\mbox{rc1\_tops}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{top\_str}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) \ \Rightarrow \ \mbox{interior}(a,b) = \mbox{subset\_complement}(\mbox{the\_carrier}(a),\mbox{topstr\_closure}(a,\mbox{subset\_complement}(\mbox{the\_carrier}(a),b))) ) )\qquad\mbox{fof}(\mbox{d1\_tops}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{topological\_space}(a)\mbox{ and }\mbox{top\_str}(a) ) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) \ \Rightarrow \ \mbox{open\_subset}(\mbox{interior}(a,b),a) ) )\qquad\mbox{fof}(\mbox{t51\_tops}_1,\mbox{conjecture})$

\medskip\textbf{SEU814$\wedge$2.p} Transitive Sets

(! X:i.transitiveset X $\to$ (! A:i.! x:i.in A X $\to$ in x A $\to$
 in x X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{transitiveset} = ( \lambda a{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ ( \subseteq @ x@ a ) ) )\qquad\mbox{thf}(\mbox{transitiveset},\mbox{definition})$

$\mbox{transitivesetOp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{transitivesetOp1\_type},\mbox{type})$

$\mbox{transitivesetOp}_1 = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{transitiveset}@ x ) \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ ( \subseteq @ a@ x ) ) ) )\qquad\mbox{thf}(\mbox{transitivesetOp}_1,\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ ( \mbox{transitivesetOp}_1 \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{transitiveset}@ x ) \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{transitivesetOp}_2,\mbox{conjecture})$

\medskip\textbf{SEU140+1.p} MPTP bushy problem t63\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_intersection}_2(a,c)\subseteq \mbox{set\_intersection}_2(b,c) )\qquad\mbox{fof}(\mbox{t26\_xboole}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t3\_xboole}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }\mbox{disjoint}(b,c) ) \ \Rightarrow \ \mbox{disjoint}(a,c) )\qquad\mbox{fof}(\mbox{t63\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU154+1.p} MPTP bushy problem l28\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{in}(a,b) \ \Rightarrow \ \mbox{disjoint}(\mbox{singleton}(a),b) )\qquad\mbox{fof}(\mbox{l28\_zfmisc}_1,\mbox{conjecture})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{t2\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SEU160+1.p} MPTP bushy problem t39\_zfmisc\_1






$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq \mbox{singleton}(b)\iff ( a = \mbox{empty\_set}\mbox{ or }a = \mbox{singleton}(b) ) )\qquad\mbox{fof}(\mbox{t39\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( a\subseteq \mbox{singleton}(b)\iff ( a = \mbox{empty\_set}\mbox{ or }a = \mbox{singleton}(b) ) )\qquad\mbox{fof}(\mbox{l4\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU970$\wedge$5.p} TPS problem from CHECKERBOARD-RELNS






$s{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{s\_type},\mbox{type})$

$\mbox{cCKB\_E}_2{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_E2\_type},\mbox{type})$

$\mbox{cCKB\_E}_2 = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx}\mbox{ and }\forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ ( s@ \mbox{xu} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cCKB\_E2\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB\_E}_2@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{cCKB\_E}_2@ \mbox{xy}@ \mbox{xz} ) \ \Rightarrow \ ( \mbox{cCKB\_E}_2@ \mbox{xx}@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{cCKB\_L34000\_pme},\mbox{conjecture})$

\medskip\textbf{SEU905$\wedge$5.p} TPS problem THM126A






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xh}_1{:}\;g\to b,\mbox{xh}_2{:}\;b\to a,\mbox{xs}_1{:}\;g\to \mbox{\$o},\mbox{xf}_1{:}\;g\to g\to g,\mbox{xs}_2{:}\;b\to \mbox{\$o},\mbox{xf}_2{:}\;b\to b\to b,\mbox{xs}_3{:}\;a\to \mbox{\$o},\mbox{xf}_3{:}\;a\to a\to a{:}\;( \neg\,\forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx}\mbox{ and }\mbox{xs}_3@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_2@ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xx} ) )@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xy} ) ) ) ) \ \Rightarrow \ \neg\,\forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx}\mbox{ and }\mbox{xs}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xf}_2@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g,\mbox{xy}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx}\mbox{ and }\mbox{xs}_1@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{xf}_2@ ( \mbox{xh}_1@ \mbox{xx} )@ ( \mbox{xh}_1@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx}\mbox{ and }\mbox{xs}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xf}_2@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx}\mbox{ and }\mbox{xs}_3@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx}@ \mbox{xy} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_2@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b,\mbox{xy}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx}\mbox{ and }\mbox{xs}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xh}_2@ ( \mbox{xf}_2@ \mbox{xx}@ \mbox{xy} ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ \mbox{xx} )@ ( \mbox{xh}_2@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM126A\_pme},\mbox{conjecture})$

\medskip\textbf{SEU157+1.p} MPTP bushy problem l55\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{conjecture})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{ordered\_pair}(a,b) = \mbox{ordered\_pair}(c,d) \ \Rightarrow \ ( a = c\mbox{ and }b = d ) )\qquad\mbox{fof}(\mbox{t33\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU472$\wedge$1.p} The transitive reflexive symmetric closure operator is inflationary






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{infl}@ \mbox{trsc}\qquad\mbox{thf}(\mbox{transitive\_reflexive\_symmetric\_closure\_op\_is\_inflationary},\mbox{conjecture})$

\medskip\textbf{SEU606$\wedge$2.p} Preliminary Notions - Operations on Sets - Set Difference

(! A:i.! B:i.! x:i.~(in x (setminus A B)) $\to$ ~(in x B) $\to$
 ~(in x A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusERneg}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusERneg\_type},\mbox{type})$

$\mbox{setminusERneg} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{setminusERneg},\mbox{definition})$

$\mbox{setminusERneg} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ a ) )\qquad\mbox{thf}(\mbox{setminusELneg},\mbox{conjecture})$

\medskip\textbf{SEU857$\wedge$5.p} TPS problem GAZING-THM43






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o},u{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( ( ( ( s@ \mbox{xz}\mbox{ and }\neg\,t@ \mbox{xz} )\mbox{ or }( t@ \mbox{xz}\mbox{ and }\neg\,s@ \mbox{xz} ) )\mbox{ and }\neg\,u@ \mbox{xz} )\mbox{ or }( u@ \mbox{xz}\mbox{ and }\neg\,( s@ \mbox{xz}\mbox{ and }\neg\,t@ \mbox{xz} )\mbox{ or }( t@ \mbox{xz}\mbox{ and }\neg\,s@ \mbox{xz} ) ) ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( s@ \mbox{xz}\mbox{ and }\neg\,( t@ \mbox{xz}\mbox{ and }\neg\,u@ \mbox{xz} )\mbox{ or }( u@ \mbox{xz}\mbox{ and }\neg\,t@ \mbox{xz} ) )\mbox{ or }( ( ( t@ \mbox{xz}\mbox{ and }\neg\,u@ \mbox{xz} )\mbox{ or }( u@ \mbox{xz}\mbox{ and }\neg\,t@ \mbox{xz} ) )\mbox{ and }\neg\,s@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM43\_pme},\mbox{conjecture})$

\medskip\textbf{SEU172+1.p} MPTP bushy problem t54\_subset\_1






$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{subset\_complement}(a,\mbox{subset\_complement}(a,b)) = b )\qquad\mbox{fof}(\mbox{involutiveness\_k3\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{element}(\mbox{subset\_complement}(a,b),\mbox{powerset}(a)) )\qquad\mbox{fof}(\mbox{dt\_k3\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{subset\_complement}(a,b) = \mbox{set\_difference}(a,b) )\qquad\mbox{fof}(\mbox{d5\_subset}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{element}(c,\mbox{powerset}(a)) \ \Rightarrow \ \neg\,\mbox{in}(b,\mbox{subset\_complement}(a,c))\mbox{ and }\mbox{in}(b,c) )\qquad\mbox{fof}(\mbox{t54\_subset}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

\medskip\textbf{SEU267+1.p} MPTP bushy problem t7\_mcart\_1






$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_mcart}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_mcart}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{pair\_first}(\mbox{ordered\_pair}(a,b)) = a\mbox{ and }\mbox{pair\_second}(\mbox{ordered\_pair}(a,b)) = b )\qquad\mbox{fof}(\mbox{t7\_mcart}_1,\mbox{conjecture})$

$\forall a{:}\;( \exists b,c{:}\;a = \mbox{ordered\_pair}(b,c) \ \Rightarrow \ \forall b{:}\;( b = \mbox{pair\_first}(a)\iff \forall c,d{:}\;( a = \mbox{ordered\_pair}(c,d) \ \Rightarrow \ b = c ) ) )\qquad\mbox{fof}(\mbox{d1\_mcart}_1,\mbox{axiom})$

$\forall a{:}\;( \exists b,c{:}\;a = \mbox{ordered\_pair}(b,c) \ \Rightarrow \ \forall b{:}\;( b = \mbox{pair\_second}(a)\iff \forall c,d{:}\;( a = \mbox{ordered\_pair}(c,d) \ \Rightarrow \ b = d ) ) )\qquad\mbox{fof}(\mbox{d2\_mcart}_1,\mbox{axiom})$

\medskip\textbf{SEU615$\wedge$2.p} Preliminary Notions - Operations on Sets - Symmetric Difference

(! A:i.! B:i.! x:i.~(in x A) $\to$ ~(in x B) $\to$ ~(in x
 (symdiff A B)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{symdiff}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{symdiff\_type},\mbox{type})$

$\mbox{symdiffE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{symdiffE\_type},\mbox{type})$

$\mbox{symdiffE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{symdiff}@ a@ b ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ \mbox{xphi} ) ) \ \Rightarrow \ ( ( \neg\,\mbox{in}@ \mbox{xx}@ a \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \mbox{xphi} ) ) \ \Rightarrow \ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{symdiffE},\mbox{definition})$

$\mbox{symdiffE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{in}@ \mbox{xx}@ a \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{symdiff}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{symdiffIneg}_2,\mbox{conjecture})$

\medskip\textbf{SEU578$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.in A (powerset A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetRefl}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetRefl\_type},\mbox{type})$

$\mbox{subsetRefl} = ( \forall a{:}\;\mbox{\$i}{:}\;( \subseteq @ a@ a ) )\qquad\mbox{thf}(\mbox{subsetRefl},\mbox{definition})$

$\mbox{powersetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI1\_type},\mbox{type})$

$\mbox{powersetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ b@ a ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI}_1,\mbox{definition})$

$\mbox{subsetRefl} \ \Rightarrow \ ( \mbox{powersetI}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( \mbox{in}@ a@ ( \mbox{powerset}@ a ) ) )\qquad\mbox{thf}(\mbox{inPowerset},\mbox{conjecture})$

\medskip\textbf{SEU359+1.p} MPTP bushy problem t30\_yellow\_0






$\forall a{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \forall b,c{:}\;( \mbox{element}(c,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{ex\_sup\_of\_relstr\_set}(a,b) \ \Rightarrow \ ( c = \mbox{join\_on\_relstr}(a,b)\iff ( \mbox{relstr\_set\_smaller}(a,b,c)\mbox{ and }\forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_set\_smaller}(a,b,d) \ \Rightarrow \ \mbox{related}(a,c,d) ) ) ) ) ) ) )\qquad\mbox{fof}(\mbox{d9\_yellow}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \mbox{element}(\mbox{join\_on\_relstr}(a,b),\mbox{the\_carrier}(a)) )\qquad\mbox{fof}(\mbox{dt\_k1\_yellow}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \mbox{one\_sorted\_str}(a) )\qquad\mbox{fof}(\mbox{dt\_l1\_orders}_2,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_l1\_struct}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_u1\_struct}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{rel\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_orders}_2,\mbox{axiom})$

$\exists a{:}\;\mbox{one\_sorted\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_struct}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{antisymmetric\_relstr}(a)\mbox{ and }\mbox{rel\_str}(a) ) \ \Rightarrow \ \forall b{:}\;( \mbox{ex\_sup\_of\_relstr\_set}(a,b)\iff \exists c{:}\;( \mbox{element}(c,\mbox{the\_carrier}(a))\mbox{ and }\mbox{relstr\_set\_smaller}(a,b,c)\mbox{ and }\forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_set\_smaller}(a,b,d) \ \Rightarrow \ \mbox{related}(a,c,d) ) ) ) ) )\qquad\mbox{fof}(\mbox{t15\_yellow}_0,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{antisymmetric\_relstr}(a)\mbox{ and }\mbox{rel\_str}(a) ) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{the\_carrier}(a)) \ \Rightarrow \ \forall c{:}\;( ( ( b = \mbox{join\_on\_relstr}(a,c)\mbox{ and }\mbox{ex\_sup\_of\_relstr\_set}(a,c) ) \ \Rightarrow \ ( \mbox{relstr\_set\_smaller}(a,c,b)\mbox{ and }\forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_set\_smaller}(a,c,d) \ \Rightarrow \ \mbox{related}(a,b,d) ) ) ) )\mbox{ and }( ( \mbox{relstr\_set\_smaller}(a,c,b)\mbox{ and }\forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_set\_smaller}(a,c,d) \ \Rightarrow \ \mbox{related}(a,b,d) ) ) ) \ \Rightarrow \ ( b = \mbox{join\_on\_relstr}(a,c)\mbox{ and }\mbox{ex\_sup\_of\_relstr\_set}(a,c) ) ) ) ) )\qquad\mbox{fof}(\mbox{t30\_yellow}_0,\mbox{conjecture})$

\medskip\textbf{SEU474$\wedge$1.p} Swapping symmetric closure and reflexive closure

Taking the symmetric closure of the reflexive closure is the same
 as taking the reflexive closure of the symmetric closure




$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{sc}@ ( \mbox{rc}@ r ) ) = ( \mbox{rc}@ ( \mbox{sc}@ r ) )\qquad\mbox{thf}(\mbox{composing\_symmetric\_closure\_and\_reflexive\_closure},\mbox{conjecture})$

\medskip\textbf{SEU467$\wedge$1.p} The transitive closure operator is inflationary






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{infl}@ \mbox{tc}\qquad\mbox{thf}(\mbox{transitive\_closure\_op\_is\_inflationary},\mbox{conjecture})$

\medskip\textbf{SEU470$\wedge$1.p} Transitive reflexive symmetric closure properties

The transitive reflexive symmetric closure of a binary relation
 is transitive, reflexive, and symmetric.




$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{trans}@ ( \mbox{trsc}@ r )\mbox{ and }\mbox{refl}@ ( \mbox{trsc}@ r )\mbox{ and }\mbox{symm}@ ( \mbox{trsc}@ r ) )\qquad\mbox{thf}(\mbox{transitive\_reflexive\_symmetric\_closure\_is\_transitive\_reflexive\_symmetric},\mbox{conjecture})$

\medskip\textbf{SEU355+1.p} MPTP bushy problem t6\_yellow\_0






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{finite}(a) )\qquad\mbox{fof}(\mbox{cc1\_finset}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \forall b,c{:}\;( \mbox{element}(c,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_element\_smaller}(a,b,c)\iff \forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{in}(d,b) \ \Rightarrow \ \mbox{related}(a,c,d) ) ) ) ) )\qquad\mbox{fof}(\mbox{d8\_lattice}_3,\mbox{axiom})$

$\forall a{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \forall b,c{:}\;( \mbox{element}(c,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_set\_smaller}(a,b,c)\iff \forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{in}(d,b) \ \Rightarrow \ \mbox{related}(a,d,c) ) ) ) ) )\qquad\mbox{fof}(\mbox{d9\_lattice}_3,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \mbox{one\_sorted\_str}(a) )\qquad\mbox{fof}(\mbox{dt\_l1\_orders}_2,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_l1\_struct}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_u1\_struct}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{rel\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_orders}_2,\mbox{axiom})$

$\exists a{:}\;\mbox{one\_sorted\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_struct}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{finite}(a) )\qquad\mbox{fof}(\mbox{rc1\_finset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_set\_smaller}(a,\mbox{empty\_set},b)\mbox{ and }\mbox{relstr\_element\_smaller}(a,\mbox{empty\_set},b) ) ) )\qquad\mbox{fof}(\mbox{t6\_yellow}_0,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU648$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.! z:i.! u:i.kpair x y = kpair z u $\to$ x = z)




$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{setukpairinjL}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjL2\_type},\mbox{type})$

$\mbox{setukpairinjL}_2 = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xu}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) )\qquad\mbox{thf}(\mbox{setukpairinjL}_2,\mbox{definition})$

$\mbox{setukpairinjL}_2 \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} )\qquad\mbox{thf}(\mbox{setukpairinjL},\mbox{conjecture})$

\medskip\textbf{SEU155+1.p} MPTP bushy problem l50\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ a\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{l50\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

\medskip\textbf{SEU897$\wedge$5.p} TPS problem THM30






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cS}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\mbox{cR}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cR}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xx} ) )\iff \forall f{:}\;a\to a,\mbox{xx}{:}\;a{:}\;( \exists \mbox{xt}{:}\;a{:}\;( \mbox{cR}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( f@ \mbox{xt} ) ) \ \Rightarrow \ \exists \mbox{xt}{:}\;a{:}\;( \mbox{cS}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( f@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cTHM30\_pme},\mbox{conjecture})$

\medskip\textbf{SEU145+1.p} MPTP bushy problem l3\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ ( \mbox{in}(c,a)\mbox{ or }a\subseteq \mbox{set\_difference}(b,\mbox{singleton}(c)) ) )\qquad\mbox{fof}(\mbox{l3\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

\medskip\textbf{SEU495$\wedge$1.p} The inverse of a total relation is again a total relation






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{total}@ r ) \ \Rightarrow \ ( \mbox{total}@ ( \mbox{inv}@ r ) ) )\qquad\mbox{thf}(\mbox{inverse\_of\_total\_relation\_is\_total\_relation},\mbox{conjecture})$

\medskip\textbf{SEU794$\wedge$2.p} More about Functions - Images of Functions

(! A:i.! f:i$>$i.exu ($\wedge$ B:i.! x:i.in x B $<$$\to$
 (? y:i.in y A \& x = f y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{setextsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextsub\_type},\mbox{type})$

$\mbox{setextsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ a ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setextsub},\mbox{definition})$

$\mbox{image1Ex}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{image1Ex\_type},\mbox{type})$

$\mbox{image1Ex} = ( \forall a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b )\iff \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xx} = ( \mbox{xf}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{image1Ex},\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{setextsub} \ \Rightarrow \ ( \mbox{image1Ex} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{exu}@ \lambda b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b )\iff \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xx} = ( \mbox{xf}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{image1Ex}_1,\mbox{conjecture})$

\medskip\textbf{SEU953$\wedge$5.p} TPS problem from FUNCTION-THMS






$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx} ) = ( g@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ ( g@ \mbox{xx} ) ) ) = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx} )\qquad\mbox{thf}(\mbox{cTHM591\_pme},\mbox{conjecture})$

\medskip\textbf{SEU739$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ ~(in x (binunion X Y)) $\to$ ~(in x Y))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIR\_type},\mbox{type})$

$\mbox{binunionIR} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIR},\mbox{definition})$

$\mbox{binunionIR} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ y ) ) ) )\qquad\mbox{thf}(\mbox{binunionTIRcontra},\mbox{conjecture})$

\medskip\textbf{SEU569$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.subset A A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{subsetI}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( \subseteq @ a@ a )\qquad\mbox{thf}(\mbox{subsetRefl},\mbox{conjecture})$

\medskip\textbf{SEU703$\wedge$2.p} Conditionals

(! A:i.! phi:o.! x:i.in x A $\to$ (! y:i.in y A $\to$ phi $\to$
 dsetconstr A ($\wedge$ z:i.phi \& z = x | ~phi \& z = y) =
 setadjoin x emptyset))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{eqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{eqinunit\_type},\mbox{type})$

$\mbox{eqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{eqinunit},\mbox{definition})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\mbox{iftrueProp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{iftrueProp1\_type},\mbox{type})$

$\mbox{iftrueProp}_1 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{xphi} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iftrueProp}_1,\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{setext} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ ( \mbox{eqinunit} \ \Rightarrow \ ( \mbox{in\_\_Cong} \ \Rightarrow \ ( \mbox{iftrueProp}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{xphi} \ \Rightarrow \ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iftrueProp}_2,\mbox{conjecture})$

\medskip\textbf{SEU985$\wedge$5.p} TPS problem from FINITE-SETS-RELNS-THMS






$\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xw}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xp} ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xw}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xq} ) ) \ \Rightarrow \ \forall \mbox{xw}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xs}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \mbox{xs} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xs}@ \mbox{xu}@ \mbox{xv}\mbox{ or }( \mbox{xu} = \mbox{xx}\mbox{ and }\mbox{xv} = \mbox{xy} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx}\mbox{ and }\mbox{xq}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM162A\_pme},\mbox{conjecture})$

\medskip\textbf{SEU888$\wedge$5.p} TPS problem THM500C-WFF






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$g{:}\;b\to a\qquad\mbox{thf}(g,\mbox{type})$

$z{:}\;a\qquad\mbox{thf}(z,\mbox{type})$

$y{:}\;b\qquad\mbox{thf}(y,\mbox{type})$

$x{:}\;b\qquad\mbox{thf}(x,\mbox{type})$

$( z = ( g@ x )\mbox{ or }z = ( g@ y ) ) \ \Rightarrow \ \exists \mbox{xt}{:}\;b{:}\;( ( \mbox{xt} = x\mbox{ or }\mbox{xt} = y )\mbox{ and }z = ( g@ \mbox{xt} ) )\qquad\mbox{thf}(\mbox{cTHM500C\_WFF\_pme},\mbox{conjecture})$

\medskip\textbf{SEU159+1.p} MPTP bushy problem t38\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{unordered\_pair}(a,b)\subseteq c\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,c) ) )\qquad\mbox{fof}(\mbox{t38\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU675$\wedge$2.p} Functions - Application

(! A:i.! B:i.! f:i.in f (funcSet A B) $\to$ func A B f)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{funcSet} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{powerset}@ ( \mbox{cartprod}@ a@ b ) )@ \lambda \mbox{xf}{:}\;\mbox{\$i}{:}\;( \mbox{func}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{funcSet},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ ( \mbox{func}@ a@ b@ \mbox{xf} ) )\qquad\mbox{thf}(\mbox{infuncsetfunc},\mbox{conjecture})$

\medskip\textbf{SEU566$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.! x:i.in x A $\to$ ~(in x B) $\to$ ~(subset A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ \neg\,\subseteq @ a@ b ) )\qquad\mbox{thf}(\mbox{notsubsetI},\mbox{conjecture})$

\medskip\textbf{SEU822$\wedge$2.p} Ordinals

(! X:i.ordinal X $\to$ ~(in X X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{stricttotalorderedByIn\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\mbox{in}@ x@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{in}@ x@ y\mbox{ or }\mbox{in}@ y@ x ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \neg\,\mbox{in}@ x@ x ) ) )\qquad\mbox{thf}(\mbox{stricttotalorderedByIn},\mbox{definition})$

$\mbox{wellorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{wellorderedByIn\_type},\mbox{type})$

$\mbox{wellorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{stricttotalorderedByIn}@ a\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{nonempty}@ x ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ x ) \ \Rightarrow \ ( \mbox{xx} = y\mbox{ or }\mbox{in}@ \mbox{xx}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{wellorderedByIn},\mbox{definition})$

$\mbox{ordinal}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ordinal\_type},\mbox{type})$

$\mbox{ordinal} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{transitiveset}@ \mbox{xx}\mbox{ and }\mbox{wellorderedByIn}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{ordinal},\mbox{definition})$

$\mbox{ordinalIrrefl}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ordinalIrrefl\_type},\mbox{type})$

$\mbox{ordinalIrrefl} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ \neg\,\mbox{in}@ a@ a ) ) )\qquad\mbox{thf}(\mbox{ordinalIrrefl},\mbox{definition})$

$\mbox{ordinalIrrefl} \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \neg\,\mbox{in}@ x@ x )\qquad\mbox{thf}(\mbox{ordinalIrrefl}_2,\mbox{conjecture})$

\medskip\textbf{SEU841$\wedge$5.p} TPS problem GAZING-THM7






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o},u{:}\;a\to \mbox{\$o}{:}\;( ( s = t\mbox{ and }t = u ) \ \Rightarrow \ s = u )\qquad\mbox{thf}(\mbox{cGAZING\_THM}_7,\mbox{conjecture})$

\medskip\textbf{SEU972$\wedge$5.p} TPS problem from COINDUCTIVE-PU-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xt} = \mbox{cZ}\iff \mbox{xu} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ \mbox{xt} = \mbox{xu} ) ) ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;\exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ \mbox{cZ}@ \mbox{xt} )\mbox{ and }\forall \mbox{xt}_0{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}_0@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xu} = \mbox{cZ} \ \Rightarrow \ \mbox{xt}_0 = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt}_0 )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt}_0 )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_LEM2E\_pme},\mbox{conjecture})$

\medskip\textbf{SEU658$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! A:i.! B:i.! u:i.in u (cartprod A B) $\to$ in (ksnd u) B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprodmempair}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodmempair1\_type},\mbox{type})$

$\mbox{cartprodmempair}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprodmempair}_1,\mbox{definition})$

$\mbox{ksnd}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ksnd\_type},\mbox{type})$

$\mbox{ksndpairEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ksndpairEq\_type},\mbox{type})$

$\mbox{ksndpairEq} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{ksnd}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{ksndpairEq},\mbox{definition})$

$\mbox{cartprodmempair}_1 \ \Rightarrow \ ( \mbox{ksndpairEq} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{ksnd}@ \mbox{xu} )@ b ) ) )\qquad\mbox{thf}(\mbox{cartprodsndin},\mbox{conjecture})$

\medskip\textbf{SEU858$\wedge$5.p} TPS problem THM163

A direct consequence of the definition of FINITE1.




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) )\qquad\mbox{thf}(\mbox{cTHM163\_pme},\mbox{conjecture})$

\medskip\textbf{SEU544$\wedge$2.p} Preliminary Notions - Equivalence Laws

(! phi:i$>$o.(? x:i.phi x \& (! y:i.phi y $\to$ x = y)) $\to$
 exu ($\wedge$ x:i.phi x))




$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu},\mbox{definition})$

$\forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{exuI}_1,\mbox{conjecture})$

\medskip\textbf{SEU681$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! R:i.breln A B R $\to$ (! phi:i$>$o.(! x:i.in x A $\to$
 (! y:i.in y B $\to$ in (kpair x y) R $\to$ phi (kpair x y))) $\to$
 (! x:i.in x R $\to$ phi x)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprodmempair}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodmempair1\_type},\mbox{type})$

$\mbox{cartprodmempair}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprodmempair}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ ( \mbox{cartprodmempair}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ r ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{xphi}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ r ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{brelnall}_1,\mbox{conjecture})$

\medskip\textbf{SEU521$\wedge$2.p} Preliminary Notions - Power Sets and Unions

(! A:i.! B:i.! x:i.in B (powerset A) $\to$ in x B $\to$ in x A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetAx\_type},\mbox{type})$

$\mbox{powersetAx} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) )\iff \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetAx},\mbox{definition})$

$\mbox{powersetAx} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{powersetE},\mbox{conjecture})$

\medskip\textbf{SEU820$\wedge$2.p} Ordinals

ordinal emptyset




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{nonempty} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\ne \mbox{emptyset} )\qquad\mbox{thf}(\mbox{nonempty},\mbox{definition})$

$\mbox{vacuousDall}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{vacuousDall\_type},\mbox{type})$

$\mbox{vacuousDall} = ( \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{vacuousDall},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetemptysetimpeq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetemptysetimpeq\_type},\mbox{type})$

$\mbox{subsetemptysetimpeq} = ( \forall a{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ \mbox{emptyset} ) \ \Rightarrow \ a = \mbox{emptyset} ) )\qquad\mbox{thf}(\mbox{subsetemptysetimpeq},\mbox{definition})$

$\mbox{powersetE}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE1\_type},\mbox{type})$

$\mbox{powersetE}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \subseteq @ b@ a ) ) )\qquad\mbox{thf}(\mbox{powersetE}_1,\mbox{definition})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{transitiveset} = ( \lambda a{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ ( \subseteq @ x@ a ) ) )\qquad\mbox{thf}(\mbox{transitiveset},\mbox{definition})$

$\mbox{stricttotalorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{stricttotalorderedByIn\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\mbox{in}@ x@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{in}@ x@ y\mbox{ or }\mbox{in}@ y@ x ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \neg\,\mbox{in}@ x@ x ) ) )\qquad\mbox{thf}(\mbox{stricttotalorderedByIn},\mbox{definition})$

$\mbox{wellorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{wellorderedByIn\_type},\mbox{type})$

$\mbox{wellorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{stricttotalorderedByIn}@ a\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{nonempty}@ x ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ x ) \ \Rightarrow \ ( \mbox{xx} = y\mbox{ or }\mbox{in}@ \mbox{xx}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{wellorderedByIn},\mbox{definition})$

$\mbox{ordinal}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ordinal\_type},\mbox{type})$

$\mbox{ordinal} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{transitiveset}@ \mbox{xx}\mbox{ and }\mbox{wellorderedByIn}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{ordinal},\mbox{definition})$

$\mbox{vacuousDall} \ \Rightarrow \ ( \mbox{subsetemptysetimpeq} \ \Rightarrow \ ( \mbox{powersetE}_1 \ \Rightarrow \ ( \mbox{ordinal}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{emptysetOrdinal},\mbox{conjecture})$

\medskip\textbf{SEU609$\wedge$2.p} Preliminary Notions - Operations on Sets - Set Difference

(! A:i.! B:i.subset (setminus A B) A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusEL\_type},\mbox{type})$

$\mbox{setminusEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{setminusEL},\mbox{definition})$

$\mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{setminusEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \subseteq @ ( \mbox{setminus}@ a@ b )@ a ) )\qquad\mbox{thf}(\mbox{setminusLsub},\mbox{conjecture})$

\medskip\textbf{SEU667$\wedge$2.p} Ordered Pairs - Sets of Pairs

(! A:i.! B:i.! phi:i$>$(i$>$o).breln A B (dpsetconstr A B
 ($\wedge$ x,y:i.phi x y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstr} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{dsetconstr}@ ( \mbox{cartprod}@ a@ b )@ \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xphi}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstr},\mbox{definition})$

$\mbox{dpsetconstrSub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dpsetconstrSub\_type},\mbox{type})$

$\mbox{dpsetconstrSub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \subseteq @ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) )@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrSub},\mbox{definition})$

$\mbox{dpsetconstrSub} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{breln}@ a@ b@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setOfPairsIsBReln},\mbox{conjecture})$

\medskip\textbf{SEU685$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i.in f (funcSet A B) $\to$ (! x:i.in x A $\to$
 in (kpair x (ap A B f x)) f))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{infuncsetfunc}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{infuncsetfunc\_type},\mbox{type})$

$\mbox{infuncsetfunc} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ ( \mbox{func}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{infuncsetfunc},\mbox{definition})$

$\mbox{funcGraphProp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcGraphProp1\_type},\mbox{type})$

$\mbox{funcGraphProp}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) )@ \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{funcGraphProp}_1,\mbox{definition})$

$\mbox{infuncsetfunc} \ \Rightarrow \ ( \mbox{funcGraphProp}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) )@ \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{funcGraphProp}_3,\mbox{conjecture})$

\medskip\textbf{SEU281+1.p} MPTP bushy problem s1\_xboole\_0\_\_e4\_5\_1\_\_funct\_1






$\forall a,b{:}\;\exists c{:}\;\forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,\mbox{cartesian\_product}_2(a,b))\mbox{ and }\exists e,f{:}\;( \mbox{ordered\_pair}(e,f) = d\mbox{ and }\mbox{in}(e,a)\mbox{ and }f = \mbox{singleton}(e) ) ) )\qquad\mbox{fof}(\mbox{s1\_xboole\_0\_\_e16\_22\_\_wellord2\_}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) )\qquad\mbox{fof}(\mbox{cc1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) \ \Rightarrow \ \mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{cc2\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc1\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc3\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) ) )\qquad\mbox{fof}(\mbox{cc3\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \forall c,d,e{:}\;( ( c = d\mbox{ and }\exists f,g{:}\;( \mbox{ordered\_pair}(f,g) = d\mbox{ and }\mbox{in}(f,a)\mbox{ and }g = \mbox{singleton}(f) )\mbox{ and }c = e\mbox{ and }\exists h,i{:}\;( \mbox{ordered\_pair}(h,i) = e\mbox{ and }\mbox{in}(h,a)\mbox{ and }i = \mbox{singleton}(h) ) ) \ \Rightarrow \ d = e ) \ \Rightarrow \ \exists c{:}\;\forall d{:}\;( \mbox{in}(d,c)\iff \exists e{:}\;( \mbox{in}(e,\mbox{cartesian\_product}_2(a,b))\mbox{ and }e = d\mbox{ and }\exists j,k{:}\;( \mbox{ordered\_pair}(j,k) = d\mbox{ and }\mbox{in}(j,a)\mbox{ and }k = \mbox{singleton}(j) ) ) ) )\qquad\mbox{fof}(\mbox{s1\_tarski\_\_e16\_22\_\_wellord2\_}_2,\mbox{axiom})$

\medskip\textbf{SEU456$\wedge$1.p} The reflexive closure operator is monotonic






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{mono}@ \mbox{rc}\qquad\mbox{thf}(\mbox{reflexive\_closure\_op\_is\_monotonic},\mbox{conjecture})$

\medskip\textbf{SEU650$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.x = y $\to$ setadjoin (setadjoin x emptyset) (setadjoin
 (setadjoin x (setadjoin y emptyset)) emptyset) = setadjoin
 (setadjoin x emptyset) emptyset)




$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setukpairinjR}_{11}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjR11\_type},\mbox{type})$

$\mbox{setukpairinjR}_{11} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{setukpairinjR}_{11},\mbox{definition})$

$\mbox{setukpairinjR}_{11} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ \mbox{emptyset} ) )\qquad\mbox{thf}(\mbox{setukpairinjR}_{12},\mbox{conjecture})$

\medskip\textbf{SEU813$\wedge$2.p} Transitive Sets

(! X:i.transitiveset X $\to$ (! Y:i.transitiveset Y $\to$
 transitiveset (binintersect X Y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectSubset}_5{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectSubset5\_type},\mbox{type})$

$\mbox{binintersectSubset}_5 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( ( \subseteq @ c@ a ) \ \Rightarrow \ ( ( \subseteq @ c@ b ) \ \Rightarrow \ ( \subseteq @ c@ ( \mbox{binintersect}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectSubset}_5,\mbox{definition})$

$\mbox{binintersectEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectEL\_type},\mbox{type})$

$\mbox{binintersectEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{definition})$

$\mbox{binintersectER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectER\_type},\mbox{type})$

$\mbox{binintersectER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersectER},\mbox{definition})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{transitiveset} = ( \lambda a{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ ( \subseteq @ x@ a ) ) )\qquad\mbox{thf}(\mbox{transitiveset},\mbox{definition})$

$\mbox{transitivesetOp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{transitivesetOp1\_type},\mbox{type})$

$\mbox{transitivesetOp}_1 = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{transitiveset}@ x ) \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ ( \subseteq @ a@ x ) ) ) )\qquad\mbox{thf}(\mbox{transitivesetOp}_1,\mbox{definition})$

$\mbox{binintersectSubset}_5 \ \Rightarrow \ ( \mbox{binintersectEL} \ \Rightarrow \ ( \mbox{binintersectER} \ \Rightarrow \ ( \mbox{transitivesetOp}_1 \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{transitiveset}@ x ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{transitiveset}@ y ) \ \Rightarrow \ ( \mbox{transitiveset}@ ( \mbox{binintersect}@ x@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{binintTransitive},\mbox{conjecture})$

\medskip\textbf{SEU995$\wedge$5.p} TPS problem THM24






$\mbox{cV}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cV},\mbox{type})$

$\mbox{cU}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{cU},\mbox{type})$

$\mbox{cU}\ne \mbox{cV} \ \Rightarrow \ \exists g{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}{:}\;\neg\,\forall \mbox{xx}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( g@ \mbox{xx}@ \mbox{xy} ) = ( g@ \mbox{xy}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cTHM24\_pme},\mbox{conjecture})$

\medskip\textbf{SEU661$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! A:i.! B:i.! x:i.in x A $\to$ (! y:i.in y B $\to$
 kpair x y = kpair x y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair},\mbox{type})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cartprodmempaircEq},\mbox{conjecture})$

\medskip\textbf{SEU147+1.p} MPTP bushy problem t1\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{powerset}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c\subseteq a ) )\qquad\mbox{fof}(\mbox{d1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\mbox{powerset}(\mbox{empty\_set}) = \mbox{singleton}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{t1\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t3\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SEU508$\wedge$2.p} Preliminary Notions - Basic Laws of Logic

(! A:i.! phi:i$>$o.! x:i.in x A $\to$ (in x (dsetconstr A
 ($\wedge$ y:i.phi y)) $<$$\to$ phi x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{dsetconstrER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) )\iff ( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{setbeta},\mbox{conjecture})$

\medskip\textbf{SEU761$\wedge$2.p} Typed Set Theory - First Wizard of Oz Examples - WoZ1 Lemmas

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! Z:i.in Z (powerset A) $\to$ subset X Y $\to$ subset X Z $\to$
 subset X (binintersect Y Z))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectI\_type},\mbox{type})$

$\mbox{binintersectI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectI},\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ ( \mbox{binintersectI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ x@ y ) \ \Rightarrow \ ( ( \subseteq @ x@ z ) \ \Rightarrow \ ( \subseteq @ x@ ( \mbox{binintersect}@ y@ z ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz13rule}_3,\mbox{conjecture})$

\medskip\textbf{SEU466$\wedge$1.p} The transitive closure operator is idempotent






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{idem}@ \mbox{tc}\qquad\mbox{thf}(\mbox{transitive\_closure\_op\_is\_idempotent},\mbox{conjecture})$

\medskip\textbf{SEU727$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! x:i.in x A $\to$ in x X $\to$
 in x (setminus A (setminus A X))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{setminusI} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{setminus}@ a@ x ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{doubleComplementI}_1,\mbox{conjecture})$

\medskip\textbf{SEU670$\wedge$2.p} Ordered Pairs - Sets of Pairs

(! A:i.! B:i.! phi:i$>$(i$>$o).! x:i.! y:i.in (kpair x y)
 (dpsetconstr A B ($\wedge$ z,u:i.phi z u)) $\to$ in y B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprodpairmemER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodpairmemER\_type},\mbox{type})$

$\mbox{cartprodpairmemER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ b ) ) )\qquad\mbox{thf}(\mbox{cartprodpairmemER},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstr} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{dsetconstr}@ ( \mbox{cartprod}@ a@ b )@ \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xphi}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstr},\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{cartprodpairmemER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xz}@ \mbox{xu} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ b ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrEL}_2,\mbox{conjecture})$

\medskip\textbf{SEU182+1.p} MPTP bushy problem t44\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \forall c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_composition}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff \exists f{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,f),a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(f,e),b) ) ) ) ) ) )\qquad\mbox{fof}(\mbox{d8\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_composition}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k5\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_dom}(\mbox{relation\_composition}(a,b))\subseteq \mbox{relation\_dom}(a) ) )\qquad\mbox{fof}(\mbox{t44\_relat}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU924$\wedge$5.p} TPS problem THM134

Every positive iterate of a constant function is a constant 
 function.




$\forall \mbox{xz}{:}\;\mbox{\$i},\mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xz}\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xz} ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xg}@ \mbox{xx} ) = \mbox{xz} )\qquad\mbox{thf}(\mbox{cTHM134\_pme},\mbox{conjecture})$

\medskip\textbf{SEU560$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.A = B $\to$ subset B A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\subseteq  = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{subset},\mbox{definition})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ ( \subseteq @ b@ a ) )\qquad\mbox{thf}(\mbox{eqimpsubset}_2,\mbox{conjecture})$

\medskip\textbf{SEU997$\wedge$5.p} TPS problem CD-LATTICE-THM

A complemented distributive lattice has unique complements.




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{jOIN}{:}\;a\to a\to a,\mbox{mEET}{:}\;a\to a\to a,\mbox{tOP}{:}\;a,\mbox{bOTTOM}{:}\;a{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{jOIN}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{jOIN}@ \mbox{xx}@ ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{mEET}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{mEET}@ \mbox{xx}@ ( \mbox{mEET}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{mEET}@ \mbox{xy}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{jOIN}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{xy}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{mEET}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{xy}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xz} ) ) = ( \mbox{jOIN}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ ( \mbox{mEET}@ \mbox{xy}@ \mbox{xz} ) ) = ( \mbox{mEET}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{mEET}@ \mbox{tOP}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{tOP}@ \mbox{xx} ) = \mbox{tOP}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{mEET}@ \mbox{bOTTOM}@ \mbox{xx} ) = \mbox{bOTTOM}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{bOTTOM}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;\exists \mbox{xy}{:}\;a{:}\;( ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} ) = \mbox{tOP}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} ) = \mbox{bOTTOM} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( ( ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} ) = \mbox{tOP}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} ) = \mbox{bOTTOM}\mbox{ and }( \mbox{jOIN}@ \mbox{xx}@ \mbox{xz} ) = \mbox{tOP}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xz} ) = \mbox{bOTTOM} ) \ \Rightarrow \ \mbox{xy} = \mbox{xz} ) )\qquad\mbox{thf}(\mbox{cCD\_LATTICE\_THM\_pme},\mbox{conjecture})$

\medskip\textbf{SEU778$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$
 breln1 A (breln1compset A R S)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{setOfPairsIsBReln}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setOfPairsIsBReln1\_type},\mbox{type})$

$\mbox{setOfPairsIsBReln}_1 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{breln}_1@ a@ ( \mbox{dpsetconstr}@ a@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{setOfPairsIsBReln}_1,\mbox{definition})$

$\mbox{breln1compset}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1compset\_type},\mbox{type})$

$\mbox{breln1compset} = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i},s{:}\;\mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xz}@ a\mbox{ and }\mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xz} )@ r\mbox{ and }\mbox{in}@ ( \mbox{kpair}@ \mbox{xz}@ \mbox{xy} )@ s ) ) )\qquad\mbox{thf}(\mbox{breln1compset},\mbox{definition})$

$\mbox{setOfPairsIsBReln}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \mbox{breln}_1@ a@ ( \mbox{breln1compset}@ a@ r@ s ) ) ) )\qquad\mbox{thf}(\mbox{breln1compprop},\mbox{conjecture})$

\medskip\textbf{SEU848$\wedge$5.p} TPS problem GAZING-THM38






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o}{:}\;( ( \lambda \mbox{xz}{:}\;a{:}\;( s@ \mbox{xz}\mbox{ or }t@ \mbox{xz} ) ) = s\iff \forall \mbox{xx}{:}\;a{:}\;( ( t@ \mbox{xx} ) \ \Rightarrow \ ( s@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM38\_pme},\mbox{conjecture})$

\medskip\textbf{SEU150+3.p} Basic properties of sets, theorem






$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t8\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{t9\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU999$\wedge$5.p} TPS problem PENTAGON-THM2B






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{jOIN}{:}\;a\to a\to a,\mbox{mEET}{:}\;a\to a\to a{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{jOIN}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{jOIN}@ \mbox{xx}@ ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{mEET}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{mEET}@ \mbox{xx}@ ( \mbox{mEET}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{mEET}@ \mbox{xy}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{jOIN}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{xy}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{mEET}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{xy} ) \ \Rightarrow \ ( \exists \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xa}{:}\;a,\mbox{xb}{:}\;a,\mbox{xc}{:}\;a{:}\;( \mbox{xa}\ne \mbox{xb}\mbox{ and }\mbox{xa}\ne \mbox{xc}\mbox{ and }\mbox{xa}\ne \mbox{xx}\mbox{ and }\mbox{xa}\ne \mbox{xy}\mbox{ and }\mbox{xb}\ne \mbox{xc}\mbox{ and }\mbox{xb}\ne \mbox{xx}\mbox{ and }\mbox{xb}\ne \mbox{xy}\mbox{ and }\mbox{xc}\ne \mbox{xx}\mbox{ and }\mbox{xc}\ne \mbox{xy}\mbox{ and }\mbox{xx}\ne \mbox{xy}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xa} ) = \mbox{xa}\mbox{ and }( \mbox{jOIN}@ \mbox{xx}@ \mbox{xa} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xb} ) = \mbox{xb}\mbox{ and }( \mbox{jOIN}@ \mbox{xx}@ \mbox{xb} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xx}@ \mbox{xc} ) = \mbox{xc}\mbox{ and }( \mbox{jOIN}@ \mbox{xx}@ \mbox{xc} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xa}@ \mbox{xb} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xa}@ \mbox{xc} ) = \mbox{xa}\mbox{ and }( \mbox{jOIN}@ \mbox{xa}@ \mbox{xc} ) = \mbox{xc}\mbox{ and }( \mbox{mEET}@ \mbox{xa}@ \mbox{xy} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xa}@ \mbox{xy} ) = \mbox{xa}\mbox{ and }( \mbox{mEET}@ \mbox{xb}@ \mbox{xc} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xb}@ \mbox{xc} ) = \mbox{xx}\mbox{ and }( \mbox{mEET}@ \mbox{xb}@ \mbox{xy} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xb}@ \mbox{xy} ) = \mbox{xb}\mbox{ and }( \mbox{mEET}@ \mbox{xc}@ \mbox{xy} ) = \mbox{xy}\mbox{ and }( \mbox{jOIN}@ \mbox{xc}@ \mbox{xy} ) = \mbox{xc} ) \ \Rightarrow \ \neg\,\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ ( \mbox{mEET}@ \mbox{xy}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xz} ) ) ) = ( \mbox{mEET}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xz} ) ) ) )\qquad\mbox{thf}(\mbox{cPENTAGON\_THM2B\_pme},\mbox{conjecture})$

\medskip\textbf{SEU931$\wedge$5.p} TPS problem THM171

If g commutes with f, and g has a unique fixed point, then f has
 a fixed point.




$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( f@ ( g@ \mbox{xx} ) ) ) = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( g@ ( f@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xz} ) = \mbox{xz} \ \Rightarrow \ \mbox{xz} = \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( f@ \mbox{xy} ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM171\_pme},\mbox{conjecture})$

\medskip\textbf{SEU124+2.p} MPTP chainy problem t7\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{lemma})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{t2\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t3\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\neg\,\exists c{:}\;\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t4\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;a\subseteq \mbox{set\_union}_2(a,b)\qquad\mbox{fof}(\mbox{t7\_xboole}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU255+1.p} MPTP bushy problem t25\_wellord1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;\mbox{relation\_restriction}(a,b) = \mbox{set\_intersection}_2(a,\mbox{cartesian\_product}_2(b,b)) )\qquad\mbox{fof}(\mbox{d6\_wellord}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k2\_wellord}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{antisymmetric}(a)\iff \forall b,c{:}\;( ( \mbox{in}(\mbox{ordered\_pair}(b,c),a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(c,b),a) ) \ \Rightarrow \ b = c ) ) )\qquad\mbox{fof}(\mbox{l3\_wellord}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_restriction}(c,b))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(a,\mbox{cartesian\_product}_2(b,b)) ) ) )\qquad\mbox{fof}(\mbox{t16\_wellord}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( \mbox{antisymmetric}(b) \ \Rightarrow \ \mbox{antisymmetric}(\mbox{relation\_restriction}(b,a)) ) )\qquad\mbox{fof}(\mbox{t25\_wellord}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU783$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ in (kpair x y) R $\to$
 in (kpair x y) (binunion R S)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIL\_type},\mbox{type})$

$\mbox{binunionIL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIL},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{binunionIL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionIL},\mbox{conjecture})$

\medskip\textbf{SEU834$\wedge$5.p} TPS problem GAZING-THM21






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx}\mbox{ and }t@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM21\_pme},\mbox{conjecture})$

\medskip\textbf{SEU148+3.p} Basic properties of sets, theorem 6






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{singleton}(a)\ne \mbox{empty\_set}\qquad\mbox{fof}(\mbox{l1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq \mbox{singleton}(b)\iff ( a = \mbox{empty\_set}\mbox{ or }a = \mbox{singleton}(b) ) )\qquad\mbox{fof}(\mbox{l4\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq \mbox{singleton}(b) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t6\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU536$\wedge$2.p} Preliminary Notions - Bounded Quantifier Laws

(! A:i.! phi:i$>$o.~(! x:i.in x A $\to$ phi x) $\to$
 (? x:i.in x A \& ~(phi x)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in},\mbox{type})$

$\forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\neg\,\mbox{xphi}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{quantDeMorgan}_1,\mbox{conjecture})$

\medskip\textbf{SEU497$\wedge$1.p} Every strict (partial) order induces a partial order






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{so}@ r ) \ \Rightarrow \ ( \mbox{po}@ ( \mbox{rc}@ r ) ) )\qquad\mbox{thf}(\mbox{strict\_order\_induces\_partial\_order},\mbox{conjecture})$

\medskip\textbf{SEU722$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ ~(in x Y) $\to$ ~(in x (binintersect X Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectER\_type},\mbox{type})$

$\mbox{binintersectER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersectER},\mbox{definition})$

$\mbox{binintersectER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ y \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectTERcontra},\mbox{conjecture})$

\medskip\textbf{SEU139+1.p} MPTP bushy problem t60\_xboole\_1






$\forall a,b{:}\;( \mbox{proper\_subset}(a,b) \ \Rightarrow \ \neg\,\mbox{proper\_subset}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{proper\_subset}(a,b)\iff ( a\subseteq b\mbox{ and }a\ne b ) )\qquad\mbox{fof}(\mbox{d8\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{proper\_subset}(a,a)\qquad\mbox{fof}(\mbox{irreflexivity\_r2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,a\subseteq b\mbox{ and }\mbox{proper\_subset}(b,a)\qquad\mbox{fof}(\mbox{t60\_xboole}_1,\mbox{conjecture})$

\medskip\textbf{SEU868$\wedge$5.p} TPS problem from FINITE-SET-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cC}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$\forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{cC} ) ) \ \Rightarrow \ \forall p{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \forall e{:}\;a\to \mbox{\$o}{:}\;( \neg\,\exists \mbox{xt}{:}\;a{:}\;( e@ \mbox{xt} ) \ \Rightarrow \ ( p@ e ) )\mbox{ and }\forall y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a,z{:}\;a\to \mbox{\$o}{:}\;( ( p@ y\mbox{ and }\forall \mbox{xx}_{16}{:}\;a{:}\;( ( z@ \mbox{xx}_{16} )\iff ( y@ \mbox{xx}_{16}\mbox{ or }\mbox{xx}_{16} = \mbox{xx} ) ) ) \ \Rightarrow \ ( p@ z ) ) ) \ \Rightarrow \ ( p@ \mbox{cC} ) )\qquad\mbox{thf}(\mbox{cTHM551\_pme},\mbox{conjecture})$

\medskip\textbf{SEU876$\wedge$5.p} TPS problem from SET-TOPOLOGY-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cE}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cE},\mbox{type})$

$\mbox{cA}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{cE}@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{cE} ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xx}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{cE}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ ( \mbox{cA}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xx}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{cE}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ \exists \mbox{xe}{:}\;a\to \mbox{\$o}{:}\;( \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}_0{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx}_0 ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}_0@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xe} ) )\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xx}_0 ) )\mbox{ and }\forall \mbox{xy}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xy}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xy}@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ ( \mbox{cA}@ \mbox{xy}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{cE}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xy}@ \mbox{xx}_0 ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cDOMLEMMA4\_pme},\mbox{conjecture})$

\medskip\textbf{SEU303+1.p} MPTP bushy problem t26\_finset\_1






$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k9\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{finite}(b) ) \ \Rightarrow \ \mbox{finite}(\mbox{relation\_image}(a,b)) )\qquad\mbox{fof}(\mbox{fc13\_finset}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation\_image}(a,\mbox{relation\_dom}(a)) = \mbox{relation\_rng}(a) )\qquad\mbox{fof}(\mbox{t146\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(b)\mbox{ and }\mbox{function}(b) ) \ \Rightarrow \ ( \mbox{finite}(a) \ \Rightarrow \ \mbox{finite}(\mbox{relation\_image}(b,a)) ) )\qquad\mbox{fof}(\mbox{t17\_finset}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{finite}(\mbox{relation\_dom}(a)) \ \Rightarrow \ \mbox{finite}(\mbox{relation\_rng}(a)) ) )\qquad\mbox{fof}(\mbox{t26\_finset}_1,\mbox{conjecture})$

\medskip\textbf{SEU695$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i$>$i.(! x:i.in x A $\to$ in (f x) B) $\to$
 (! x:i.in x A $\to$ ap A B (lam A B ($\wedge$ y:i.f y)) x = f x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dpsetconstrI\_type},\mbox{type})$

$\mbox{dpsetconstrI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xz}@ \mbox{xu} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrI},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{lam}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{lam\_type},\mbox{type})$

$\mbox{lam} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{lam},\mbox{definition})$

$\mbox{lamp}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{lamp\_type},\mbox{type})$

$\mbox{lamp} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ ( \mbox{func}@ a@ b@ ( \mbox{lam}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{lamp},\mbox{definition})$

$\mbox{funcGraphProp}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcGraphProp2\_type},\mbox{type})$

$\mbox{funcGraphProp}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{funcGraphProp}_2,\mbox{definition})$

$\mbox{dpsetconstrI} \ \Rightarrow \ ( \mbox{lamp} \ \Rightarrow \ ( \mbox{funcGraphProp}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ ( \mbox{lam}@ a@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) )@ \mbox{xx} ) = ( \mbox{xf}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{beta}_1,\mbox{conjecture})$

\medskip\textbf{SEU319+1.p} MPTP bushy problem t29\_tops\_1






$\forall a{:}\;( \mbox{top\_str}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) \ \Rightarrow \ ( \mbox{closed\_subset}(b,a)\iff \mbox{open\_subset}(\mbox{subset\_difference}(\mbox{the\_carrier}(a),\mbox{cast\_as\_carrier\_subset}(a),b),a) ) ) )\qquad\mbox{fof}(\mbox{d6\_pre\_topc},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{one\_sorted\_str}(a) \ \Rightarrow \ \mbox{element}(\mbox{cast\_as\_carrier\_subset}(a),\mbox{powerset}(\mbox{the\_carrier}(a))) )\qquad\mbox{fof}(\mbox{dt\_k2\_pre\_topc},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{element}(\mbox{subset\_complement}(a,b),\mbox{powerset}(a)) )\qquad\mbox{fof}(\mbox{dt\_k3\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{element}(c,\mbox{powerset}(a)) ) \ \Rightarrow \ \mbox{element}(\mbox{subset\_difference}(a,b,c),\mbox{powerset}(a)) )\qquad\mbox{fof}(\mbox{dt\_k6\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{top\_str}(a) \ \Rightarrow \ \mbox{one\_sorted\_str}(a) )\qquad\mbox{fof}(\mbox{dt\_l1\_pre\_topc},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_l1\_struct}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_u1\_struct}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{top\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_pre\_topc},\mbox{axiom})$

$\exists a{:}\;\mbox{one\_sorted\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_struct}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{subset\_complement}(a,\mbox{subset\_complement}(a,b)) = b )\qquad\mbox{fof}(\mbox{involutiveness\_k3\_subset}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{element}(c,\mbox{powerset}(a)) ) \ \Rightarrow \ \mbox{subset\_difference}(a,b,c) = \mbox{set\_difference}(b,c) )\qquad\mbox{fof}(\mbox{redefinition\_k6\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{one\_sorted\_str}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) \ \Rightarrow \ \mbox{subset\_complement}(\mbox{the\_carrier}(a),b) = \mbox{subset\_difference}(\mbox{the\_carrier}(a),\mbox{cast\_as\_carrier\_subset}(a),b) ) )\qquad\mbox{fof}(\mbox{t17\_pre\_topc},\mbox{axiom})$

$\forall a{:}\;( \mbox{top\_str}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{the\_carrier}(a))) \ \Rightarrow \ ( \mbox{closed\_subset}(b,a)\iff \mbox{open\_subset}(\mbox{subset\_complement}(\mbox{the\_carrier}(a),b),a) ) ) )\qquad\mbox{fof}(\mbox{t29\_tops}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

\medskip\textbf{SEU481$\wedge$1.p} Satisfying the induction principle implies termination






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{ind}@ r ) \ \Rightarrow \ ( \mbox{term}@ r ) )\qquad\mbox{thf}(\mbox{induction\_implies\_termination},\mbox{conjecture})$

\medskip\textbf{SEU608$\wedge$2.p} Preliminary Notions - Operations on Sets - Set Difference

(! A:i.! B:i.! x:i.in x B $\to$ ~(in x (setminus A B)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{setminusER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) )\qquad\mbox{thf}(\mbox{setminusIRneg},\mbox{conjecture})$

\medskip\textbf{SEU191+1.p} MPTP bushy problem t74\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( b = \mbox{identity\_relation}(a)\iff \forall c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(c,d),b)\iff ( \mbox{in}(c,a)\mbox{ and }c = d ) ) ) )\qquad\mbox{fof}(\mbox{d10\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \forall c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_composition}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff \exists f{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,f),a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(f,e),b) ) ) ) ) ) )\qquad\mbox{fof}(\mbox{d8\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_composition}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{relation}(\mbox{identity\_relation}(a))\qquad\mbox{fof}(\mbox{dt\_k6\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{empty}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_composition}(b,a))\mbox{ and }\mbox{relation}(\mbox{relation\_composition}(b,a)) ) )\qquad\mbox{fof}(\mbox{fc10\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{empty}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_composition}(a,b))\mbox{ and }\mbox{relation}(\mbox{relation\_composition}(a,b)) ) )\qquad\mbox{fof}(\mbox{fc9\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{relation}(d) \ \Rightarrow \ ( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{relation\_composition}(\mbox{identity\_relation}(c),d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(a,b),d) ) ) )\qquad\mbox{fof}(\mbox{t74\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU124+1.p} MPTP bushy problem t7\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;a\subseteq \mbox{set\_union}_2(a,b)\qquad\mbox{fof}(\mbox{t7\_xboole}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU193+1.p} MPTP bushy problem t88\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_dom\_restriction}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff ( \mbox{in}(d,b)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,e),a) ) ) ) ) )\qquad\mbox{fof}(\mbox{d11\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( a\subseteq b\iff \forall c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(c,d),a) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(c,d),b) ) ) ) )\qquad\mbox{fof}(\mbox{d3\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_dom\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k7\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_dom\_restriction}(b,a)\subseteq b )\qquad\mbox{fof}(\mbox{t88\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU613$\wedge$2.p} Preliminary Notions - Operations on Sets - Symmetric Difference

(! A:i.! B:i.! x:i.~(in x A) $\to$ in x B $\to$ in x (symdiff A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIR\_type},\mbox{type})$

$\mbox{binunionIR} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIR},\mbox{definition})$

$\mbox{symdiff}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{symdiff\_type},\mbox{type})$

$\mbox{symdiff} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{binunion}@ a@ b )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{in}@ \mbox{xx}@ a\mbox{ or }\neg\,\mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{symdiff},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{binunionIR} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{in}@ \mbox{xx}@ a \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{symdiff}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{symdiffI}_2,\mbox{conjecture})$

\medskip\textbf{SEU925$\wedge$5.p} TPS problem THM7-TPS2

Unitset is injective.




$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \lambda \mbox{xy}_0{:}\;\mbox{\$i}{:}\;\mbox{xx} = \mbox{xy}_0 ) = ( \lambda \mbox{xy}_2{:}\;\mbox{\$i}{:}\;\mbox{xy} = \mbox{xy}_2 ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM7\_TPS2\_pme},\mbox{conjecture})$

\medskip\textbf{SEU212+1.p} MPTP bushy problem t8\_funct\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ \forall b,c{:}\;( ( \mbox{in}(b,\mbox{relation\_dom}(a)) \ \Rightarrow \ ( c = \mbox{apply}(a,b)\iff \mbox{in}(\mbox{ordered\_pair}(b,c),a) ) )\mbox{ and }( \neg\,\mbox{in}(b,\mbox{relation\_dom}(a)) \ \Rightarrow \ ( c = \mbox{apply}(a,b)\iff c = \mbox{empty\_set} ) ) ) )\qquad\mbox{fof}(\mbox{d4\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_funct}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{relation}(c)\mbox{ and }\mbox{function}(c) ) \ \Rightarrow \ ( \mbox{in}(\mbox{ordered\_pair}(a,b),c)\iff ( \mbox{in}(a,\mbox{relation\_dom}(c))\mbox{ and }b = \mbox{apply}(c,a) ) ) )\qquad\mbox{fof}(\mbox{t8\_funct}_1,\mbox{conjecture})$

\medskip\textbf{SEU574$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.subset A B $\to$ subset B A $\to$ A = B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{setext} \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ a ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setextsub},\mbox{conjecture})$

\medskip\textbf{SEU123+2.p} MPTP chainy problem t3\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{lemma})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{t2\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t3\_xboole}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\neg\,\exists c{:}\;\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t4\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU898$\wedge$5.p} TPS problem THM132






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xh}{:}\;a\to a,\mbox{xs}{:}\;a\to \mbox{\$o},\mbox{xf}{:}\;a\to a{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}@ ( \mbox{xf}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}@ ( \mbox{xf}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}@ ( \mbox{xh}@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xh}@ ( \mbox{xf}@ \mbox{xx} ) ) = ( \mbox{xf}@ ( \mbox{xh}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xs}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xh}@ ( \mbox{xh}@ ( \mbox{xf}@ \mbox{xx} ) ) ) = ( \mbox{xf}@ ( \mbox{xh}@ ( \mbox{xh}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM132\_pme},\mbox{conjecture})$

\medskip\textbf{SEU922$\wedge$5.p} TPS problem THM128






$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( f@ ( g@ \mbox{xx} ) ) ) = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( g@ ( f@ \mbox{xx} ) ) ) \ \Rightarrow \ ( ( \mbox{cP}@ ( f@ ( f@ ( g@ a ) ) ) ) \ \Rightarrow \ ( \mbox{cP}@ ( g@ ( f@ ( f@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM128\_pme},\mbox{conjecture})$

\medskip\textbf{SEU971$\wedge$5.p} TPS problem from CHECKERBOARD-RELNS






$\mbox{cCKB\_INF}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_INF\_type},\mbox{type})$

$\mbox{cCKB\_INJ}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_INJ\_type},\mbox{type})$

$\mbox{cCKB\_XPL}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_XPL\_type},\mbox{type})$

$\mbox{cCKB\_INJ} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xy}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i},\mbox{xy}_2{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{xh}@ \mbox{xx}_1@ \mbox{xy}_1@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xh}@ \mbox{xx}_2@ \mbox{xy}_2@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{xx}_1 = \mbox{xx}_2\mbox{ and }\mbox{xy}_1 = \mbox{xy}_2 ) ) )\qquad\mbox{thf}(\mbox{cCKB\_INJ\_def},\mbox{definition})$

$\mbox{cCKB\_XPL} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ \mbox{xm}@ \mbox{xn}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xk}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xk}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\neg\,\mbox{xu} = \mbox{xm}\mbox{ and }\mbox{xv} = \mbox{xn} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_XPL\_def},\mbox{definition})$

$\mbox{cCKB\_INF} = ( \lambda \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB\_INJ}@ \mbox{xh}\mbox{ and }\mbox{cCKB\_XPL}@ \mbox{xh}@ \mbox{xk}@ \mbox{xm}@ \mbox{xn} ) )\qquad\mbox{thf}(\mbox{cCKB\_INF\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{cCKB\_INF}@ \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xu}@ \mbox{xv}\mbox{ or }( \mbox{xu} = \mbox{xx}\mbox{ and }\mbox{xv} = \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{cCKB\_INF}@ \mbox{xr} ) )\qquad\mbox{thf}(\mbox{cCKB6\_L80000\_pme},\mbox{conjecture})$

\medskip\textbf{SEU121+1.p} MPTP bushy problem t1\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU239+1.p} MPTP bushy problem l1\_wellord1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{is\_reflexive\_in}(a,b)\iff \forall c{:}\;( \mbox{in}(c,b) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(c,c),a) ) ) )\qquad\mbox{fof}(\mbox{d1\_relat}_2,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation\_field}(a) = \mbox{set\_union}_2(\mbox{relation\_dom}(a),\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{d6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{reflexive}(a)\iff \mbox{is\_reflexive\_in}(a,\mbox{relation\_field}(a)) ) )\qquad\mbox{fof}(\mbox{d9\_relat}_2,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{reflexive}(a)\iff \forall b{:}\;( \mbox{in}(b,\mbox{relation\_field}(a)) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(b,b),a) ) ) )\qquad\mbox{fof}(\mbox{l1\_wellord}_1,\mbox{conjecture})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU589$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.! x:i.in x A $\to$ in x B $\to$ in x (binintersect A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersect} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersect},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binintersectI},\mbox{conjecture})$

\medskip\textbf{SEU726$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 subset (setminus A Y) (setminus A X) $\to$ subset X Y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{subsetTI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetTI\_type},\mbox{type})$

$\mbox{subsetTI} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \subseteq @ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{subsetTI},\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ ( \mbox{setminusI} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ ( \mbox{subsetTI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ ( \mbox{setminus}@ a@ y )@ ( \mbox{setminus}@ a@ x ) ) \ \Rightarrow \ ( \subseteq @ x@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{contrasubsetT}_3,\mbox{conjecture})$

\medskip\textbf{SEU883$\wedge$5.p} TPS problem X5212






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$f{:}\;b\to a\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(g,\mbox{type})$

$( \lambda \mbox{xz}{:}\;a{:}\;\exists \mbox{xx}{:}\;b{:}\;( g@ \mbox{xx}\mbox{ and }\mbox{xz} = ( f@ \mbox{xx} ) ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;\exists \mbox{xt}{:}\;b{:}\;( g@ \mbox{xt}\mbox{ and }\mbox{xz} = ( f@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cX5212\_pme},\mbox{conjecture})$

\medskip\textbf{SEU657$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! u:i.iskpair u $\to$ kpair (kfst u) (ksnd u) = u)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{iskpair}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{iskpair\_type},\mbox{type})$

$\mbox{iskpair} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setunion}@ a )\mbox{ and }a = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{iskpair},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{kfst}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kfst\_type},\mbox{type})$

$\mbox{kfstpairEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{kfstpairEq\_type},\mbox{type})$

$\mbox{kfstpairEq} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{kfst}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{kfstpairEq},\mbox{definition})$

$\mbox{ksnd}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ksnd\_type},\mbox{type})$

$\mbox{ksndpairEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ksndpairEq\_type},\mbox{type})$

$\mbox{ksndpairEq} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{ksnd}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{ksndpairEq},\mbox{definition})$

$\mbox{kfstpairEq} \ \Rightarrow \ ( \mbox{ksndpairEq} \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{iskpair}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{kpair}@ ( \mbox{kfst}@ \mbox{xu} )@ ( \mbox{ksnd}@ \mbox{xu} ) ) = \mbox{xu} ) )\qquad\mbox{thf}(\mbox{kpairsurjEq},\mbox{conjecture})$

\medskip\textbf{SEU715$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x X $\to$ in x Y) $\to$ (! x:i.in x A $\to$ in x Y $\to$
 in x X) $\to$ X = Y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{powersetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE\_type},\mbox{type})$

$\mbox{powersetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetE},\mbox{definition})$

$\mbox{setext} \ \Rightarrow \ ( \mbox{powersetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) \ \Rightarrow \ x = y ) ) ) ) )\qquad\mbox{thf}(\mbox{setextT},\mbox{conjecture})$

\medskip\textbf{SEU854$\wedge$5.p} TPS problem GAZING-THM34






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o},u{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( t@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx} ) )\iff ( \lambda \mbox{xz}{:}\;a{:}\;( ( u@ \mbox{xz}\mbox{ and }\neg\,s@ \mbox{xz} )\mbox{ or }t@ \mbox{xz} ) ) = u ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM34\_pme},\mbox{conjecture})$

\medskip\textbf{SEU909$\wedge$5.p} TPS problem from SET-TOP-CATEGORY-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cA}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cA}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xe}{:}\;a\to \mbox{\$o}{:}\;( ( \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xe} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx} ) \ \Rightarrow \ \exists s{:}\;a\to \mbox{\$o}{:}\;( \mbox{cA}@ s\mbox{ and }s@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xx}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ ( \mbox{cA}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xx}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ \exists \mbox{xe}_0{:}\;a\to \mbox{\$o}{:}\;( \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}_0{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx}_0 ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{xe}_0@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}_0@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xe}_0 ) )\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}_0@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xx}_0 ) )\mbox{ and }\forall \mbox{xy}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xy}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}_0@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xy}@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ ( \mbox{cA}@ \mbox{xy}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xy}@ \mbox{xx}_0 ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cDOMTHM8\_pme},\mbox{conjecture})$

\medskip\textbf{SEU801$\wedge$2.p} More about Functions - Surjective Functions

(! A:i.! B:i.! f:i.in f (surjFuncSet A B) $\to$ in f (funcSet A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{surjective}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{surjective\_type},\mbox{type})$

$\mbox{surjective} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xy} ) = \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{surjective},\mbox{definition})$

$\mbox{surjFuncSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{surjFuncSet\_type},\mbox{type})$

$\mbox{surjFuncSet} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{funcSet}@ a@ b )@ \lambda \mbox{xf}{:}\;\mbox{\$i}{:}\;( \mbox{surjective}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{surjFuncSet},\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{surjFuncSet}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{surjFuncSetFuncIn},\mbox{conjecture})$

\medskip\textbf{SEU652$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.! z:i.setadjoin x (setadjoin y emptyset) =
 setadjoin z emptyset $\to$ x = y)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{secondinupair}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{secondinupair\_type},\mbox{type})$

$\mbox{secondinupair} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{secondinupair},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ ( \mbox{secondinupair} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{upairequniteq},\mbox{conjecture})$

\medskip\textbf{SEU485$\wedge$1.p} In a confluent relation every element has at most one normal form






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{confl}@ r ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( ( \mbox{nfof}@ r@ y@ x\mbox{ and }\mbox{nfof}@ r@ z@ x ) \ \Rightarrow \ y = z ) )\qquad\mbox{thf}(\mbox{confluent\_implies\_at\_most\_one\_nf},\mbox{conjecture})$

\medskip\textbf{SEU264+1.p} MPTP bushy problem t16\_relset\_1






$\forall a,b,c{:}\;( \mbox{element}(c,\mbox{powerset}(\mbox{cartesian\_product}_2(a,b))) \ \Rightarrow \ \mbox{relation}(c) )\qquad\mbox{fof}(\mbox{cc1\_relset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_relset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation\_of2\_as\_subset}(c,a,b) \ \Rightarrow \ \mbox{element}(c,\mbox{powerset}(\mbox{cartesian\_product}_2(a,b))) )\qquad\mbox{fof}(\mbox{dt\_m2\_relset}_1,\mbox{axiom})$

$\forall a,b{:}\;\exists c{:}\;\mbox{relation\_of}_2(c,a,b)\qquad\mbox{fof}(\mbox{existence\_m1\_relset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\exists c{:}\;\mbox{relation\_of2\_as\_subset}(c,a,b)\qquad\mbox{fof}(\mbox{existence\_m2\_relset}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation\_of2\_as\_subset}(c,a,b)\iff \mbox{relation\_of}_2(c,a,b) )\qquad\mbox{fof}(\mbox{redefinition\_m2\_relset}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation\_of2\_as\_subset}(c,a,b) \ \Rightarrow \ ( \mbox{relation\_dom}(c)\subseteq a\mbox{ and }\mbox{relation\_rng}(c)\subseteq b ) )\qquad\mbox{fof}(\mbox{t12\_relset}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{relation\_of2\_as\_subset}(d,c,a) \ \Rightarrow \ ( \mbox{relation\_rng}(d)\subseteq b \ \Rightarrow \ \mbox{relation\_of2\_as\_subset}(d,c,b) ) )\qquad\mbox{fof}(\mbox{t14\_relset}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{relation\_of2\_as\_subset}(d,c,a) \ \Rightarrow \ ( a\subseteq b \ \Rightarrow \ \mbox{relation\_of2\_as\_subset}(d,c,b) ) )\qquad\mbox{fof}(\mbox{t16\_relset}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

\medskip\textbf{SEU458$\wedge$1.p} The symmetric closure operator is idempotent






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{idem}@ \mbox{sc}\qquad\mbox{thf}(\mbox{symmetric\_closure\_op\_is\_idempotent},\mbox{conjecture})$

\medskip\textbf{SEU174+1.p} MPTP bushy problem t46\_setfam\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{powerset}(a))) \ \Rightarrow \ \forall c{:}\;( \mbox{element}(c,\mbox{powerset}(\mbox{powerset}(a))) \ \Rightarrow \ ( c = \mbox{complements\_of\_subsets}(a,b)\iff \forall d{:}\;( \mbox{element}(d,\mbox{powerset}(a)) \ \Rightarrow \ ( \mbox{in}(d,c)\iff \mbox{in}(\mbox{subset\_complement}(a,d),b) ) ) ) ) )\qquad\mbox{fof}(\mbox{d8\_setfam}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{element}(\mbox{subset\_complement}(a,b),\mbox{powerset}(a)) )\qquad\mbox{fof}(\mbox{dt\_k3\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{powerset}(a))) \ \Rightarrow \ \mbox{element}(\mbox{complements\_of\_subsets}(a,b),\mbox{powerset}(\mbox{powerset}(a))) )\qquad\mbox{fof}(\mbox{dt\_k7\_setfam}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(a)) \ \Rightarrow \ \mbox{subset\_complement}(a,\mbox{subset\_complement}(a,b)) = b )\qquad\mbox{fof}(\mbox{involutiveness\_k3\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{powerset}(a))) \ \Rightarrow \ \mbox{complements\_of\_subsets}(a,\mbox{complements\_of\_subsets}(a,b)) = b )\qquad\mbox{fof}(\mbox{involutiveness\_k7\_setfam}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(b,\mbox{powerset}(\mbox{powerset}(a))) \ \Rightarrow \ \neg\,b\ne \mbox{empty\_set}\mbox{ and }\mbox{complements\_of\_subsets}(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t46\_setfam}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU639$\wedge$2.p} Ordered Pairs - Singletons

(! A:i.! phi:i$>$o.! x:i.in x A $\to$ phi x $\to$ (! y:i.in y A $\to$
 phi y $\to$ y = x) $\to$ ex1 A ($\wedge$ y:i.phi y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{eqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{eqinunit\_type},\mbox{type})$

$\mbox{eqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{eqinunit},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{setext} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ ( \mbox{eqinunit} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xy} = \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{ex}_1@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ex1I},\mbox{conjecture})$

\medskip\textbf{SEU787$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ in (kpair x y) (binunion R S) $\to$
 (! phi:o.(in (kpair x y) R $\to$ phi) $\to$
 (in (kpair x y) S $\to$ phi) $\to$ phi)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln1unionE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1unionE\_type},\mbox{type})$

$\mbox{breln1unionE} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r\mbox{ or }\mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionE},\mbox{definition})$

$\mbox{breln1unionE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionEcases},\mbox{conjecture})$

\medskip\textbf{SEU563$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.subset emptyset A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptysetimpfalse}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetimpfalse\_type},\mbox{type})$

$\mbox{emptysetimpfalse} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \mbox{\$false} ) )\qquad\mbox{thf}(\mbox{emptysetimpfalse},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\subseteq  = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{subset},\mbox{definition})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{emptysetimpfalse} \ \Rightarrow \ ( \mbox{subsetI}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( \subseteq @ \mbox{emptyset}@ a ) )\qquad\mbox{thf}(\mbox{emptysetsubset},\mbox{conjecture})$

\medskip\textbf{SEU771$\wedge$2.p} Binary Relations on a Set

(! A:i.! phi:i$>$(i$>$o).breln1 A (dpsetconstr A A ($\wedge$ x,y:i.phi x y)))




$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{setOfPairsIsBReln}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setOfPairsIsBReln\_type},\mbox{type})$

$\mbox{setOfPairsIsBReln} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{breln}@ a@ b@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{setOfPairsIsBReln},\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{setOfPairsIsBReln} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{breln}_1@ a@ ( \mbox{dpsetconstr}@ a@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setOfPairsIsBReln}_1,\mbox{conjecture})$

\medskip\textbf{SEU121+2.p} MPTP chainy problem t1\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\neg\,\exists c{:}\;\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t4\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU119+2.p} MPTP chainy problem t3\_xboole\_0






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{conjecture})$

\medskip\textbf{SEU584$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.! x:i.in x B $\to$ in x (binunion A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{setunionI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionI\_type},\mbox{type})$

$\mbox{setunionI} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{setunionI},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunion} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{binunion},\mbox{definition})$

$\mbox{upairset2IR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{upairset2IR\_type},\mbox{type})$

$\mbox{upairset2IR} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{upairset2IR},\mbox{definition})$

$\mbox{setunionI} \ \Rightarrow \ ( \mbox{upairset2IR} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIR},\mbox{conjecture})$

\medskip\textbf{SEU949$\wedge$5.p} TPS problem THM589






$f{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(g,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx} ) = ( g@ \mbox{xy} ) \ \Rightarrow \ ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xh}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ ( g@ \mbox{xx} ) ) ) = f\qquad\mbox{thf}(\mbox{cTHM589\_pme},\mbox{conjecture})$

\medskip\textbf{SEU579$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.subset A B $\to$ subset (powerset A) (powerset B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\mbox{powersetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE\_type},\mbox{type})$

$\mbox{powersetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetE},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{powersetI} \ \Rightarrow \ ( \mbox{powersetE} \ \Rightarrow \ ( \mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( \subseteq @ ( \mbox{powerset}@ a )@ ( \mbox{powerset}@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{powersetsubset},\mbox{conjecture})$

\medskip\textbf{SEU716$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x X $\to$ in x Y) $\to$ subset X Y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE\_type},\mbox{type})$

$\mbox{powersetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetE},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{powersetE} \ \Rightarrow \ ( \mbox{subsetI}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \subseteq @ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{subsetTI},\mbox{conjecture})$

\medskip\textbf{SEU978$\wedge$5.p} TPS problem from COINDUCTIVE-PU-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xt} = \mbox{cZ}\iff \mbox{xu} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ \mbox{xt} = \mbox{xu} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xu}{:}\;a,\mbox{xv}{:}\;a{:}\;( ( \exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}_0{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu}_0 ) ) \ \Rightarrow \ ( ( \mbox{xu}_0 = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu}_0 ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu}_0 ) ) ) ) )\mbox{ and }\exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ \mbox{xu}@ \mbox{xv} )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}_0{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu}_0 ) ) \ \Rightarrow \ ( ( \mbox{xu}_0 = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu}_0 ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu}_0 ) ) ) ) ) ) \ \Rightarrow \ \exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xu} )@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xv} ) )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}_0{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu}_0 ) ) \ \Rightarrow \ ( ( \mbox{xu}_0 = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu}_0 ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu}_0 ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_LEM2D\_pme},\mbox{conjecture})$

\medskip\textbf{SEU144+1.p} MPTP bushy problem l2\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq b\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l2\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

\medskip\textbf{SEU666$\wedge$2.p} Ordered Pairs - Sets of Pairs

(! A:i.! B:i.! phi:i$>$(i$>$o).subset (dpsetconstr A B
 ($\wedge$ x,y:i.phi x y)) (cartprod A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstr} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{dsetconstr}@ ( \mbox{cartprod}@ a@ b )@ \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xphi}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstr},\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{subsetI}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \subseteq @ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) )@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrSub},\mbox{conjecture})$

\medskip\textbf{SEU468$\wedge$1.p} The transitive closure operator is monotonic






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{mono}@ \mbox{tc}\qquad\mbox{thf}(\mbox{transitive\_closure\_op\_is\_monotonic},\mbox{conjecture})$

\medskip\textbf{SEU673$\wedge$2.p} Functions - Application

(! A:i.! B:i.! f:i.func A B f $\to$ (! x:i.in x A $\to$ in (setunion
 (dsetconstr B ($\wedge$ y:i.in (kpair x y) f))) B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{theprop}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{theprop\_type},\mbox{type})$

$\mbox{theprop} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{singleton}@ x ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setunion}@ x )@ x ) ) )\qquad\mbox{thf}(\mbox{theprop},\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{funcImageSingleton}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcImageSingleton\_type},\mbox{type})$

$\mbox{funcImageSingleton} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{funcImageSingleton},\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{theprop} \ \Rightarrow \ ( \mbox{funcImageSingleton} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setunion}@ ( \mbox{dsetconstr}@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) ) )@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{apProp},\mbox{conjecture})$

\medskip\textbf{SEU165+1.p} MPTP bushy problem t106\_zfmisc\_1






$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{t106\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU918$\wedge$5.p} TPS problem THM197

If there are at least two individuals, then WHAT?




$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$a\ne b \ \Rightarrow \ \neg\,\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ ( \mbox{xg}@ \mbox{xx} ) ) ) = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xg}@ ( \mbox{xf}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cTHM197\_pme},\mbox{conjecture})$

\medskip\textbf{SEU180+1.p} MPTP bushy problem t30\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) ) )\qquad\mbox{fof}(\mbox{d5\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation\_field}(a) = \mbox{set\_union}_2(\mbox{relation\_dom}(a),\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{d6\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(\mbox{ordered\_pair}(a,b),c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_field}(c))\mbox{ and }\mbox{in}(b,\mbox{relation\_field}(c)) ) ) )\qquad\mbox{fof}(\mbox{t30\_relat}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU177+1.p} MPTP bushy problem t20\_relat\_1






$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(\mbox{ordered\_pair}(a,b),c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_dom}(c))\mbox{ and }\mbox{in}(b,\mbox{relation\_rng}(c)) ) ) )\qquad\mbox{fof}(\mbox{t20\_relat}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) ) )\qquad\mbox{fof}(\mbox{d5\_relat}_1,\mbox{axiom})$

\medskip\textbf{SEU948$\wedge$5.p} TPS problem THM135

The composition of iterates of a function is also an iterate of
 that function. 




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xf}{:}\;a\to a,\mbox{xg}_1{:}\;a\to a,\mbox{xg}_2{:}\;a\to a{:}\;( ( \forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg}_1 ) )\mbox{ and }\forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg}_2 ) ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xg}_1@ ( \mbox{xg}_2@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM135\_pme},\mbox{conjecture})$

\medskip\textbf{SEU586$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.! x:i.in x (binunion A B) $\to$ in x A | in x B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionEcases}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionEcases\_type},\mbox{type})$

$\mbox{binunionEcases} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{binunionEcases},\mbox{definition})$

$\mbox{binunionEcases} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a\mbox{ or }\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{binunionE},\mbox{conjecture})$

\medskip\textbf{SEU824$\wedge$2.p} Ordinals

(! X:i.ordinal X $\to$ (! x:i.! A:i.! B:i.in x A $\to$ in A B $\to$
 in B X $\to$ in x B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{stricttotalorderedByIn\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\mbox{in}@ x@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{in}@ x@ y\mbox{ or }\mbox{in}@ y@ x ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \neg\,\mbox{in}@ x@ x ) ) )\qquad\mbox{thf}(\mbox{stricttotalorderedByIn},\mbox{definition})$

$\mbox{wellorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{wellorderedByIn\_type},\mbox{type})$

$\mbox{wellorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{stricttotalorderedByIn}@ a\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{nonempty}@ x ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ x ) \ \Rightarrow \ ( \mbox{xx} = y\mbox{ or }\mbox{in}@ \mbox{xx}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{wellorderedByIn},\mbox{definition})$

$\mbox{ordinal}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ordinal\_type},\mbox{type})$

$\mbox{ordinal} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{transitiveset}@ \mbox{xx}\mbox{ and }\mbox{wellorderedByIn}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{ordinal},\mbox{definition})$

$\mbox{ordinalTransSet}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ordinalTransSet\_type},\mbox{type})$

$\mbox{ordinalTransSet} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{ordinalTransSet},\mbox{definition})$

$\mbox{ordinalTransSet} \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{ordinalTransIn},\mbox{conjecture})$

\medskip\textbf{SEU194+1.p} MPTP bushy problem t90\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_dom\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k7\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{set\_intersection}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc1\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_dom}(\mbox{relation\_dom\_restriction}(c,b)))\iff ( \mbox{in}(a,b)\mbox{ and }\mbox{in}(a,\mbox{relation\_dom}(c)) ) ) )\qquad\mbox{fof}(\mbox{t86\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_dom}(\mbox{relation\_dom\_restriction}(b,a)) = \mbox{set\_intersection}_2(\mbox{relation\_dom}(b),a) )\qquad\mbox{fof}(\mbox{t90\_relat}_1,\mbox{conjecture})$

\medskip\textbf{SEU808$\wedge$2.p} Omega and Peano

(! x:i.in x omega $\to$ in (omegaS x) omega)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{omega}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{omega\_type},\mbox{type})$

$\mbox{omegaSAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{omegaSAx\_type},\mbox{type})$

$\mbox{omegaSAx} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{omega} ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xx} )@ \mbox{omega} ) ) )\qquad\mbox{thf}(\mbox{omegaSAx},\mbox{definition})$

$\mbox{omegaS}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{omegaS\_type},\mbox{type})$

$\mbox{omegaS} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{omegaS},\mbox{definition})$

$\mbox{omegaSAx} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{omega} ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{omegaS}@ \mbox{xx} )@ \mbox{omega} ) )\qquad\mbox{thf}(\mbox{omegaSp},\mbox{conjecture})$

\medskip\textbf{SEU168+3.p} Basic properties of sets, theorem 136






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{powerset}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c\subseteq a ) )\qquad\mbox{fof}(\mbox{d1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{in}(a,b)\mbox{ and }\forall c,d{:}\;( ( \mbox{in}(c,b)\mbox{ and }d\subseteq c ) \ \Rightarrow \ \mbox{in}(d,b) )\mbox{ and }\forall c{:}\;( \mbox{in}(c,b) \ \Rightarrow \ \mbox{in}(\mbox{powerset}(c),b) )\mbox{ and }\forall c{:}\;\neg\,c\subseteq b\mbox{ and }\neg\,\mbox{are\_equipotent}(c,b)\mbox{ and }\neg\,\mbox{in}(c,b) )\qquad\mbox{fof}(\mbox{t136\_zfmisc}_1,\mbox{conjecture})$

$\forall a{:}\;\exists b{:}\;( \mbox{in}(a,b)\mbox{ and }\forall c,d{:}\;( ( \mbox{in}(c,b)\mbox{ and }d\subseteq c ) \ \Rightarrow \ \mbox{in}(d,b) )\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\forall d{:}\;\neg\,\mbox{in}(d,b)\mbox{ and }\forall e{:}\;( e\subseteq c \ \Rightarrow \ \mbox{in}(e,d) )\mbox{ and }\forall c{:}\;\neg\,c\subseteq b\mbox{ and }\neg\,\mbox{are\_equipotent}(c,b)\mbox{ and }\neg\,\mbox{in}(c,b) )\qquad\mbox{fof}(\mbox{t9\_tarski},\mbox{axiom})$

\medskip\textbf{SEU614$\wedge$2.p} Preliminary Notions - Operations on Sets - Symmetric Difference

(! A:i.! B:i.! x:i.in x A $\to$ in x B $\to$ ~(in x (symdiff A B)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{symdiff}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{symdiff\_type},\mbox{type})$

$\mbox{symdiffE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{symdiffE\_type},\mbox{type})$

$\mbox{symdiffE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{symdiff}@ a@ b ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ \mbox{xphi} ) ) \ \Rightarrow \ ( ( \neg\,\mbox{in}@ \mbox{xx}@ a \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \mbox{xphi} ) ) \ \Rightarrow \ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{symdiffE},\mbox{definition})$

$\mbox{symdiffE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{symdiff}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{symdiffIneg}_1,\mbox{conjecture})$

\medskip\textbf{SEU500$\wedge$2.p} Preliminary Notions - Propositions as Sets

(! phi:o.! x:i.in x (prop2set phi) $\to$ phi)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{prop2set}{:}\;\mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{prop2set\_type},\mbox{type})$

$\mbox{prop2set} = ( \lambda \mbox{xphi}{:}\;\mbox{\$o}{:}\;( \mbox{dsetconstr}@ ( \mbox{powerset}@ \mbox{emptyset} )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{prop2set},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{prop2set}@ \mbox{xphi} ) ) \ \Rightarrow \ \mbox{xphi} )\qquad\mbox{thf}(\mbox{prop2setE},\mbox{conjecture})$

\medskip\textbf{SEU800$\wedge$2.p} More about Functions - Injective Functions

(! x:i.! y:i.! f:i.in f (injFuncSet x y) $\to$ injective x y f)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{injective}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{injective\_type},\mbox{type})$

$\mbox{injFuncSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{injFuncSet\_type},\mbox{type})$

$\mbox{injFuncSet} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{funcSet}@ a@ b )@ \lambda \mbox{xf}{:}\;\mbox{\$i}{:}\;( \mbox{injective}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{injFuncSet},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{injFuncSet}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{injective}@ \mbox{xx}@ \mbox{xy}@ \mbox{xf} ) )\qquad\mbox{thf}(\mbox{injFuncSetFuncInj},\mbox{conjecture})$

\medskip\textbf{SEU627$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! B:i.! x:i.in x A $\to$ (! y:i.in y B $\to$ subset (setadjoin
 (setadjoin x emptyset) (setadjoin (setadjoin x
 (setadjoin y emptyset)) emptyset)) (powerset (binunion A B))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{singletoninpowunion}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{singletoninpowunion\_type},\mbox{type})$

$\mbox{singletoninpowunion} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{singletoninpowunion},\mbox{definition})$

$\mbox{upairset2E}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{upairset2E\_type},\mbox{type})$

$\mbox{upairset2E} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) \ \Rightarrow \ ( \mbox{xz} = \mbox{xx}\mbox{ or }\mbox{xz} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{upairset2E},\mbox{definition})$

$\mbox{upairinpowunion}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{upairinpowunion\_type},\mbox{type})$

$\mbox{upairinpowunion} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{upairinpowunion},\mbox{definition})$

$\mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{singletoninpowunion} \ \Rightarrow \ ( \mbox{upairset2E} \ \Rightarrow \ ( \mbox{upairinpowunion} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \subseteq @ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) )@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ubforcartprodlem}_1,\mbox{conjecture})$

\medskip\textbf{SEU632$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! B:i.! u:i.in u (cartprod A B) $\to$ iskpair u)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{iskpair}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{iskpair\_type},\mbox{type})$

$\mbox{iskpair} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setunion}@ a )\mbox{ and }a = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{iskpair},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{kpairp}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{kpairp\_type},\mbox{type})$

$\mbox{kpairp} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{iskpair}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{kpairp},\mbox{definition})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprodmempair}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodmempair1\_type},\mbox{type})$

$\mbox{cartprodmempair}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprodmempair}_1,\mbox{definition})$

$\mbox{kpairp} \ \Rightarrow \ ( \mbox{cartprodmempair}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ ( \mbox{iskpair}@ \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{cartprodmempair},\mbox{conjecture})$

\medskip\textbf{SEU679$\wedge$2.p} Functions - Lambda Abstraction

(! A:i.! B:i.! f:i$>$i.(! x:i.in x A $\to$ in (f x) B) $\to$
 func A B (lam A B ($\wedge$ x:i.f x)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{lamProp}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{lamProp\_type},\mbox{type})$

$\mbox{lamProp} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ ( \mbox{func}@ a@ b@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{lamProp},\mbox{definition})$

$\mbox{lam}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{lam\_type},\mbox{type})$

$\mbox{lam} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{lam},\mbox{definition})$

$\mbox{lamProp} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ ( \mbox{func}@ a@ b@ ( \mbox{lam}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{lamp},\mbox{conjecture})$

\medskip\textbf{SEU960$\wedge$5.p} TPS problem from FUNCTION-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xf}{:}\;a\to a,\mbox{xg}_1{:}\;a\to a,\mbox{xg}_2{:}\;a\to a{:}\;( ( \forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xf}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg}_1 ) )\mbox{ and }\forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xf}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg}_2 ) ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xf}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xg}_1@ ( \mbox{xg}_2@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM135D\_pme},\mbox{conjecture})$

\medskip\textbf{SEU943$\wedge$5.p} TPS problem THM172

If g is an iterate of f, and g has a unique fixed point, then f
 has a fixed point.




$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ f\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( f@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ g ) ) \ \Rightarrow \ ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xz} ) = \mbox{xz} \ \Rightarrow \ \mbox{xz} = \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( f@ \mbox{xy} ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM172\_pme},\mbox{conjecture})$

\medskip\textbf{SEU906$\wedge$5.p} TPS problem from FUNS-AND-SETS-THMS






$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$\mbox{cS}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cS}@ \mbox{xx} )\iff ( \mbox{xu}@ ( f@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cSV5\_pme},\mbox{conjecture})$

\medskip\textbf{SEU605$\wedge$2.p} Preliminary Notions - Operations on Sets - Set Difference

(! A:i.! B:i.! x:i.~(in x (setminus A B)) $\to$ in x A $\to$ in x B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{setminusI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{setminusERneg},\mbox{conjecture})$

\medskip\textbf{SEU198+1.p} MPTP bushy problem t116\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_rng\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k8\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{fc6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_rng}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_rng}(a)) ) )\qquad\mbox{fof}(\mbox{fc8\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_rng}(\mbox{relation\_rng\_restriction}(b,c)))\iff ( \mbox{in}(a,b)\mbox{ and }\mbox{in}(a,\mbox{relation\_rng}(c)) ) ) )\qquad\mbox{fof}(\mbox{t115\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_rng}(\mbox{relation\_rng\_restriction}(a,b))\subseteq a )\qquad\mbox{fof}(\mbox{t116\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU942$\wedge$5.p} TPS problem THM15B






$\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \exists \mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xf}\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg} ) )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xg}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xg}@ \mbox{xz} ) = \mbox{xz} \ \Rightarrow \ \mbox{xz} = \mbox{xx} ) ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM15B\_pme},\mbox{conjecture})$

\medskip\textbf{SEU754$\wedge$2.p} Typed Set Theory - Laws for Typed Sets - DeMorgan Laws

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x (setminus A (binunion X Y)) $\to$
 in x (binintersect (setminus A X) (setminus A Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectI\_type},\mbox{type})$

$\mbox{binintersectI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectI},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{binunionTILcontra}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionTILcontra\_type},\mbox{type})$

$\mbox{binunionTILcontra} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionTILcontra},\mbox{definition})$

$\mbox{binunionTIRcontra}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionTIRcontra\_type},\mbox{type})$

$\mbox{binunionTIRcontra} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionTIRcontra},\mbox{definition})$

$\mbox{binintersectI} \ \Rightarrow \ ( \mbox{setminusI} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ ( \mbox{binunionTILcontra} \ \Rightarrow \ ( \mbox{binunionTIRcontra} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{setminus}@ a@ y ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{demorgan2a},\mbox{conjecture})$

\medskip\textbf{SEU889$\wedge$5.p} TPS problem THM29A






$c{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cG}{:}\;c\to b\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cF}{:}\;b\to a\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cS}{:}\;c\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\forall \mbox{xx}{:}\;a{:}\;( \exists \mbox{xt}{:}\;b{:}\;( \exists \mbox{xt}_0{:}\;c{:}\;( \mbox{cS}@ \mbox{xt}_0\mbox{ and }\mbox{xt} = ( \mbox{cG}@ \mbox{xt}_0 ) )\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) ) \ \Rightarrow \ \exists \mbox{xt}{:}\;c{:}\;( \mbox{cS}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( \mbox{cF}@ ( \mbox{cG}@ \mbox{xt} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM29A\_pme},\mbox{conjecture})$

\medskip\textbf{SEU697$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i$>$i.(! x:i.in x A $\to$ in (f x) B) $\to$
 lam A B ($\wedge$ x:i.f x) = lam A B ($\wedge$ x:i.f x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{lam}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{lam\_type},\mbox{type})$

$\mbox{lam} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{lam},\mbox{definition})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ ( \mbox{lam}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) ) = ( \mbox{lam}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{lam2lamEq},\mbox{conjecture})$

\medskip\textbf{SEU166+3.p} Basic properties of sets, theorem 118






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ ( \mbox{cartesian\_product}_2(a,c)\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }\mbox{cartesian\_product}_2(c,a)\subseteq \mbox{cartesian\_product}_2(c,b) ) )\qquad\mbox{fof}(\mbox{t118\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU700$\wedge$2.p} Conditionals

(! A:i.! phi:o.! x:i.in x A $\to$ (! y:i.in y A $\to$ ~phi $\to$
 in y (dsetconstr A ($\wedge$ z:i.phi \& z = x | ~phi \& z = y))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \neg\,\mbox{xphi} \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iffalseProp}_1,\mbox{conjecture})$

\medskip\textbf{SEU847$\wedge$5.p} TPS problem GAZING-THM41






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( ( s@ \mbox{xz}\mbox{ and }\neg\,\mbox{\$false} )\mbox{ or }( \mbox{\$false}\mbox{ and }\neg\,s@ \mbox{xz} ) ) ) = s\qquad\mbox{thf}(\mbox{cGAZING\_THM41\_pme},\mbox{conjecture})$

\medskip\textbf{SEU638$\wedge$2.p} Ordered Pairs - Singletons

(! A:i.! phi:i$>$o.ex1 A ($\wedge$ x:i.phi x) $\to$ (? x:i.in x A \& phi x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{dsetconstrER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{ex}_1@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{ex1E}_1,\mbox{conjecture})$

\medskip\textbf{SEU244+1.p} MPTP bushy problem t8\_wellord1






$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{antisymmetric}(a)\iff \mbox{is\_antisymmetric\_in}(a,\mbox{relation\_field}(a)) ) )\qquad\mbox{fof}(\mbox{d12\_relat}_2,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{connected}(a)\iff \mbox{is\_connected\_in}(a,\mbox{relation\_field}(a)) ) )\qquad\mbox{fof}(\mbox{d14\_relat}_2,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{transitive}(a)\iff \mbox{is\_transitive\_in}(a,\mbox{relation\_field}(a)) ) )\qquad\mbox{fof}(\mbox{d16\_relat}_2,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{well\_ordering}(a)\iff ( \mbox{reflexive}(a)\mbox{ and }\mbox{transitive}(a)\mbox{ and }\mbox{antisymmetric}(a)\mbox{ and }\mbox{connected}(a)\mbox{ and }\mbox{well\_founded\_relation}(a) ) ) )\qquad\mbox{fof}(\mbox{d4\_wellord}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{well\_orders}(a,b)\iff ( \mbox{is\_reflexive\_in}(a,b)\mbox{ and }\mbox{is\_transitive\_in}(a,b)\mbox{ and }\mbox{is\_antisymmetric\_in}(a,b)\mbox{ and }\mbox{is\_connected\_in}(a,b)\mbox{ and }\mbox{is\_well\_founded\_in}(a,b) ) ) )\qquad\mbox{fof}(\mbox{d5\_wellord}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation\_field}(a) = \mbox{set\_union}_2(\mbox{relation\_dom}(a),\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{d6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{reflexive}(a)\iff \mbox{is\_reflexive\_in}(a,\mbox{relation\_field}(a)) ) )\qquad\mbox{fof}(\mbox{d9\_relat}_2,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{well\_founded\_relation}(a)\iff \mbox{is\_well\_founded\_in}(a,\mbox{relation\_field}(a)) ) )\qquad\mbox{fof}(\mbox{t5\_wellord}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{well\_orders}(a,\mbox{relation\_field}(a))\iff \mbox{well\_ordering}(a) ) )\qquad\mbox{fof}(\mbox{t8\_wellord}_1,\mbox{conjecture})$

\medskip\textbf{SEU494$\wedge$1.p} Inverse of a strict (partial) order is a strict (partial) order






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{so}@ r ) \ \Rightarrow \ ( \mbox{so}@ ( \mbox{inv}@ r ) ) )\qquad\mbox{thf}(\mbox{inverse\_of\_strict\_order\_is\_strict\_order},\mbox{conjecture})$

\medskip\textbf{SEU731$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ X = setminus A (setminus A X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{complementT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementT\_lem\_type},\mbox{type})$

$\mbox{complementT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{complementT\_lem},\mbox{definition})$

$\mbox{setextT}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextT\_type},\mbox{type})$

$\mbox{setextT} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) \ \Rightarrow \ x = y ) ) ) ) )\qquad\mbox{thf}(\mbox{setextT},\mbox{definition})$

$\mbox{doubleComplementI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{doubleComplementI1\_type},\mbox{type})$

$\mbox{doubleComplementI}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{setminus}@ a@ x ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{doubleComplementI}_1,\mbox{definition})$

$\mbox{doubleComplementE}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{doubleComplementE1\_type},\mbox{type})$

$\mbox{doubleComplementE}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{setminus}@ a@ x ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{doubleComplementE}_1,\mbox{definition})$

$\mbox{complementT\_lem} \ \Rightarrow \ ( \mbox{setextT} \ \Rightarrow \ ( \mbox{doubleComplementI}_1 \ \Rightarrow \ ( \mbox{doubleComplementE}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ x = ( \mbox{setminus}@ a@ ( \mbox{setminus}@ a@ x ) ) ) ) ) )\qquad\mbox{thf}(\mbox{doubleComplementEq},\mbox{conjecture})$

\medskip\textbf{SEU161+1.p} MPTP bushy problem t46\_zfmisc\_1






$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_union}_2(\mbox{singleton}(a),b) = b )\qquad\mbox{fof}(\mbox{t46\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_union}_2(\mbox{singleton}(a),b) = b )\qquad\mbox{fof}(\mbox{l23\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU993$\wedge$5.p} TPS problem from GRAPHS-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\mbox{cR}{:}\;a\to a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cS}{:}\;b\to b\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$( \forall \mbox{xx}{:}\;a{:}\;( \mbox{cR}@ \mbox{xx}@ \mbox{xx} )\mbox{ and }\forall \mbox{xu}{:}\;a,\mbox{xv}{:}\;a,\mbox{xw}{:}\;a{:}\;( ( \mbox{cR}@ \mbox{xu}@ \mbox{xw}\mbox{ and }\mbox{cR}@ \mbox{xv}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xu}@ \mbox{xv} ) )\mbox{ and }\exists \mbox{xf}{:}\;a\to b\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;\exists \mbox{xy}{:}\;b{:}\;( \mbox{xf}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}_1{:}\;b,\mbox{xy}_2{:}\;b{:}\;( ( \mbox{xf}@ \mbox{xx}@ \mbox{xy}_1\mbox{ and }\mbox{xf}@ \mbox{xx}@ \mbox{xy}_2 ) \ \Rightarrow \ ( \mbox{cS}@ \mbox{xy}_1@ \mbox{xy}_2 ) )\mbox{ and }\forall \mbox{xx}_1{:}\;a,\mbox{xx}_2{:}\;a,\mbox{xy}{:}\;b{:}\;( ( \mbox{xf}@ \mbox{xx}_1@ \mbox{xy}\mbox{ and }\mbox{xf}@ \mbox{xx}_2@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx}_1@ \mbox{xx}_2 ) ) ) ) \ \Rightarrow \ \exists \mbox{xg}{:}\;b\to a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;\exists \mbox{xy}{:}\;b{:}\;( \mbox{xg}@ \mbox{xy}@ \mbox{xx} )\mbox{ and }\forall \mbox{xy}{:}\;b,\mbox{xx}_1{:}\;a,\mbox{xx}_2{:}\;a{:}\;( ( \mbox{xg}@ \mbox{xy}@ \mbox{xx}_1\mbox{ and }\mbox{xg}@ \mbox{xy}@ \mbox{xx}_2 ) \ \Rightarrow \ ( \mbox{cR}@ \mbox{xx}_1@ \mbox{xx}_2 ) )\mbox{ and }\forall \mbox{xy}{:}\;b{:}\;\exists \mbox{xx}{:}\;a{:}\;( \mbox{xg}@ \mbox{xy}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTHM552\_pme},\mbox{conjecture})$

\medskip\textbf{SEU135+1.p} MPTP bushy problem t39\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,\mbox{set\_difference}(b,a)) = \mbox{set\_union}_2(a,b)\qquad\mbox{fof}(\mbox{t39\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU357+1.p} MPTP bushy problem t16\_yellow\_0






$\forall a{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{ex\_inf\_of\_relstr\_set}(a,b)\iff \exists c{:}\;( \mbox{element}(c,\mbox{the\_carrier}(a))\mbox{ and }\mbox{relstr\_element\_smaller}(a,b,c)\mbox{ and }\forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_element\_smaller}(a,b,d) \ \Rightarrow \ \mbox{related}(a,d,c) ) )\mbox{ and }\forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( ( \mbox{relstr\_element\_smaller}(a,b,d)\mbox{ and }\forall e{:}\;( \mbox{element}(e,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_element\_smaller}(a,b,e) \ \Rightarrow \ \mbox{related}(a,e,d) ) ) ) \ \Rightarrow \ d = c ) ) ) ) )\qquad\mbox{fof}(\mbox{d8\_yellow}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{rel\_str}(a) \ \Rightarrow \ \mbox{one\_sorted\_str}(a) )\qquad\mbox{fof}(\mbox{dt\_l1\_orders}_2,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_l1\_struct}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_u1\_struct}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{rel\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_orders}_2,\mbox{axiom})$

$\exists a{:}\;\mbox{one\_sorted\_str}(a)\qquad\mbox{fof}(\mbox{existence\_l1\_struct}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{antisymmetric\_relstr}(a)\mbox{ and }\mbox{rel\_str}(a) ) \ \Rightarrow \ \forall b{:}\;( \mbox{ex\_inf\_of\_relstr\_set}(a,b)\iff \exists c{:}\;( \mbox{element}(c,\mbox{the\_carrier}(a))\mbox{ and }\mbox{relstr\_element\_smaller}(a,b,c)\mbox{ and }\forall d{:}\;( \mbox{element}(d,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( \mbox{relstr\_element\_smaller}(a,b,d) \ \Rightarrow \ \mbox{related}(a,d,c) ) ) ) ) )\qquad\mbox{fof}(\mbox{t16\_yellow}_0,\mbox{conjecture})$

$\forall a{:}\;( ( \mbox{antisymmetric\_relstr}(a)\mbox{ and }\mbox{rel\_str}(a) ) \ \Rightarrow \ \forall b{:}\;( \mbox{element}(b,\mbox{the\_carrier}(a)) \ \Rightarrow \ \forall c{:}\;( \mbox{element}(c,\mbox{the\_carrier}(a)) \ \Rightarrow \ ( ( \mbox{related}(a,b,c)\mbox{ and }\mbox{related}(a,c,b) ) \ \Rightarrow \ b = c ) ) ) )\qquad\mbox{fof}(\mbox{t25\_orders}_2,\mbox{axiom})$

\medskip\textbf{SEU690$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! R:i.breln A B R $\to$ (! S:i.breln A B S $\to$
 (! x:i.in x A $\to$ (! y:i.in y B $\to$ in (kpair x y) R $\to$
 in (kpair x y) S)) $\to$ (! x:i.in x A $\to$ (! y:i.in y B $\to$
 in (kpair x y) S $\to$ in (kpair x y) R)) $\to$ R = S))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{setextsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextsub\_type},\mbox{type})$

$\mbox{setextsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ a ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setextsub},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{subbreln}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subbreln\_type},\mbox{type})$

$\mbox{subbreln} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ s ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) \ \Rightarrow \ ( \subseteq @ r@ s ) ) ) ) )\qquad\mbox{thf}(\mbox{subbreln},\mbox{definition})$

$\mbox{setextsub} \ \Rightarrow \ ( \mbox{subbreln} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ s ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) \ \Rightarrow \ r = s ) ) ) ) )\qquad\mbox{thf}(\mbox{eqbreln},\mbox{conjecture})$

\medskip\textbf{SEU806$\wedge$2.p} The Foundation Axiom

(! A:i.~(in A A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{foundationAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{foundationAx\_type},\mbox{type})$

$\mbox{foundationAx} = ( \forall a{:}\;\mbox{\$i}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \exists b{:}\;\mbox{\$i}{:}\;( \mbox{in}@ b@ a\mbox{ and }\neg\,\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b\mbox{ and }\mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{foundationAx},\mbox{definition})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\mbox{foundationAx} \ \Rightarrow \ ( \mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ ( \mbox{in\_\_Cong} \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;\neg\,\mbox{in}@ a@ a ) ) )\qquad\mbox{thf}(\mbox{notinself},\mbox{conjecture})$

\medskip\textbf{SEU825$\wedge$3.p} setextAx and powersetAx and notinemptyset are consistent






$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setextAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextAx\_type},\mbox{type})$

$\mbox{setextAx} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ a = b ) )\qquad\mbox{thf}(\mbox{setextAx},\mbox{definition})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetAx\_type},\mbox{type})$

$\mbox{powersetAx} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) )\iff \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetAx},\mbox{definition})$

$\mbox{notinemptyset}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{notinemptyset\_type},\mbox{type})$

$\mbox{notinemptyset} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{in}@ \mbox{xx}@ \mbox{emptyset} )\qquad\mbox{thf}(\mbox{notinemptyset},\mbox{definition})$

$\mbox{setextAx} \ \Rightarrow \ ( \mbox{powersetAx} \ \Rightarrow \ ( \mbox{notinemptyset} \ \Rightarrow \ \mbox{\$false} ) )\qquad\mbox{thf}(\mbox{setext},\mbox{conjecture})$

\medskip\textbf{SEU947$\wedge$5.p} TPS problem THM15B-V2






$\mbox{cFIXPOINT}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cFIXPOINT\_type},\mbox{type})$

$\mbox{cUNIQUE\_FIXPOINT}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cUNIQUE\_FIXPOINT\_type},\mbox{type})$

$\mbox{cFIXPOINT} = ( \lambda \mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xg}@ \mbox{xx} ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{cFIXPOINT\_def},\mbox{definition})$

$\mbox{cUNIQUE\_FIXPOINT} = ( \lambda \mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cFIXPOINT}@ \mbox{xg}@ \mbox{xx}\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cFIXPOINT}@ \mbox{xg}@ \mbox{xz} ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cUNIQUE\_FIXPOINT\_def},\mbox{definition})$

$\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \exists \mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xf}\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg} ) )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cUNIQUE\_FIXPOINT}@ \mbox{xg}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{cFIXPOINT}@ \mbox{xf}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cTHM15B\_V2\_pme},\mbox{conjecture})$

\medskip\textbf{SEU598$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.subset B A $\to$ binintersect A B = B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{setextsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextsub\_type},\mbox{type})$

$\mbox{setextsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ a ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setextsub},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectI\_type},\mbox{type})$

$\mbox{binintersectI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectI},\mbox{definition})$

$\mbox{binintersectRsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectRsub\_type},\mbox{type})$

$\mbox{binintersectRsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \subseteq @ ( \mbox{binintersect}@ a@ b )@ b ) )\qquad\mbox{thf}(\mbox{binintersectRsub},\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ ( \mbox{setextsub} \ \Rightarrow \ ( \mbox{binintersectI} \ \Rightarrow \ ( \mbox{binintersectRsub} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ b@ a ) \ \Rightarrow \ ( \mbox{binintersect}@ a@ b ) = b ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectSubset}_4,\mbox{conjecture})$

\medskip\textbf{SEU738$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ ~(in x (binunion X Y)) $\to$ ~(in x X))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIL\_type},\mbox{type})$

$\mbox{binunionIL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIL},\mbox{definition})$

$\mbox{binunionIL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ x ) ) ) )\qquad\mbox{thf}(\mbox{binunionTILcontra},\mbox{conjecture})$

\medskip\textbf{SEU867$\wedge$5.p} TPS problem from FINITE-SET-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xp}{:}\;a\to \mbox{\$o},\mbox{xq}{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xp} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xq}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xq} ) ) )\qquad\mbox{thf}(\mbox{cTHM160\_pme},\mbox{conjecture})$

\medskip\textbf{SEU740$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! Z:i.in Z (powerset A) $\to$ (! x:i.in x A $\to$
 in x (binintersect X Y) $\to$ in x (binunion X Z)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIL\_type},\mbox{type})$

$\mbox{binunionIL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIL},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectEL\_type},\mbox{type})$

$\mbox{binintersectEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{definition})$

$\mbox{binunionIL} \ \Rightarrow \ ( \mbox{binintersectEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ z ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{inIntersectImpInUnion},\mbox{conjecture})$

\medskip\textbf{SEU996$\wedge$5.p} TPS problem MODULAR-THM

Every distributive lattice is modular.




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{jOIN}{:}\;a\to a\to a,\mbox{mEET}{:}\;a\to a\to a{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{jOIN}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{jOIN}@ \mbox{xx}@ ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{mEET}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) = ( \mbox{mEET}@ \mbox{xx}@ ( \mbox{mEET}@ \mbox{xy}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{mEET}@ \mbox{xy}@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{jOIN}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{xy}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{mEET}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xy} ) = \mbox{xy}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{jOIN}@ \mbox{xx}@ ( \mbox{mEET}@ \mbox{xy}@ \mbox{xz} ) ) = ( \mbox{mEET}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xz} ) )\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( \mbox{mEET}@ \mbox{xx}@ ( \mbox{jOIN}@ \mbox{xy}@ \mbox{xz} ) ) = ( \mbox{jOIN}@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{mEET}@ \mbox{xx}@ \mbox{xz} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a,\mbox{xz}{:}\;a{:}\;( ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xz} ) = \mbox{xz} \ \Rightarrow \ ( \mbox{jOIN}@ \mbox{xx}@ ( \mbox{mEET}@ \mbox{xy}@ \mbox{xz} ) ) = ( \mbox{mEET}@ ( \mbox{jOIN}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cMODULAR\_THM\_pme},\mbox{conjecture})$

\medskip\textbf{SEU704$\wedge$2.p} Conditionals

(! A:i.! phi:o.! x:i.in x A $\to$ (! y:i.in y A $\to$ singleton
 (dsetconstr A ($\wedge$ z:i.phi \& z = x | ~phi \& z = y))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{iffalseProp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{iffalseProp1\_type},\mbox{type})$

$\mbox{iffalseProp}_1 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \neg\,\mbox{xphi} \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iffalseProp}_1,\mbox{definition})$

$\mbox{iffalseProp}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{iffalseProp2\_type},\mbox{type})$

$\mbox{iffalseProp}_2 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \neg\,\mbox{xphi} \ \Rightarrow \ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) = ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{iffalseProp}_2,\mbox{definition})$

$\mbox{iftrueProp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{iftrueProp1\_type},\mbox{type})$

$\mbox{iftrueProp}_1 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{xphi} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iftrueProp}_1,\mbox{definition})$

$\mbox{iftrueProp}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{iftrueProp2\_type},\mbox{type})$

$\mbox{iftrueProp}_2 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{xphi} \ \Rightarrow \ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{iftrueProp}_2,\mbox{definition})$

$\mbox{iffalseProp}_1 \ \Rightarrow \ ( \mbox{iffalseProp}_2 \ \Rightarrow \ ( \mbox{iftrueProp}_1 \ \Rightarrow \ ( \mbox{iftrueProp}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ifSingleton},\mbox{conjecture})$

\medskip\textbf{SEU745$\wedge$2.p} Typed Set Theory - Laws for Typed Sets - DeMorgan Laws

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x (setminus A (binunion X Y)) $\to$
 in x (setminus A X))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{inComplementUnionImpNotIn}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{inComplementUnionImpNotIn1\_type},\mbox{type})$

$\mbox{inComplementUnionImpNotIn}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{inComplementUnionImpNotIn}_1,\mbox{definition})$

$\mbox{setminusI} \ \Rightarrow \ ( \mbox{inComplementUnionImpNotIn}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{inComplementUnionImpInComplement}_1,\mbox{conjecture})$

\medskip\textbf{SEU860$\wedge$5.p} TPS problem from FINITE-SET-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xp}{:}\;a\to \mbox{\$o},\mbox{xq}{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xp} ) )\mbox{ and }\exists \mbox{xt}{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xq}@ \mbox{xz}\mbox{ or }\mbox{xt}@ \mbox{xz} ) ) = \mbox{xp} ) \ \Rightarrow \ \forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xq} ) ) )\qquad\mbox{thf}(\mbox{cTHM160B\_pme},\mbox{conjecture})$

\medskip\textbf{SEU593$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.subset A B $\to$ binintersect A B = A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{setextsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextsub\_type},\mbox{type})$

$\mbox{setextsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ a ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setextsub},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersect} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersect},\mbox{definition})$

$\mbox{binintersectI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectI\_type},\mbox{type})$

$\mbox{binintersectI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectI},\mbox{definition})$

$\mbox{binintersectLsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectLsub\_type},\mbox{type})$

$\mbox{binintersectLsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \subseteq @ ( \mbox{binintersect}@ a@ b )@ a ) )\qquad\mbox{thf}(\mbox{binintersectLsub},\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ ( \mbox{setextsub} \ \Rightarrow \ ( \mbox{binintersectI} \ \Rightarrow \ ( \mbox{binintersectLsub} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( \mbox{binintersect}@ a@ b ) = a ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectSubset}_2,\mbox{conjecture})$

\medskip\textbf{SEU577$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.in B (powerset A) $\to$ subset B A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE\_type},\mbox{type})$

$\mbox{powersetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetE},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{powersetE} \ \Rightarrow \ ( \mbox{subsetI}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \subseteq @ b@ a ) ) )\qquad\mbox{thf}(\mbox{powersetE}_1,\mbox{conjecture})$

\medskip\textbf{SEU567$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.~(subset A B) $\to$ ~(A = B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \neg\,\subseteq @ a@ b \ \Rightarrow \ a\ne b )\qquad\mbox{thf}(\mbox{notequalI}_1,\mbox{conjecture})$

\medskip\textbf{SEU461$\wedge$1.p} The transitive closure of a binary relation is transitive, part 1






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{trans}@ ( \mbox{tc}@ r ) )\qquad\mbox{thf}(\mbox{transitive\_closure\_is\_transitive},\mbox{conjecture})$

\medskip\textbf{SEU487$\wedge$1.p} Local confluence does NOT imply confluence






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\exists w{:}\;\mbox{\$i},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( w\ne x\mbox{ and }w\ne y\mbox{ and }w\ne z\mbox{ and }x\ne y\mbox{ and }x\ne z\mbox{ and }y\ne z )\qquad\mbox{thf}(\mbox{four\_individuals},\mbox{hypothesis})$

$\neg\,\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{lconfl}@ r ) \ \Rightarrow \ ( \mbox{confl}@ r ) )\qquad\mbox{thf}(\mbox{local\_confluence\_does\_not\_imply\_confluence},\mbox{conjecture})$

\medskip\textbf{SEU939$\wedge$5.p} TPS problem THM112B






$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xm}_9{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}_{10}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xw}_1{:}\;\mbox{\$i}{:}\;( \mbox{xm}_9@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}@ \mbox{xw}_1\mbox{ or }\mbox{xm}_{10}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}@ \mbox{xw}_1 )\mbox{ and }\forall g{:}\;\mbox{\$i}\to \mbox{\$i},h{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \forall \mbox{xw}_1{:}\;\mbox{\$i}{:}\;( \mbox{xm}_9@ g@ \mbox{xw}_1\mbox{ or }\mbox{xm}_{10}@ g@ \mbox{xw}_1 )\mbox{ and }\forall \mbox{xw}_1{:}\;\mbox{\$i}{:}\;( \mbox{xm}_9@ h@ \mbox{xw}_1\mbox{ or }\mbox{xm}_{10}@ h@ \mbox{xw}_1 ) ) \ \Rightarrow \ ( \forall \mbox{xw}_1{:}\;\mbox{\$i}{:}\;( \mbox{xm}_9@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( g@ ( h@ \mbox{xx} ) )@ \mbox{xw}_1\mbox{ or }\mbox{xm}_{10}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( g@ ( h@ \mbox{xx} ) )@ \mbox{xw}_1 )\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( p@ y ) \ \Rightarrow \ ( p@ ( g@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM112B},\mbox{conjecture})$

\medskip\textbf{SEU920$\wedge$5.p} TPS problem FN-THM-4






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$c{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\forall f{:}\;a\to b,g{:}\;b\to c{:}\;( \forall y{:}\;c{:}\;\exists x{:}\;a{:}\;( g@ ( f@ x ) ) = y \ \Rightarrow \ \forall y{:}\;c{:}\;\exists x{:}\;b{:}\;( g@ x ) = y )\qquad\mbox{thf}(\mbox{cFN\_THM\_4\_pme},\mbox{conjecture})$

\medskip\textbf{SEU680$\wedge$2.p} Functions - Lambda Abstraction

(! A:i.! B:i.! f:i$>$i.(! x:i.in x A $\to$ in (f x) B) $\to$
 in (lam A B ($\wedge$ x:i.f x)) (funcSet A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{funcinfuncset}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcinfuncset\_type},\mbox{type})$

$\mbox{funcinfuncset} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{funcinfuncset},\mbox{definition})$

$\mbox{lam}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{lam\_type},\mbox{type})$

$\mbox{lam} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{lam},\mbox{definition})$

$\mbox{lamp}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{lamp\_type},\mbox{type})$

$\mbox{lamp} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ ( \mbox{func}@ a@ b@ ( \mbox{lam}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{lamp},\mbox{definition})$

$\mbox{funcinfuncset} \ \Rightarrow \ ( \mbox{lamp} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{lam}@ a@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) )@ ( \mbox{funcSet}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{lam2p},\mbox{conjecture})$

\medskip\textbf{SEU526$\wedge$2.p} Preliminary Notions - Equality Laws

(! A:i.nonempty A $\to$ (? x:i.in x A \& true))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{nonempty} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xx}\ne \mbox{emptyset} )\qquad\mbox{thf}(\mbox{nonempty},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ ( \mbox{setext} \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{nonempty}@ a ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\mbox{\$true} ) ) )\qquad\mbox{thf}(\mbox{nonemptyImpWitness},\mbox{conjecture})$

\medskip\textbf{SEU217+1.p} MPTP bushy problem t35\_funct\_1






$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_funct}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{relation}(\mbox{identity\_relation}(a))\qquad\mbox{fof}(\mbox{dt\_k6\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(\mbox{identity\_relation}(a))\mbox{ and }\mbox{function}(\mbox{identity\_relation}(a)) )\qquad\mbox{fof}(\mbox{fc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(b,a) \ \Rightarrow \ \mbox{apply}(\mbox{identity\_relation}(a),b) = b )\qquad\mbox{fof}(\mbox{t35\_funct}_1,\mbox{conjecture})$

$\forall a,b{:}\;( ( \mbox{relation}(b)\mbox{ and }\mbox{function}(b) ) \ \Rightarrow \ ( b = \mbox{identity\_relation}(a)\iff ( \mbox{relation\_dom}(b) = a\mbox{ and }\forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{apply}(b,c) = c ) ) ) )\qquad\mbox{fof}(\mbox{t34\_funct}_1,\mbox{axiom})$

\medskip\textbf{SEU261+1.p} MPTP bushy problem t54\_wellord1






$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{well\_ordering}(a)\iff ( \mbox{reflexive}(a)\mbox{ and }\mbox{transitive}(a)\mbox{ and }\mbox{antisymmetric}(a)\mbox{ and }\mbox{connected}(a)\mbox{ and }\mbox{well\_founded\_relation}(a) ) ) )\qquad\mbox{fof}(\mbox{d4\_wellord}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \forall c{:}\;( ( \mbox{relation}(c)\mbox{ and }\mbox{function}(c) ) \ \Rightarrow \ ( \mbox{relation\_isomorphism}(a,b,c) \ \Rightarrow \ ( ( \mbox{reflexive}(a) \ \Rightarrow \ \mbox{reflexive}(b) )\mbox{ and }( \mbox{transitive}(a) \ \Rightarrow \ \mbox{transitive}(b) )\mbox{ and }( \mbox{connected}(a) \ \Rightarrow \ \mbox{connected}(b) )\mbox{ and }( \mbox{antisymmetric}(a) \ \Rightarrow \ \mbox{antisymmetric}(b) )\mbox{ and }( \mbox{well\_founded\_relation}(a) \ \Rightarrow \ \mbox{well\_founded\_relation}(b) ) ) ) ) ) )\qquad\mbox{fof}(\mbox{t53\_wellord}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \forall c{:}\;( ( \mbox{relation}(c)\mbox{ and }\mbox{function}(c) ) \ \Rightarrow \ ( ( \mbox{well\_ordering}(a)\mbox{ and }\mbox{relation\_isomorphism}(a,b,c) ) \ \Rightarrow \ \mbox{well\_ordering}(b) ) ) ) )\qquad\mbox{fof}(\mbox{t54\_wellord}_1,\mbox{conjecture})$

\medskip\textbf{SEU869$\wedge$5.p} TPS problem from FINITE-SET-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cB}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cC}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$( \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{cC}@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{cC} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cB}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cC}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a{:}\;( ( \mbox{cB}@ \mbox{xt} ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}@ \mbox{xz}\mbox{ or }\mbox{xt} = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{cB} ) )\qquad\mbox{thf}(\mbox{cTHM531C\_pme},\mbox{conjecture})$

\medskip\textbf{SEU465$\wedge$1.p} The transitive closure of a binary relation is transitive, part 5






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},s{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{trans}@ s\mbox{ and }\mbox{subrel}@ r@ s\mbox{ and }( ( \mbox{trans}@ s\mbox{ and }\mbox{subrel}@ r@ s ) \ \Rightarrow \ ( s@ x@ y ) ) ) \ \Rightarrow \ ( s@ x@ y ) )\qquad\mbox{thf}(\mbox{transitive\_closure\_is\_transitive}_5,\mbox{conjecture})$

\medskip\textbf{SEU153+1.p} MPTP bushy problem l25\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{disjoint}(\mbox{singleton}(a),b)\mbox{ and }\mbox{in}(a,b)\qquad\mbox{fof}(\mbox{l25\_zfmisc}_1,\mbox{conjecture})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

\medskip\textbf{SEU967$\wedge$5.p} TPS problem from FUNCTION-THMS






$\mbox{cY}_0{:}\;( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cY}_0,\mbox{type})$

$\mbox{cG}_0{:}\;( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cG}_0,\mbox{type})$

$\mbox{cP}_0{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP}_0,\mbox{type})$

$\mbox{cH}_0{:}\;( ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cH}_0,\mbox{type})$

$\neg\,\forall \mbox{xm}_5{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$o}{:}\;( \exists \mbox{xw}_2{:}\;\mbox{\$i}{:}\;( \mbox{xm}_5@ ( \mbox{cG}_0@ \lambda \mbox{xw}_{1144}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists \mbox{xw}_{20}{:}\;\mbox{\$i}{:}\;( \mbox{xm}_5@ \mbox{xw}_{1144}@ \mbox{xw}_{20} ) )@ \mbox{xw}_2 )\mbox{ and }\exists \mbox{xw}_2{:}\;\mbox{\$i}{:}\;( \mbox{xm}_5@ ( \mbox{cH}_0@ \lambda \mbox{xw}_{1144}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists \mbox{xw}_{20}{:}\;\mbox{\$i}{:}\;( \mbox{xm}_5@ \mbox{xw}_{1144}@ \mbox{xw}_{20} ) )@ \mbox{xw}_2 )\mbox{ and }( \neg\,\exists \mbox{xw}_2{:}\;\mbox{\$i}{:}\;( \mbox{xm}_5@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{cG}_0@ \lambda \mbox{xw}_{1144}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists \mbox{xw}_{20}{:}\;\mbox{\$i}{:}\;( \mbox{xm}_5@ \mbox{xw}_{1144}@ \mbox{xw}_{20} )@ ( \mbox{cH}_0@ \lambda \mbox{xw}_{1144}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists \mbox{xw}_{20}{:}\;\mbox{\$i}{:}\;( \mbox{xm}_5@ \mbox{xw}_{1144}@ \mbox{xw}_{20} )@ \mbox{xx} ) )@ \mbox{xw}_2 )\mbox{ or }( \mbox{cP}_0@ ( \mbox{cY}_0@ \lambda \mbox{xw}_{1144}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists \mbox{xw}_2{:}\;\mbox{\$i}{:}\;( \mbox{xm}_5@ \mbox{xw}_{1144}@ \mbox{xw}_2 ) )\mbox{ and }\neg\,\mbox{cP}_0@ ( \mbox{cG}_0@ \lambda \mbox{xw}_{1144}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists \mbox{xw}_2{:}\;\mbox{\$i}{:}\;( \mbox{xm}_5@ \mbox{xw}_{1144}@ \mbox{xw}_2 )@ ( \mbox{cY}_0@ \lambda \mbox{xw}_{1144}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\exists \mbox{xw}_2{:}\;\mbox{\$i}{:}\;( \mbox{xm}_5@ \mbox{xw}_{1144}@ \mbox{xw}_2 ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cBUGWFFA\_pme},\mbox{conjecture})$

\medskip\textbf{SEU921$\wedge$5.p} TPS problem THM588LEM2

Another possible lemma for THM588, for manipulating composite 
 functions.




$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$h{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(h,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx} ) = \mbox{xy} \ \Rightarrow \ ( h@ \mbox{xy} ) = ( f@ \mbox{xx} ) ) \ \Rightarrow \ ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( h@ ( g@ \mbox{xx} ) ) ) = f\qquad\mbox{thf}(\mbox{cTHM588LEM2\_pme},\mbox{conjecture})$

\medskip\textbf{SEU136+1.p} MPTP bushy problem t40\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_difference}(\mbox{set\_union}_2(a,b),b) = \mbox{set\_difference}(a,b)\qquad\mbox{fof}(\mbox{t40\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU537$\wedge$2.p} Preliminary Notions - Bounded Quantifier Laws

(! A:i.! phi:i$>$o.(! x:i.in x A $\to$ ~(phi x)) $\to$
 ~(? x:i.in x A \& phi x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in},\mbox{type})$

$\forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \neg\,\mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ \neg\,\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\mbox{xphi}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{quantDeMorgan}_2,\mbox{conjecture})$

\medskip\textbf{SEU553$\wedge$2.p} A simple congruence property of powerset

(forall A:i.forall B:i.A = B $\to$ powerset A = powerset B)




$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset},\mbox{type})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ ( \mbox{powerset}@ a ) = ( \mbox{powerset}@ b ) )\qquad\mbox{thf}(\mbox{powerset\_\_Cong},\mbox{conjecture})$

\medskip\textbf{SEU743$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! Z:i.in Z (powerset A) $\to$ in (binintersect X Y)
 (powerset (binintersect (binunion X Z) (binunion Y Z))))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectT\_lem\_type},\mbox{type})$

$\mbox{binintersectT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{binintersect}@ x@ y )@ ( \mbox{powerset}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectT\_lem},\mbox{definition})$

$\mbox{binunionT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionT\_lem\_type},\mbox{type})$

$\mbox{binunionT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{binunion}@ x@ y )@ ( \mbox{powerset}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionT\_lem},\mbox{definition})$

$\mbox{powersetTI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetTI1\_type},\mbox{type})$

$\mbox{powersetTI}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \mbox{in}@ x@ ( \mbox{powerset}@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{powersetTI}_1,\mbox{definition})$

$\mbox{inIntersectImpInIntersectUnions}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{inIntersectImpInIntersectUnions\_type},\mbox{type})$

$\mbox{inIntersectImpInIntersectUnions} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ ( \mbox{binunion}@ x@ z )@ ( \mbox{binunion}@ y@ z ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{inIntersectImpInIntersectUnions},\mbox{definition})$

$\mbox{binintersectT\_lem} \ \Rightarrow \ ( \mbox{binunionT\_lem} \ \Rightarrow \ ( \mbox{powersetTI}_1 \ \Rightarrow \ ( \mbox{inIntersectImpInIntersectUnions} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{binintersect}@ x@ y )@ ( \mbox{powerset}@ ( \mbox{binintersect}@ ( \mbox{binunion}@ x@ z )@ ( \mbox{binunion}@ y@ z ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{intersectInPowersetIntersectUnions},\mbox{conjecture})$

\medskip\textbf{SEU833$\wedge$5.p} TPS problem GAZING-THM24

Trybulec's 61st Boolean property of sets




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( s@ \mbox{xx}\mbox{ and }\mbox{\$false} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} )\qquad\mbox{thf}(\mbox{cGAZING\_THM24\_pme},\mbox{conjecture})$

\medskip\textbf{SEU190+1.p} MPTP bushy problem t71\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( b = \mbox{identity\_relation}(a)\iff \forall c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(c,d),b)\iff ( \mbox{in}(c,a)\mbox{ and }c = d ) ) ) )\qquad\mbox{fof}(\mbox{d10\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) ) )\qquad\mbox{fof}(\mbox{d5\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{relation}(\mbox{identity\_relation}(a))\qquad\mbox{fof}(\mbox{dt\_k6\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{fc6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_rng}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_rng}(a)) ) )\qquad\mbox{fof}(\mbox{fc8\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation\_dom}(\mbox{identity\_relation}(a)) = a\mbox{ and }\mbox{relation\_rng}(\mbox{identity\_relation}(a)) = a )\qquad\mbox{fof}(\mbox{t71\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU674$\wedge$2.p} Functions - Application

(! A:i.! B:i.! f:i.func A B f $\to$ (! x:i.in x A $\to$
 in (ap A B f x) B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{apProp}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{apProp\_type},\mbox{type})$

$\mbox{apProp} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setunion}@ ( \mbox{dsetconstr}@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) ) )@ b ) ) ) )\qquad\mbox{thf}(\mbox{apProp},\mbox{definition})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{ap} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{dsetconstr}@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{ap},\mbox{definition})$

$\mbox{apProp} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} )@ b ) ) )\qquad\mbox{thf}(\mbox{app},\mbox{conjecture})$

\medskip\textbf{SEU663$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! A:i.! B:i.! x:i.in x A $\to$ (! y:i.in y B $\to$
 ksnd (kpair x y) = y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{ksnd}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ksnd\_type},\mbox{type})$

$\mbox{ksndpairEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ksndpairEq\_type},\mbox{type})$

$\mbox{ksndpairEq} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{ksnd}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{ksndpairEq},\mbox{definition})$

$\mbox{cartprodmempaircEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodmempaircEq\_type},\mbox{type})$

$\mbox{cartprodmempaircEq} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{cartprodmempaircEq},\mbox{definition})$

$\mbox{ksndpairEq} \ \Rightarrow \ ( \mbox{cartprodmempaircEq} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{ksnd}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cartprodsndpairEq},\mbox{conjecture})$

\medskip\textbf{SEU851$\wedge$5.p} TPS problem GAZING-THM42






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( ( s@ \mbox{xz}\mbox{ and }\neg\,t@ \mbox{xz} )\mbox{ or }( t@ \mbox{xz}\mbox{ and }\neg\,s@ \mbox{xz} ) ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( t@ \mbox{xz}\mbox{ and }\neg\,s@ \mbox{xz} )\mbox{ or }( s@ \mbox{xz}\mbox{ and }\neg\,t@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM42\_pme},\mbox{conjecture})$

\medskip\textbf{SEU636$\wedge$2.p} Ordered Pairs - Singletons

(! x:i.setunion (setadjoin x emptyset) = x)




$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{setextsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextsub\_type},\mbox{type})$

$\mbox{setextsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ a ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setextsub},\mbox{definition})$

$\mbox{setunionsingleton}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionsingleton1\_type},\mbox{type})$

$\mbox{setunionsingleton}_1 = ( \forall a{:}\;\mbox{\$i}{:}\;( \subseteq @ ( \mbox{setunion}@ ( \mbox{setadjoin}@ a@ \mbox{emptyset} ) )@ a ) )\qquad\mbox{thf}(\mbox{setunionsingleton}_1,\mbox{definition})$

$\mbox{setunionsingleton}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionsingleton2\_type},\mbox{type})$

$\mbox{setunionsingleton}_2 = ( \forall a{:}\;\mbox{\$i}{:}\;( \subseteq @ a@ ( \mbox{setunion}@ ( \mbox{setadjoin}@ a@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{setunionsingleton}_2,\mbox{definition})$

$\mbox{setextsub} \ \Rightarrow \ ( \mbox{setunionsingleton}_1 \ \Rightarrow \ ( \mbox{setunionsingleton}_2 \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) = \mbox{xx} ) )\qquad\mbox{thf}(\mbox{setunionsingleton},\mbox{conjecture})$

\medskip\textbf{SEU571$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! x:i.! A:i.subset A (setadjoin x A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIR\_type},\mbox{type})$

$\mbox{setadjoinIR} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{setadjoinIR},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{setadjoinIR} \ \Rightarrow \ ( \mbox{subsetI}_1 \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i}{:}\;( \subseteq @ a@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{setadjoinSub},\mbox{conjecture})$

\medskip\textbf{SEU019+1.p} Functions and their basic properties, theorem 52






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{one\_to\_one}(a)\iff \forall b,c{:}\;( ( \mbox{in}(b,\mbox{relation\_dom}(a))\mbox{ and }\mbox{in}(c,\mbox{relation\_dom}(a))\mbox{ and }\mbox{apply}(a,b) = \mbox{apply}(a,c) ) \ \Rightarrow \ b = c ) ) )\qquad\mbox{fof}(\mbox{d8\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{relation}(\mbox{identity\_relation}(a))\qquad\mbox{fof}(\mbox{dt\_k6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(\mbox{identity\_relation}(a))\mbox{ and }\mbox{function}(\mbox{identity\_relation}(a)) )\qquad\mbox{fof}(\mbox{fc2\_funct}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(b)\mbox{ and }\mbox{function}(b) ) \ \Rightarrow \ ( b = \mbox{identity\_relation}(a)\iff ( \mbox{relation\_dom}(b) = a\mbox{ and }\forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{apply}(b,c) = c ) ) ) )\qquad\mbox{fof}(\mbox{t34\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a{:}\;\mbox{one\_to\_one}(\mbox{identity\_relation}(a))\qquad\mbox{fof}(\mbox{t52\_funct}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU581$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! phi:i$>$o.subset (dsetconstr A ($\wedge$ x:i.phi x)) A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{powersetE}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE1\_type},\mbox{type})$

$\mbox{powersetE}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \subseteq @ b@ a ) ) )\qquad\mbox{thf}(\mbox{powersetE}_1,\mbox{definition})$

$\mbox{sepInPowerset}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{sepInPowerset\_type},\mbox{type})$

$\mbox{sepInPowerset} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{in}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) )@ ( \mbox{powerset}@ a ) ) )\qquad\mbox{thf}(\mbox{sepInPowerset},\mbox{definition})$

$\mbox{powersetE}_1 \ \Rightarrow \ ( \mbox{sepInPowerset} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \subseteq @ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) )@ a ) )\qquad\mbox{thf}(\mbox{sepSubset},\mbox{conjecture})$

\medskip\textbf{SEU871$\wedge$5.p} TPS problem from FINITE-SET-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cB}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$\mbox{cC}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$( \forall p{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \forall e{:}\;a\to \mbox{\$o}{:}\;( \neg\,\exists \mbox{xt}{:}\;a{:}\;( e@ \mbox{xt} ) \ \Rightarrow \ ( p@ e ) )\mbox{ and }\forall y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a,z{:}\;a\to \mbox{\$o}{:}\;( ( p@ y\mbox{ and }\forall \mbox{xx}_{30}{:}\;a{:}\;( ( z@ \mbox{xx}_{30} )\iff ( y@ \mbox{xx}_{30}\mbox{ or }\mbox{xx}_{30} = \mbox{xx} ) ) ) \ \Rightarrow \ ( p@ z ) ) ) \ \Rightarrow \ ( p@ \mbox{cC} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{cB}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cC}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall p{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \forall e{:}\;a\to \mbox{\$o}{:}\;( \neg\,\exists \mbox{xt}{:}\;a{:}\;( e@ \mbox{xt} ) \ \Rightarrow \ ( p@ e ) )\mbox{ and }\forall y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a,z{:}\;a\to \mbox{\$o}{:}\;( ( p@ y\mbox{ and }\forall \mbox{xx}_{31}{:}\;a{:}\;( ( z@ \mbox{xx}_{31} )\iff ( y@ \mbox{xx}_{31}\mbox{ or }\mbox{xx}_{31} = \mbox{xx} ) ) ) \ \Rightarrow \ ( p@ z ) ) ) \ \Rightarrow \ ( p@ \mbox{cB} ) )\qquad\mbox{thf}(\mbox{cTHM531D\_pme},\mbox{conjecture})$

\medskip\textbf{SEU907$\wedge$5.p} TPS problem from FUNS-AND-SETS-THMS






$\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \exists \mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xf}\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg} ) )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xg}@ \mbox{xx} ) = \mbox{xx} ) = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\mbox{xx} = \mbox{xy}@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM15\_pme},\mbox{conjecture})$

\medskip\textbf{SEU777$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! x:i.in x A $\to$ (! y:i.in y A $\to$
 in (kpair y x) (breln1invset A R) $\to$ in (kpair x y) R)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dpsetconstrER\_type},\mbox{type})$

$\mbox{dpsetconstrER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xz}@ \mbox{xu} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrER},\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{breln1invset}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1invset\_type},\mbox{type})$

$\mbox{breln1invset} = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xy}@ \mbox{xx} )@ r ) ) )\qquad\mbox{thf}(\mbox{breln1invset},\mbox{definition})$

$\mbox{dpsetconstrER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xy}@ \mbox{xx} )@ ( \mbox{breln1invset}@ a@ r ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1invE},\mbox{conjecture})$

\medskip\textbf{SEU815$\wedge$2.p} Transitive Sets

(! X:i.(! x:i.in x X $\to$ transitiveset x) $\to$
 transitiveset (setunion X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{setunionI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionI\_type},\mbox{type})$

$\mbox{setunionI} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{setunionI},\mbox{definition})$

$\mbox{setunionE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionE\_type},\mbox{type})$

$\mbox{setunionE} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( \forall b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ \mbox{xphi} ) ) \ \Rightarrow \ \mbox{xphi} ) ) )\qquad\mbox{thf}(\mbox{setunionE},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{transitiveset} = ( \lambda a{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ ( \subseteq @ x@ a ) ) )\qquad\mbox{thf}(\mbox{transitiveset},\mbox{definition})$

$\mbox{transitivesetOp}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{transitivesetOp2\_type},\mbox{type})$

$\mbox{transitivesetOp}_2 = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{transitiveset}@ x ) \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{transitivesetOp}_2,\mbox{definition})$

$\mbox{setunionI} \ \Rightarrow \ ( \mbox{setunionE} \ \Rightarrow \ ( \mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{transitivesetOp}_2 \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{transitiveset}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{transitiveset}@ ( \mbox{setunion}@ x ) ) ) ) ) )\qquad\mbox{thf}(\mbox{setunionTransitive},\mbox{conjecture})$

\medskip\textbf{SEU523$\wedge$2.p} Preliminary Notions - Power Sets and Unions

(! A:i.! x:i.in x (setunion A) $\to$ (! phi:o.(! B:i.in x B $\to$
 in B A $\to$ phi) $\to$ phi))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{setunionAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionAx\_type},\mbox{type})$

$\mbox{setunionAx} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) )\iff \exists b{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b\mbox{ and }\mbox{in}@ b@ a ) ) )\qquad\mbox{thf}(\mbox{setunionAx},\mbox{definition})$

$\mbox{setunionAx} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( \forall b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ \mbox{xphi} ) ) \ \Rightarrow \ \mbox{xphi} ) )\qquad\mbox{thf}(\mbox{setunionE},\mbox{conjecture})$

\medskip\textbf{SEU522$\wedge$2.p} Preliminary Notions - Power Sets and Unions

(! A:i.! x:i.! B:i.in x B $\to$ in B A $\to$ in x (setunion A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{setunionAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionAx\_type},\mbox{type})$

$\mbox{setunionAx} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) )\iff \exists b{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b\mbox{ and }\mbox{in}@ b@ a ) ) )\qquad\mbox{thf}(\mbox{setunionAx},\mbox{definition})$

$\mbox{setunionAx} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) ) )\qquad\mbox{thf}(\mbox{setunionI},\mbox{conjecture})$

\medskip\textbf{SEU148+1.p} MPTP bushy problem t6\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{singleton}(a)\ne \mbox{empty\_set}\qquad\mbox{fof}(\mbox{l1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq \mbox{singleton}(b)\iff ( a = \mbox{empty\_set}\mbox{ or }a = \mbox{singleton}(b) ) )\qquad\mbox{fof}(\mbox{l4\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq \mbox{singleton}(b) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t6\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU807$\wedge$2.p} The Foundation Axiom

(! A:i.! B:i.in A B $\to$ ~(in B A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{foundationAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{foundationAx\_type},\mbox{type})$

$\mbox{foundationAx} = ( \forall a{:}\;\mbox{\$i}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \exists b{:}\;\mbox{\$i}{:}\;( \mbox{in}@ b@ a\mbox{ and }\neg\,\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b\mbox{ and }\mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{foundationAx},\mbox{definition})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{setadjoinIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIR\_type},\mbox{type})$

$\mbox{setadjoinIR} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{setadjoinIR},\mbox{definition})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\mbox{upairset2E}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{upairset2E\_type},\mbox{type})$

$\mbox{upairset2E} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) \ \Rightarrow \ ( \mbox{xz} = \mbox{xx}\mbox{ or }\mbox{xz} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{upairset2E},\mbox{definition})$

$\mbox{foundationAx} \ \Rightarrow \ ( \mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{setadjoinIR} \ \Rightarrow \ ( \mbox{in\_\_Cong} \ \Rightarrow \ ( \mbox{upairset2E} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ b ) \ \Rightarrow \ \neg\,\mbox{in}@ b@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{notinself}_2,\mbox{conjecture})$

\medskip\textbf{SEU168+1.p} MPTP bushy problem t136\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{powerset}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c\subseteq a ) )\qquad\mbox{fof}(\mbox{d1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{in}(a,b)\mbox{ and }\forall c,d{:}\;( ( \mbox{in}(c,b)\mbox{ and }d\subseteq c ) \ \Rightarrow \ \mbox{in}(d,b) )\mbox{ and }\forall c{:}\;( \mbox{in}(c,b) \ \Rightarrow \ \mbox{in}(\mbox{powerset}(c),b) )\mbox{ and }\forall c{:}\;\neg\,c\subseteq b\mbox{ and }\neg\,\mbox{are\_equipotent}(c,b)\mbox{ and }\neg\,\mbox{in}(c,b) )\qquad\mbox{fof}(\mbox{t136\_zfmisc}_1,\mbox{conjecture})$

$\forall a{:}\;\exists b{:}\;( \mbox{in}(a,b)\mbox{ and }\forall c,d{:}\;( ( \mbox{in}(c,b)\mbox{ and }d\subseteq c ) \ \Rightarrow \ \mbox{in}(d,b) )\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\forall d{:}\;\neg\,\mbox{in}(d,b)\mbox{ and }\forall e{:}\;( e\subseteq c \ \Rightarrow \ \mbox{in}(e,d) )\mbox{ and }\forall c{:}\;\neg\,c\subseteq b\mbox{ and }\neg\,\mbox{are\_equipotent}(c,b)\mbox{ and }\neg\,\mbox{in}(c,b) )\qquad\mbox{fof}(\mbox{t9\_tarski},\mbox{axiom})$

\medskip\textbf{SEU406+2.p} The Operation of Addition of Relational Structures T01






$\mbox{include}(\mbox{'Axioms/SET007/SET007+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+1.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+2.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+3.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+4.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+6.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+7.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+9.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+10.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+11.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+13.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+14.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+16.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+17.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+19.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+20.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+24.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+25.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+26.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+31.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+35.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+54.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+55.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+59.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+60.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+64.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+80.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+200.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+205.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+218.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+242.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+295.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+335.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+480.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+481.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+483.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+484.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+485.ax'})$

$\mbox{include}(\mbox{'Axioms/SET007/SET007+492.ax'})$

$\forall a,b{:}\;( ( \mbox{l1\_orders}_2(a)\mbox{ and }\mbox{l1\_orders}_2(b) ) \ \Rightarrow \ ( \mbox{v1\_orders}_2(\mbox{k1\_latsum}_1(a,b))\mbox{ and }\mbox{l1\_orders}_2(\mbox{k1\_latsum}_1(a,b)) ) )\qquad\mbox{fof}(\mbox{dt\_k1\_latsum}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;\neg\,\mbox{r2\_hidden}(a,\mbox{k2\_xboole}_0(c,d))\mbox{ and }\mbox{r2\_hidden}(b,\mbox{k2\_xboole}_0(c,d))\mbox{ and }\neg\,\mbox{r2\_hidden}(a,\mbox{k4\_xboole}_0(c,d))\mbox{ and }\mbox{r2\_hidden}(b,\mbox{k4\_xboole}_0(c,d))\mbox{ and }\neg\,\mbox{r2\_hidden}(a,d)\mbox{ and }\mbox{r2\_hidden}(b,d)\mbox{ and }\neg\,\mbox{r2\_hidden}(a,\mbox{k4\_xboole}_0(c,d))\mbox{ and }\mbox{r2\_hidden}(b,d)\mbox{ and }\neg\,\mbox{r2\_hidden}(a,d)\mbox{ and }\mbox{r2\_hidden}(b,\mbox{k4\_xboole}_0(c,d))\qquad\mbox{fof}(\mbox{t1\_latsum}_1,\mbox{conjecture})$

\medskip\textbf{SEU229+1.p} MPTP bushy problem t3\_ordinal1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( d = \mbox{unordered\_triple}(a,b,c)\iff \forall e{:}\;( \mbox{in}(e,d)\iff \neg\,e\ne a\mbox{ and }e\ne b\mbox{ and }e\ne c ) )\qquad\mbox{fof}(\mbox{d1\_enumset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_enumset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc4\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{in}(b,c)\mbox{ and }\mbox{in}(c,a)\qquad\mbox{fof}(\mbox{t3\_ordinal}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\forall d{:}\;\neg\,\mbox{in}(d,b)\mbox{ and }\mbox{in}(d,c)\qquad\mbox{fof}(\mbox{t7\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU188+1.p} MPTP bushy problem t64\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) ) )\qquad\mbox{fof}(\mbox{d5\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{fc6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_rng}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_rng}(a)) ) )\qquad\mbox{fof}(\mbox{fc8\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \forall b,c{:}\;\neg\,\mbox{in}(\mbox{ordered\_pair}(b,c),a) \ \Rightarrow \ a = \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{t56\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( ( \mbox{relation\_dom}(a) = \mbox{empty\_set}\mbox{ or }\mbox{relation\_rng}(a) = \mbox{empty\_set} ) \ \Rightarrow \ a = \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{t64\_relat}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU826$\wedge$1.p} About sets 1






$\mbox{seteq}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{seteq\_type},\mbox{type})$

$\mbox{seteq} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall u{:}\;\mbox{\$i}{:}\;( ( x@ u )\iff ( y@ u ) ) )\qquad\mbox{thf}(\mbox{seteq},\mbox{definition})$

$u{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{u\_type},\mbox{type})$

$u = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},u{:}\;\mbox{\$i}{:}\;( x@ u\mbox{ or }y@ u ) )\qquad\mbox{thf}(u,\mbox{definition})$

$n{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{n\_type},\mbox{type})$

$n = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},u{:}\;\mbox{\$i}{:}\;( x@ u\mbox{ and }y@ u ) )\qquad\mbox{thf}(n,\mbox{definition})$

$a{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$c{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\mbox{seteq}@ ( u@ a@ ( n@ b@ c ) )@ ( n@ ( u@ a@ b )@ ( u@ a@ c ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SEU682$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! R:i.breln A B R $\to$ (! phi:i$>$o.(! x:i.in x A $\to$
 (! y:i.in y B $\to$ in (kpair x y) R $\to$ phi (kpair x y))) $\to$
 (! x:i.in x R $\to$ phi x)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{brelnall}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{brelnall1\_type},\mbox{type})$

$\mbox{brelnall}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ r ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{xphi}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ r ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{brelnall}_1,\mbox{definition})$

$\mbox{brelnall}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ r ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{xphi}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ r ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{brelnall}_2,\mbox{conjecture})$

\medskip\textbf{SEU165+3.p} Basic properties of sets, theorem 106






$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{t106\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU802$\wedge$2.p} More about Functions - Surjective Functions

(! x:i.! y:i.! f:i.in f (surjFuncSet x y) $\to$ surjective x y f)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{surjective}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{surjective\_type},\mbox{type})$

$\mbox{surjective} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xy} ) = \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{surjective},\mbox{definition})$

$\mbox{surjFuncSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{surjFuncSet\_type},\mbox{type})$

$\mbox{surjFuncSet} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{funcSet}@ a@ b )@ \lambda \mbox{xf}{:}\;\mbox{\$i}{:}\;( \mbox{surjective}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{surjFuncSet},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{surjFuncSet}@ \mbox{xx}@ \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{surjective}@ \mbox{xx}@ \mbox{xy}@ \mbox{xf} ) )\qquad\mbox{thf}(\mbox{surjFuncSetFuncSurj},\mbox{conjecture})$

\medskip\textbf{SEU141+1.p} MPTP bushy problem t83\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\neg\,\exists c{:}\;\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t4\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_difference}(a,b) = a )\qquad\mbox{fof}(\mbox{t83\_xboole}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU539$\wedge$2.p} Preliminary Notions - Bounded Quantifier Laws

(! A:i.! phi:i$>$o.(? x:i.in x A \& ~(phi x)) $\to$ ~(! x:i.in x A $\to$
 phi x))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in},\mbox{type})$

$\forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\neg\,\mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ \neg\,\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{quantDeMorgan}_4,\mbox{conjecture})$

\medskip\textbf{SEU525$\wedge$2.p} Preliminary Notions - Equality Laws

(! phi:i$>$o.exu ($\wedge$ x:i.phi x) $\to$ (? x:i.! y:i.phi y $<$$\to$ y = x))




$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu},\mbox{definition})$

$\mbox{exuE}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{exuE1\_type},\mbox{type})$

$\mbox{exuE}_1 = ( \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{exuE}_1,\mbox{definition})$

$\mbox{exuE}_1 \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} )\iff \mbox{xy} = \mbox{xx} ) )\qquad\mbox{thf}(\mbox{exuE}_2,\mbox{conjecture})$

\medskip\textbf{SEU559$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.(! x:i.in x A $\to$ in x B) $\to$ subset A B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\subseteq  = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{subset},\mbox{definition})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{conjecture})$

\medskip\textbf{SEU159+3.p} Basic properties of sets, theorem 38






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{unordered\_pair}(a,b)\subseteq c\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,c) ) )\qquad\mbox{fof}(\mbox{t38\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU498$\wedge$1.p} Every partial order induces a strict (partial) order






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{po}@ r ) \ \Rightarrow \ ( \mbox{so}@ \lambda x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( r@ x@ y\mbox{ and }x\ne y ) ) )\qquad\mbox{thf}(\mbox{partial\_order\_induces\_strict\_order},\mbox{conjecture})$

\medskip\textbf{SEU762$\wedge$2.p} Typed Set Theory - First Wizard of Oz Examples - WoZ1 Lemmas

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! Z:i.in Z (powerset A) $\to$ (! W:i.in W (powerset A) $\to$
 subset X Z $\to$ subset Y W $\to$ subset (binintersect X Y)
 (binintersect Z W)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectT\_lem\_type},\mbox{type})$

$\mbox{binintersectT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{binintersect}@ x@ y )@ ( \mbox{powerset}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectT\_lem},\mbox{definition})$

$\mbox{woz13rule}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{woz13rule1\_type},\mbox{type})$

$\mbox{woz13rule}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ x@ z ) \ \Rightarrow \ ( \subseteq @ ( \mbox{binintersect}@ x@ y )@ z ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz13rule}_1,\mbox{definition})$

$\mbox{woz13rule}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{woz13rule2\_type},\mbox{type})$

$\mbox{woz13rule}_2 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ y@ z ) \ \Rightarrow \ ( \subseteq @ ( \mbox{binintersect}@ x@ y )@ z ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz13rule}_2,\mbox{definition})$

$\mbox{woz13rule}_3{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{woz13rule3\_type},\mbox{type})$

$\mbox{woz13rule}_3 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ x@ y ) \ \Rightarrow \ ( ( \subseteq @ x@ z ) \ \Rightarrow \ ( \subseteq @ x@ ( \mbox{binintersect}@ y@ z ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz13rule}_3,\mbox{definition})$

$\mbox{binintersectT\_lem} \ \Rightarrow \ ( \mbox{woz13rule}_1 \ \Rightarrow \ ( \mbox{woz13rule}_2 \ \Rightarrow \ ( \mbox{woz13rule}_3 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall w{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ w@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ x@ z ) \ \Rightarrow \ ( ( \subseteq @ y@ w ) \ \Rightarrow \ ( \subseteq @ ( \mbox{binintersect}@ x@ y )@ ( \mbox{binintersect}@ z@ w ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz13rule}_4,\mbox{conjecture})$

\medskip\textbf{SEU576$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.subset B A $\to$ in B (powerset A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{powersetI} \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ b@ a ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI}_1,\mbox{conjecture})$

\medskip\textbf{SEU664$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! A:i.! B:i.! u:i.in u (cartprod A B) $\to$ kpair (kfst u)
 (ksnd u) = u)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprodmempair}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodmempair1\_type},\mbox{type})$

$\mbox{cartprodmempair}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprodmempair}_1,\mbox{definition})$

$\mbox{kfst}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kfst\_type},\mbox{type})$

$\mbox{ksnd}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ksnd\_type},\mbox{type})$

$\mbox{cartprodfstpairEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodfstpairEq\_type},\mbox{type})$

$\mbox{cartprodfstpairEq} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{kfst}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cartprodfstpairEq},\mbox{definition})$

$\mbox{cartprodsndpairEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{cartprodsndpairEq\_type},\mbox{type})$

$\mbox{cartprodsndpairEq} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{ksnd}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cartprodsndpairEq},\mbox{definition})$

$\mbox{cartprodmempair}_1 \ \Rightarrow \ ( \mbox{cartprodfstpairEq} \ \Rightarrow \ ( \mbox{cartprodsndpairEq} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xu}@ ( \mbox{cartprod}@ a@ b ) ) \ \Rightarrow \ ( \mbox{kpair}@ ( \mbox{kfst}@ \mbox{xu} )@ ( \mbox{ksnd}@ \mbox{xu} ) ) = \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{cartprodpairsurjEq},\mbox{conjecture})$

\medskip\textbf{SEU720$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! x:i.in x A $\to$
 ~(in x (setminus A X)) $\to$ in x X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{setminusI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) )\qquad\mbox{thf}(\mbox{complementTE}_1,\mbox{conjecture})$

\medskip\textbf{SEU585$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.! x:i.! phi:o.in x (binunion A B) $\to$
 (in x A $\to$ phi) $\to$ (in x B $\to$ phi) $\to$ phi)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{setadjoinE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinE\_type},\mbox{type})$

$\mbox{setadjoinE} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( \mbox{xy} = \mbox{xx} \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{setadjoinE},\mbox{definition})$

$\mbox{setunionE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionE\_type},\mbox{type})$

$\mbox{setunionE} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( \forall b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ \mbox{xphi} ) ) \ \Rightarrow \ \mbox{xphi} ) ) )\qquad\mbox{thf}(\mbox{setunionE},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunion} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{binunion},\mbox{definition})$

$\mbox{setadjoinE} \ \Rightarrow \ ( \mbox{setunionE} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionEcases},\mbox{conjecture})$

\medskip\textbf{SEU515$\wedge$2.p} Preliminary Notions - Adjoining Elements to Sets

(! x:i.! A:i.! y:i.in y (setadjoin x A) $\to$ (! phi:o.(y = x $\to$ phi)
 $\to$ (in y A $\to$ phi) $\to$ phi))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinAx\_type},\mbox{type})$

$\mbox{setadjoinAx} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) )\iff ( \mbox{xy} = \mbox{xx}\mbox{ or }\mbox{in}@ \mbox{xy}@ a ) ) )\qquad\mbox{thf}(\mbox{setadjoinAx},\mbox{definition})$

$\mbox{setadjoinAx} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( \mbox{xy} = \mbox{xx} \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) )\qquad\mbox{thf}(\mbox{setadjoinE},\mbox{conjecture})$

\medskip\textbf{SEU744$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x (setminus A (binunion X Y)) $\to$ ~(in x X))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{binunionTILcontra}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionTILcontra\_type},\mbox{type})$

$\mbox{binunionTILcontra} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionTILcontra},\mbox{definition})$

$\mbox{setminusER} \ \Rightarrow \ ( \mbox{binunionTILcontra} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{inComplementUnionImpNotIn}_1,\mbox{conjecture})$

\medskip\textbf{SEU890$\wedge$5.p} TPS problem THM29






$c{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cG}{:}\;c\to b\qquad\mbox{thf}(\mbox{cG},\mbox{type})$

$\mbox{cF}{:}\;b\to a\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\mbox{cS}{:}\;c\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cS},\mbox{type})$

$\forall \mbox{xx}{:}\;a{:}\;( \exists \mbox{xt}{:}\;b{:}\;( \exists \mbox{xt}_0{:}\;c{:}\;( \mbox{cS}@ \mbox{xt}_0\mbox{ and }\mbox{xt} = ( \mbox{cG}@ \mbox{xt}_0 ) )\mbox{ and }\mbox{xx} = ( \mbox{cF}@ \mbox{xt} ) )\iff \exists \mbox{xt}{:}\;c{:}\;( \mbox{cS}@ \mbox{xt}\mbox{ and }\mbox{xx} = ( \mbox{cF}@ ( \mbox{cG}@ \mbox{xt} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM29\_pme},\mbox{conjecture})$

\medskip\textbf{SEU242+1.p} MPTP bushy problem l4\_wellord1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{connected}(a)\iff \mbox{is\_connected\_in}(a,\mbox{relation\_field}(a)) ) )\qquad\mbox{fof}(\mbox{d14\_relat}_2,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation\_field}(a) = \mbox{set\_union}_2(\mbox{relation\_dom}(a),\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{d6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{is\_connected\_in}(a,b)\iff \forall c,d{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\mbox{in}(d,b)\mbox{ and }c\ne d\mbox{ and }\neg\,\mbox{in}(\mbox{ordered\_pair}(c,d),a)\mbox{ and }\neg\,\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) )\qquad\mbox{fof}(\mbox{d6\_relat}_2,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{connected}(a)\iff \forall b,c{:}\;\neg\,\mbox{in}(b,\mbox{relation\_field}(a))\mbox{ and }\mbox{in}(c,\mbox{relation\_field}(a))\mbox{ and }b\ne c\mbox{ and }\neg\,\mbox{in}(\mbox{ordered\_pair}(b,c),a)\mbox{ and }\neg\,\mbox{in}(\mbox{ordered\_pair}(c,b),a) ) )\qquad\mbox{fof}(\mbox{l4\_wellord}_1,\mbox{conjecture})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU247+1.p} MPTP bushy problem t18\_wellord1






$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;\mbox{relation\_restriction}(a,b) = \mbox{set\_intersection}_2(a,\mbox{cartesian\_product}_2(b,b)) )\qquad\mbox{fof}(\mbox{d6\_wellord}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k2\_wellord}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_dom\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k7\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_rng\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k8\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ \mbox{relation\_dom\_restriction}(\mbox{relation\_rng\_restriction}(a,c),b) = \mbox{relation\_rng\_restriction}(a,\mbox{relation\_dom\_restriction}(c,b)) )\qquad\mbox{fof}(\mbox{t140\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_restriction}(b,a) = \mbox{relation\_dom\_restriction}(\mbox{relation\_rng\_restriction}(a,b),a) )\qquad\mbox{fof}(\mbox{t17\_wellord}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_restriction}(b,a) = \mbox{relation\_rng\_restriction}(a,\mbox{relation\_dom\_restriction}(b,a)) )\qquad\mbox{fof}(\mbox{t18\_wellord}_1,\mbox{conjecture})$

\medskip\textbf{SEU280+1.p} MPTP bushy problem s1\_xboole\_0\_\_e6\_22\_\_wellord2






$\forall a{:}\;\exists b{:}\;\forall c{:}\;( \mbox{in}(c,b)\iff ( \mbox{in}(c,a)\mbox{ and }\mbox{ordinal}(c) ) )\qquad\mbox{fof}(\mbox{s1\_xboole\_0\_\_e6\_22\_\_wellord}_2,\mbox{conjecture})$

$\forall a{:}\;( ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) \ \Rightarrow \ \mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{cc2\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc1\_ordinal}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) )\qquad\mbox{fof}(\mbox{cc1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( \forall b,c,d{:}\;( ( b = c\mbox{ and }\mbox{ordinal}(c)\mbox{ and }b = d\mbox{ and }\mbox{ordinal}(d) ) \ \Rightarrow \ c = d ) \ \Rightarrow \ \exists b{:}\;\forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(d,a)\mbox{ and }d = c\mbox{ and }\mbox{ordinal}(c) ) ) )\qquad\mbox{fof}(\mbox{s1\_tarski\_\_e6\_22\_\_wellord2\_}_1,\mbox{axiom})$

\medskip\textbf{SEU804$\wedge$2.p} More Functions - Surjective Functions - Surjective Cantor Theorem

(! A:i.! f:i.in f (funcSet A (powerset A)) $\to$
 ~(surjective A (powerset A) f))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{surjective}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{surjective\_type},\mbox{type})$

$\mbox{surjective} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xy} ) = \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{surjective},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{powersetI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ ( \mbox{powerset}@ a ) ) ) \ \Rightarrow \ \neg\,\mbox{surjective}@ a@ ( \mbox{powerset}@ a )@ \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{surjCantorThm},\mbox{conjecture})$

\medskip\textbf{SEU179+1.p} MPTP bushy problem t25\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) ) )\qquad\mbox{fof}(\mbox{d5\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( a\subseteq b \ \Rightarrow \ ( \mbox{relation\_dom}(a)\subseteq \mbox{relation\_dom}(b)\mbox{ and }\mbox{relation\_rng}(a)\subseteq \mbox{relation\_rng}(b) ) ) ) )\qquad\mbox{fof}(\mbox{t25\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU137+1.p} MPTP bushy problem t45\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ b = \mbox{set\_union}_2(a,\mbox{set\_difference}(b,a)) )\qquad\mbox{fof}(\mbox{t45\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU575$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.subset A emptyset $\to$ A = emptyset)




$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{emptysetsubset}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetsubset\_type},\mbox{type})$

$\mbox{emptysetsubset} = ( \forall a{:}\;\mbox{\$i}{:}\;( \subseteq @ \mbox{emptyset}@ a ) )\qquad\mbox{thf}(\mbox{emptysetsubset},\mbox{definition})$

$\mbox{setextsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextsub\_type},\mbox{type})$

$\mbox{setextsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ a ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setextsub},\mbox{definition})$

$\mbox{emptysetsubset} \ \Rightarrow \ ( \mbox{setextsub} \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ \mbox{emptyset} ) \ \Rightarrow \ a = \mbox{emptyset} ) )\qquad\mbox{thf}(\mbox{subsetemptysetimpeq},\mbox{conjecture})$

\medskip\textbf{SEU751$\wedge$2.p} Typed Set Theory - Laws for Typed Sets - DeMorgan Laws

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x (setminus A (binintersect X Y)) $\to$
 in x (binunion (setminus A X) (setminus A Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectI\_type},\mbox{type})$

$\mbox{binintersectI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectI},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{complementT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementT\_lem\_type},\mbox{type})$

$\mbox{complementT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{complementT\_lem},\mbox{definition})$

$\mbox{complementTE}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementTE1\_type},\mbox{type})$

$\mbox{complementTE}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{complementTE}_1,\mbox{definition})$

$\mbox{binunionTILcontra}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionTILcontra\_type},\mbox{type})$

$\mbox{binunionTILcontra} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionTILcontra},\mbox{definition})$

$\mbox{binunionTIRcontra}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionTIRcontra\_type},\mbox{type})$

$\mbox{binunionTIRcontra} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionTIRcontra},\mbox{definition})$

$\mbox{binintersectI} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ ( \mbox{complementT\_lem} \ \Rightarrow \ ( \mbox{complementTE}_1 \ \Rightarrow \ ( \mbox{binunionTILcontra} \ \Rightarrow \ ( \mbox{binunionTIRcontra} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binintersect}@ x@ y ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{setminus}@ a@ y ) ) ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{demorgan1a},\mbox{conjecture})$

\medskip\textbf{SEU551$\wedge$2.p} A simple congruence property of emptyset






$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset},\mbox{type})$

$\mbox{emptyset} = \mbox{emptyset}\qquad\mbox{thf}(\mbox{emptyset\_\_Cong},\mbox{conjecture})$

\medskip\textbf{SEU471$\wedge$1.p} The transitive reflexive symmetric closure operator is idempotent






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{idem}@ \mbox{trsc}\qquad\mbox{thf}(\mbox{transitive\_reflexive\_symmetric\_closure\_op\_is\_idempotent},\mbox{conjecture})$

\medskip\textbf{SEU628$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! B:i.! x:i.in x A $\to$ (! y:i.in y B $\to$ in (setadjoin
 (setadjoin x emptyset) (setadjoin (setadjoin x
 (setadjoin y emptyset)) emptyset)) (powerset (powerset
 (binunion A B)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{powersetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI1\_type},\mbox{type})$

$\mbox{powersetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ b@ a ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI}_1,\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{ubforcartprodlem}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ubforcartprodlem1\_type},\mbox{type})$

$\mbox{ubforcartprodlem}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \subseteq @ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) )@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ubforcartprodlem}_1,\mbox{definition})$

$\mbox{powersetI}_1 \ \Rightarrow \ ( \mbox{ubforcartprodlem}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) )@ ( \mbox{powerset}@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ubforcartprodlem}_2,\mbox{conjecture})$

\medskip\textbf{SEU547$\wedge$2.p} Preliminary Notions - Equality Laws

(! A:i.! B:i.A = B $\to$ (! x:i.! y:i.x = y $\to$ in x A $\to$ in y B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in},\mbox{type})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ b ) ) ) )\qquad\mbox{thf}(\mbox{inCongP},\mbox{conjecture})$

\medskip\textbf{SEU455$\wedge$1.p} The reflexive closure operator is inflationary






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{infl}@ \mbox{rc}\qquad\mbox{thf}(\mbox{reflexive\_closure\_op\_is\_inflationary},\mbox{conjecture})$

\medskip\textbf{SEU721$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ ~(in x X) $\to$ ~(in x (binintersect X Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectEL\_type},\mbox{type})$

$\mbox{binintersectEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{definition})$

$\mbox{binintersectEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ x \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectTELcontra},\mbox{conjecture})$

\medskip\textbf{SEU795$\wedge$2.p} More about Functions - Images of Functions

(! A:i.! f:i$>$i.! x:i.in x (image1 A ($\wedge$ y:i.f y)) $<$$\to$
 (? y:i.in y A \& x = f y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu},\mbox{definition})$

$\mbox{descr}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{descr\_type},\mbox{type})$

$\mbox{descrp}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{descrp\_type},\mbox{type})$

$\mbox{descrp} = ( \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{xphi}@ ( \mbox{descr}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{descrp},\mbox{definition})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\mbox{image1Ex}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{image1Ex1\_type},\mbox{type})$

$\mbox{image1Ex}_1 = ( \forall a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{exu}@ \lambda b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b )\iff \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xx} = ( \mbox{xf}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{image1Ex}_1,\mbox{definition})$

$\mbox{image}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{image1\_type},\mbox{type})$

$\mbox{image}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{descr}@ \lambda b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b )\iff \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xx} = ( \mbox{xf}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{image}_1,\mbox{definition})$

$\mbox{descrp} \ \Rightarrow \ ( \mbox{in\_\_Cong} \ \Rightarrow \ ( \mbox{image1Ex}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{image}_1@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) ) )\iff \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xx} = ( \mbox{xf}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{image1Equiv},\mbox{conjecture})$

\medskip\textbf{SEU742$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! Z:i.in Z (powerset A) $\to$ (! x:i.in x A $\to$
 in x (binintersect X Y) $\to$ in x (binintersect (binunion X Z)
 (binunion Y Z))))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectI\_type},\mbox{type})$

$\mbox{binintersectI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectI},\mbox{definition})$

$\mbox{inIntersectImpInUnion}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{inIntersectImpInUnion\_type},\mbox{type})$

$\mbox{inIntersectImpInUnion} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ z ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{inIntersectImpInUnion},\mbox{definition})$

$\mbox{inIntersectImpInUnion}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{inIntersectImpInUnion2\_type},\mbox{type})$

$\mbox{inIntersectImpInUnion}_2 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ y@ z ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{inIntersectImpInUnion}_2,\mbox{definition})$

$\mbox{binintersectI} \ \Rightarrow \ ( \mbox{inIntersectImpInUnion} \ \Rightarrow \ ( \mbox{inIntersectImpInUnion}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ x@ y ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ ( \mbox{binunion}@ x@ z )@ ( \mbox{binunion}@ y@ z ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{inIntersectImpInIntersectUnions},\mbox{conjecture})$

\medskip\textbf{SEU156+1.p} MPTP bushy problem t33\_zfmisc\_1






$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c,d{:}\;\neg\,\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(c,d)\mbox{ and }a\ne c\mbox{ and }a\ne d\qquad\mbox{fof}(\mbox{t10\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{ordered\_pair}(a,b) = \mbox{ordered\_pair}(c,d) \ \Rightarrow \ ( a = c\mbox{ and }b = d ) )\qquad\mbox{fof}(\mbox{t33\_zfmisc}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{unordered\_pair}(a,a) = \mbox{singleton}(a)\qquad\mbox{fof}(\mbox{t69\_enumset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq \mbox{singleton}(b) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t6\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t8\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{t9\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU887$\wedge$5.p} TPS problem THM28






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$c{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\forall f{:}\;b\to a,g{:}\;c\to b,s{:}\;c\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( \exists \mbox{xt}{:}\;c{:}\;( s@ \mbox{xt}\mbox{ and }\mbox{xx} = ( f@ ( g@ \mbox{xt} ) ) ) \ \Rightarrow \ \exists \mbox{xt}{:}\;b{:}\;( \exists \mbox{xt}_0{:}\;c{:}\;( s@ \mbox{xt}_0\mbox{ and }\mbox{xt} = ( g@ \mbox{xt}_0 ) )\mbox{ and }\mbox{xx} = ( f@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cTHM28\_pme},\mbox{conjecture})$

\medskip\textbf{SEU676$\wedge$2.p} Functions - Application

(! A:i.! B:i.! f:i.in f (funcSet A B) $\to$ (! x:i.in x A $\to$
 in (ap A B f x) B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{funcSet} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{powerset}@ ( \mbox{cartprod}@ a@ b ) )@ \lambda \mbox{xf}{:}\;\mbox{\$i}{:}\;( \mbox{func}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{funcSet},\mbox{definition})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{ap} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{dsetconstr}@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{ap},\mbox{definition})$

$\mbox{app}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{app\_type},\mbox{type})$

$\mbox{app} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} )@ b ) ) ) )\qquad\mbox{thf}(\mbox{app},\mbox{definition})$

$\mbox{infuncsetfunc}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{infuncsetfunc\_type},\mbox{type})$

$\mbox{infuncsetfunc} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ ( \mbox{func}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{infuncsetfunc},\mbox{definition})$

$\mbox{app} \ \Rightarrow \ ( \mbox{infuncsetfunc} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} )@ b ) ) ) )\qquad\mbox{thf}(\mbox{ap2p},\mbox{conjecture})$

\medskip\textbf{SEU479$\wedge$1.p} If a relation is terminating, then so is its transitive closure






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{term}@ r ) \ \Rightarrow \ ( \mbox{term}@ ( \mbox{tc}@ r ) ) )\qquad\mbox{thf}(\mbox{termination\_implies\_termination\_of\_tc},\mbox{conjecture})$

\medskip\textbf{SEU788$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$
 binunion R S = binunion S R))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{setextsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextsub\_type},\mbox{type})$

$\mbox{setextsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ a ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setextsub},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{subbreln}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subbreln1\_type},\mbox{type})$

$\mbox{subbreln}_1 = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) \ \Rightarrow \ ( \subseteq @ r@ s ) ) ) ) )\qquad\mbox{thf}(\mbox{subbreln}_1,\mbox{definition})$

$\mbox{breln1unionprop}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1unionprop\_type},\mbox{type})$

$\mbox{breln1unionprop} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \mbox{breln}_1@ a@ ( \mbox{binunion}@ r@ s ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionprop},\mbox{definition})$

$\mbox{breln1unionIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1unionIL\_type},\mbox{type})$

$\mbox{breln1unionIL} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionIL},\mbox{definition})$

$\mbox{breln1unionIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1unionIR\_type},\mbox{type})$

$\mbox{breln1unionIR} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionIR},\mbox{definition})$

$\mbox{breln1unionE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{breln1unionE\_type},\mbox{type})$

$\mbox{breln1unionE} = ( \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r\mbox{ or }\mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionE},\mbox{definition})$

$\mbox{setextsub} \ \Rightarrow \ ( \mbox{subbreln}_1 \ \Rightarrow \ ( \mbox{breln1unionprop} \ \Rightarrow \ ( \mbox{breln1unionIL} \ \Rightarrow \ ( \mbox{breln1unionIR} \ \Rightarrow \ ( \mbox{breln1unionE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ ( \mbox{binunion}@ r@ s ) = ( \mbox{binunion}@ s@ r ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionCommutes},\mbox{conjecture})$

\medskip\textbf{SEU823$\wedge$2.p} Ordinals

(! X:i.ordinal X $\to$ (! A:i.in X A $\to$ ~(in A X)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{stricttotalorderedByIn\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\mbox{in}@ x@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{in}@ x@ y\mbox{ or }\mbox{in}@ y@ x ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \neg\,\mbox{in}@ x@ x ) ) )\qquad\mbox{thf}(\mbox{stricttotalorderedByIn},\mbox{definition})$

$\mbox{wellorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{wellorderedByIn\_type},\mbox{type})$

$\mbox{wellorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{stricttotalorderedByIn}@ a\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{nonempty}@ x ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ x ) \ \Rightarrow \ ( \mbox{xx} = y\mbox{ or }\mbox{in}@ \mbox{xx}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{wellorderedByIn},\mbox{definition})$

$\mbox{ordinal}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ordinal\_type},\mbox{type})$

$\mbox{ordinal} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{transitiveset}@ \mbox{xx}\mbox{ and }\mbox{wellorderedByIn}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{ordinal},\mbox{definition})$

$\mbox{ordinalTransSet}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ordinalTransSet\_type},\mbox{type})$

$\mbox{ordinalTransSet} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{ordinalTransSet},\mbox{definition})$

$\mbox{ordinalIrrefl}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ordinalIrrefl\_type},\mbox{type})$

$\mbox{ordinalIrrefl} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ \neg\,\mbox{in}@ a@ a ) ) )\qquad\mbox{thf}(\mbox{ordinalIrrefl},\mbox{definition})$

$\mbox{ordinalTransSet} \ \Rightarrow \ ( \mbox{ordinalIrrefl} \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \neg\,\mbox{in}@ a@ x ) ) )\qquad\mbox{thf}(\mbox{ordinalNoCycle},\mbox{conjecture})$

\medskip\textbf{SEU799$\wedge$2.p} More about Functions - Injective Functions

(! A:i.! B:i.! f:i.in f (injFuncSet A B) $\to$ in f (funcSet A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{injective}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{injective\_type},\mbox{type})$

$\mbox{injFuncSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{injFuncSet\_type},\mbox{type})$

$\mbox{injFuncSet} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{funcSet}@ a@ b )@ \lambda \mbox{xf}{:}\;\mbox{\$i}{:}\;( \mbox{injective}@ a@ b@ \mbox{xf} ) ) )\qquad\mbox{thf}(\mbox{injFuncSet},\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xf}@ ( \mbox{injFuncSet}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xf}@ ( \mbox{funcSet}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{injFuncSetFuncIn},\mbox{conjecture})$

\medskip\textbf{SEU127+2.p} MPTP chainy problem t17\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_union}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ or }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_union}_2(a,b) = b )\qquad\mbox{fof}(\mbox{t12\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b)\subseteq a\qquad\mbox{fof}(\mbox{t17\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{lemma})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{t2\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t3\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\neg\,\exists c{:}\;\mbox{in}(c,\mbox{set\_intersection}_2(a,b))\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t4\_xboole}_0,\mbox{lemma})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;a\subseteq \mbox{set\_union}_2(a,b)\qquad\mbox{fof}(\mbox{t7\_xboole}_1,\mbox{lemma})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }c\subseteq b ) \ \Rightarrow \ \mbox{set\_union}_2(a,c)\subseteq b )\qquad\mbox{fof}(\mbox{t8\_xboole}_1,\mbox{lemma})$

\medskip\textbf{SEU974$\wedge$5.p} TPS problem from COINDUCTIVE-PU-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xt} = \mbox{cZ}\iff \mbox{xu} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ \mbox{xt} = \mbox{xu} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xu} = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) ) \ \Rightarrow \ \exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xx} )@ ( \mbox{cR}@ \mbox{xy} ) )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xu} = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_LEM2C\_pme},\mbox{conjecture})$

\medskip\textbf{SEU708$\wedge$2.p} Conditionals

(! A:i.! phi:o.! x:i.in x A $\to$ (! y:i.in y A $\to$ ~phi $\to$
 if A phi x y = y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{iffalseProp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{iffalseProp1\_type},\mbox{type})$

$\mbox{iffalseProp}_1 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \neg\,\mbox{xphi} \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iffalseProp}_1,\mbox{definition})$

$\mbox{ifSingleton}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ifSingleton\_type},\mbox{type})$

$\mbox{ifSingleton} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ifSingleton},\mbox{definition})$

$\mbox{if}{:}\;\mbox{\$i}\to \mbox{\$o}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{if\_type},\mbox{type})$

$\mbox{if} = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{if},\mbox{definition})$

$\mbox{theeq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{theeq\_type},\mbox{type})$

$\mbox{theeq} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{singleton}@ x ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{setunion}@ x ) = \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{theeq},\mbox{definition})$

$\mbox{iffalseProp}_1 \ \Rightarrow \ ( \mbox{ifSingleton} \ \Rightarrow \ ( \mbox{theeq} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \neg\,\mbox{xphi} \ \Rightarrow \ ( \mbox{if}@ a@ \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) = \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{iffalse},\mbox{conjecture})$

\medskip\textbf{SEU725$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 subset X Y $\to$ subset (setminus A Y) (setminus A X)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{complementT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementT\_lem\_type},\mbox{type})$

$\mbox{complementT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{complementT\_lem},\mbox{definition})$

$\mbox{subsetTI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetTI\_type},\mbox{type})$

$\mbox{subsetTI} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \subseteq @ x@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{subsetTI},\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ ( \mbox{setminusI} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ ( \mbox{complementT\_lem} \ \Rightarrow \ ( \mbox{subsetTI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \subseteq @ x@ y ) \ \Rightarrow \ ( \subseteq @ ( \mbox{setminus}@ a@ y )@ ( \mbox{setminus}@ a@ x ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{contrasubsetT}_2,\mbox{conjecture})$

\medskip\textbf{SEU711$\wedge$2.p} Typed Set Theory - Types of Set Operators

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 in (binunion X Y) (powerset A)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\mbox{powersetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE\_type},\mbox{type})$

$\mbox{powersetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetE},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionEcases}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionEcases\_type},\mbox{type})$

$\mbox{binunionEcases} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{binunionEcases},\mbox{definition})$

$\mbox{powersetI} \ \Rightarrow \ ( \mbox{powersetE} \ \Rightarrow \ ( \mbox{binunionEcases} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{binunion}@ x@ y )@ ( \mbox{powerset}@ a ) ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionT\_lem},\mbox{conjecture})$

\medskip\textbf{SEU821$\wedge$2.p} Ordinals

(! X:i.ordinal X $\to$ (! A:i.in A X $\to$ ~(in A A)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{nonempty}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{nonempty\_type},\mbox{type})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{stricttotalorderedByIn\_type},\mbox{type})$

$\mbox{stricttotalorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\mbox{in}@ x@ y ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ a ) \ \Rightarrow \ ( x = y\mbox{ or }\mbox{in}@ x@ y\mbox{ or }\mbox{in}@ y@ x ) ) )\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ \neg\,\mbox{in}@ x@ x ) ) )\qquad\mbox{thf}(\mbox{stricttotalorderedByIn},\mbox{definition})$

$\mbox{wellorderedByIn}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{wellorderedByIn\_type},\mbox{type})$

$\mbox{wellorderedByIn} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{stricttotalorderedByIn}@ a\mbox{ and }\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{nonempty}@ x ) \ \Rightarrow \ \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ x\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ x ) \ \Rightarrow \ ( \mbox{xx} = y\mbox{ or }\mbox{in}@ \mbox{xx}@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{wellorderedByIn},\mbox{definition})$

$\mbox{ordinal}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ordinal\_type},\mbox{type})$

$\mbox{ordinal} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{transitiveset}@ \mbox{xx}\mbox{ and }\mbox{wellorderedByIn}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{ordinal},\mbox{definition})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{ordinal}@ x ) \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ \neg\,\mbox{in}@ a@ a ) )\qquad\mbox{thf}(\mbox{ordinalIrrefl},\mbox{conjecture})$

\medskip\textbf{SEU189+1.p} MPTP bushy problem t65\_relat\_1






$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{fc6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_rng}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_rng}(a)) ) )\qquad\mbox{fof}(\mbox{fc8\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{relation\_dom}(a) = \mbox{empty\_set}\iff \mbox{relation\_rng}(a) = \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{t65\_relat}_1,\mbox{conjecture})$

$\mbox{relation\_dom}(\mbox{empty\_set}) = \mbox{empty\_set}\mbox{ and }\mbox{relation\_rng}(\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t60\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( ( \mbox{relation\_dom}(a) = \mbox{empty\_set}\mbox{ or }\mbox{relation\_rng}(a) = \mbox{empty\_set} ) \ \Rightarrow \ a = \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{t64\_relat}_1,\mbox{axiom})$

\medskip\textbf{SEU502$\wedge$2.p} Preliminary Notions - Basic Laws of Logic

(! x:i.in x emptyset $\to$ false)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \mbox{\$false} )\qquad\mbox{thf}(\mbox{emptysetimpfalse},\mbox{conjecture})$

\medskip\textbf{SEU516$\wedge$2.p} Preliminary Notions - Adjoining Elements to Sets

(! x:i.! A:i.! y:i.in y (setadjoin x A) $\to$ y = x | in y A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinAx\_type},\mbox{type})$

$\mbox{setadjoinAx} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) )\iff ( \mbox{xy} = \mbox{xx}\mbox{ or }\mbox{in}@ \mbox{xy}@ a ) ) )\qquad\mbox{thf}(\mbox{setadjoinAx},\mbox{definition})$

$\mbox{setadjoinAx} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{xy} = \mbox{xx}\mbox{ or }\mbox{in}@ \mbox{xy}@ a ) )\qquad\mbox{thf}(\mbox{setadjoinOr},\mbox{conjecture})$

\medskip\textbf{SEU853$\wedge$5.p} TPS problem GAZING-THM35






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o},u{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( t@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx} ) )\iff \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx}\mbox{ and }u@ \mbox{xx}\mbox{ and }\neg\,t@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx}\mbox{ and }\neg\,s@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM35\_pme},\mbox{conjecture})$

\medskip\textbf{SEU671$\wedge$2.p} Ordered Pairs - Sets of Pairs

(! A:i.! B:i.! phi:i$>$(i$>$o).! x:i.! y:i.in (kpair x y)
 (dpsetconstr A B ($\wedge$ z,u:i.phi z u)) $\to$ phi x y)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{setukpairinjL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjL\_type},\mbox{type})$

$\mbox{setukpairinjL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) )\qquad\mbox{thf}(\mbox{setukpairinjL},\mbox{definition})$

$\mbox{setukpairinjR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjR\_type},\mbox{type})$

$\mbox{setukpairinjR} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ \mbox{xy} = \mbox{xu} ) )\qquad\mbox{thf}(\mbox{setukpairinjR},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstr} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{dsetconstr}@ ( \mbox{cartprod}@ a@ b )@ \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ b\mbox{ and }\mbox{xphi}@ \mbox{xx}@ \mbox{xy}\mbox{ and }\mbox{xu} = ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstr},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{setukpairinjL} \ \Rightarrow \ ( \mbox{setukpairinjR} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xz}@ \mbox{xu} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrER},\mbox{conjecture})$

\medskip\textbf{SEU493$\wedge$1.p} The inverse of a partial order is again a partial order






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{po}@ r ) \ \Rightarrow \ ( \mbox{po}@ ( \mbox{inv}@ r ) ) )\qquad\mbox{thf}(\mbox{inverse\_of\_partial\_order\_is\_partial\_order},\mbox{conjecture})$

\medskip\textbf{SEU977$\wedge$5.p} TPS problem from COINDUCTIVE-PU-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$m{:}\;a\qquad\mbox{thf}(m,\mbox{type})$

$n{:}\;a\qquad\mbox{thf}(n,\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xt} = \mbox{cZ}\iff \mbox{xu} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ \mbox{xt} = \mbox{xu} ) ) ) \ \Rightarrow \ ( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{cZ} ) ) ) ) \ \Rightarrow \ ( x@ n ) ) \ \Rightarrow \ ( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{cZ} ) ) ) ) \ \Rightarrow \ ( x@ m ) ) \ \Rightarrow \ ( \exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ n@ m )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xu} = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) ) \ \Rightarrow \ ( n\ne m \ \Rightarrow \ \exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ ( \mbox{cP}@ n@ \mbox{cZ} )@ m )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xu} = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_LESS\_SUCC\_LT\_pme},\mbox{conjecture})$

\medskip\textbf{SEU592$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.subset (binintersect A B) A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersect} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersect},\mbox{definition})$

$\mbox{binintersectEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectEL\_type},\mbox{type})$

$\mbox{binintersectEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{definition})$

$\mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{binintersectEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \subseteq @ ( \mbox{binintersect}@ a@ b )@ a ) )\qquad\mbox{thf}(\mbox{binintersectLsub},\mbox{conjecture})$

\medskip\textbf{SEU580$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! phi:i$>$o.in (dsetconstr A ($\wedge$ x:i.phi x)) (powerset A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{powersetI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{in}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) )@ ( \mbox{powerset}@ a ) ) )\qquad\mbox{thf}(\mbox{sepInPowerset},\mbox{conjecture})$

\medskip\textbf{SEU864$\wedge$5.p} TPS problem from FINITE-SET-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$t{:}\;a\qquad\mbox{thf}(t,\mbox{type})$

$\forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ \forall \mbox{xt}_0{:}\;a{:}\;( t = \mbox{xt}_0 \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}@ \mbox{xz}\mbox{ or }\mbox{xt}_0 = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \lambda \mbox{xy}{:}\;a{:}\;t = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cDOMLEMMA1\_pme},\mbox{conjecture})$

\medskip\textbf{SEU464$\wedge$1.p} The transitive closure of a binary relation is transitive, part 4






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},s{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{trans}@ s\mbox{ and }\mbox{subrel}@ r@ s\mbox{ and }\forall s{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{trans}@ s\mbox{ and }\mbox{subrel}@ r@ s ) \ \Rightarrow \ ( s@ x@ y ) ) ) \ \Rightarrow \ ( s@ x@ y ) )\qquad\mbox{thf}(\mbox{transitive\_closure\_is\_transitive}_4,\mbox{conjecture})$

\medskip\textbf{SEU460$\wedge$1.p} The symmetric closure operator is monotonic






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{mono}@ \mbox{sc}\qquad\mbox{thf}(\mbox{symmetric\_closure\_op\_is\_monotonic},\mbox{conjecture})$

\medskip\textbf{SEU514$\wedge$2.p} Preliminary Notions - Adjoining Elements to Sets

(! x:i.! A:i.! y:i.in y A $\to$ in y (setadjoin x A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinAx\_type},\mbox{type})$

$\mbox{setadjoinAx} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) )\iff ( \mbox{xy} = \mbox{xx}\mbox{ or }\mbox{in}@ \mbox{xy}@ a ) ) )\qquad\mbox{thf}(\mbox{setadjoinAx},\mbox{definition})$

$\mbox{setadjoinAx} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{setadjoinIR},\mbox{conjecture})$

\medskip\textbf{SEU195+1.p} MPTP bushy problem t94\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_dom\_restriction}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff ( \mbox{in}(d,b)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,e),a) ) ) ) ) )\qquad\mbox{fof}(\mbox{d11\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( a = b\iff \forall c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(c,d),a)\iff \mbox{in}(\mbox{ordered\_pair}(c,d),b) ) ) ) )\qquad\mbox{fof}(\mbox{d2\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_composition}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{relation}(\mbox{identity\_relation}(a))\qquad\mbox{fof}(\mbox{dt\_k6\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_dom\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k7\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{empty}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_composition}(b,a))\mbox{ and }\mbox{relation}(\mbox{relation\_composition}(b,a)) ) )\qquad\mbox{fof}(\mbox{fc10\_relat}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{empty}(a)\mbox{ and }\mbox{relation}(b) ) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_composition}(a,b))\mbox{ and }\mbox{relation}(\mbox{relation\_composition}(a,b)) ) )\qquad\mbox{fof}(\mbox{fc9\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{relation}(d) \ \Rightarrow \ ( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{relation\_composition}(\mbox{identity\_relation}(c),d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(a,b),d) ) ) )\qquad\mbox{fof}(\mbox{t74\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_dom\_restriction}(b,a) = \mbox{relation\_composition}(\mbox{identity\_relation}(a),b) )\qquad\mbox{fof}(\mbox{t94\_relat}_1,\mbox{conjecture})$

\medskip\textbf{SEU926$\wedge$5.p} TPS problem THM113

There is a set of functions on P closed under composition.




$\forall p{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists m{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;\forall g{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( m@ g ) \ \Rightarrow \ ( m@ \lambda z{:}\;\mbox{\$i}{:}\;( g@ ( g@ z ) )\mbox{ and }\forall y{:}\;\mbox{\$i}{:}\;( ( p@ y ) \ \Rightarrow \ ( p@ ( g@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM}_{113},\mbox{conjecture})$

\medskip\textbf{SEU591$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.! x:i.in x (binintersect A B) $\to$ in x A)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersect} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersect},\mbox{definition})$

$\mbox{dsetconstrEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{conjecture})$

\medskip\textbf{SEU919$\wedge$5.p} TPS problem THM127






$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\mbox{cP}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( f@ ( g@ \mbox{xx} ) ) ) = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( g@ ( f@ \mbox{xx} ) ) ) \ \Rightarrow \ ( ( \mbox{cP}@ ( f@ ( g@ a ) ) ) \ \Rightarrow \ ( \mbox{cP}@ ( g@ ( f@ a ) ) ) )\qquad\mbox{thf}(\mbox{cTHM127\_pme},\mbox{conjecture})$

\medskip\textbf{SEU956$\wedge$5.p} TPS problem from FUNCTION-THMS






$\forall r{:}\;\mbox{\$i}\to \mbox{\$o},s{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( r = s \ \Rightarrow \ \forall x{:}\;\mbox{\$i}{:}\;( ( s@ x ) \ \Rightarrow \ ( r@ x ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \lambda y{:}\;\mbox{\$i}{:}\;\mbox{xx} = y ) = ( \lambda y{:}\;\mbox{\$i}{:}\;\mbox{xy} = y ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM13\_pme},\mbox{conjecture})$

\medskip\textbf{SEU946$\wedge$5.p} TPS problem THM15A

If some iterate of a function f has a unique fixed point, then f
 has a fixed point.




$\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \exists \mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xf}\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg} ) )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xg}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xg}@ \mbox{xz} ) = \mbox{xz} \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM15A\_pme},\mbox{conjecture})$

\medskip\textbf{SEU750$\wedge$2.p} Typed Set Theory - Laws for Typed Sets - DeMorgan Laws

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x (setminus A (binunion X Y)) $\to$
 in x (setminus A Y))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{binunionTIRcontra}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionTIRcontra\_type},\mbox{type})$

$\mbox{binunionTIRcontra} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionTIRcontra},\mbox{definition})$

$\mbox{setminusI} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ ( \mbox{binunionTIRcontra} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ y ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{demorgan2a}_2,\mbox{conjecture})$

\medskip\textbf{SEU769$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.in R (breln1Set A) $\to$ breln1 A R)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{breln1Set}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1Set\_type},\mbox{type})$

$\mbox{breln1Set} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{powerset}@ ( \mbox{cartprod}@ a@ a ) )@ \lambda r{:}\;\mbox{\$i}{:}\;( \mbox{breln}_1@ a@ r ) ) )\qquad\mbox{thf}(\mbox{breln1Set},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ r@ ( \mbox{breln1Set}@ a ) ) \ \Rightarrow \ ( \mbox{breln}_1@ a@ r ) )\qquad\mbox{thf}(\mbox{breln1SetBreln}_1,\mbox{conjecture})$

\medskip\textbf{SEU202+1.p} MPTP bushy problem t140\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_dom\_restriction}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff ( \mbox{in}(d,b)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,e),a) ) ) ) ) )\qquad\mbox{fof}(\mbox{d11\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \forall c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_rng\_restriction}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff ( \mbox{in}(e,a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,e),b) ) ) ) ) )\qquad\mbox{fof}(\mbox{d12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( a = b\iff \forall c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(c,d),a)\iff \mbox{in}(\mbox{ordered\_pair}(c,d),b) ) ) ) )\qquad\mbox{fof}(\mbox{d2\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_dom\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k7\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_rng\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k8\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ \mbox{relation\_dom\_restriction}(\mbox{relation\_rng\_restriction}(a,c),b) = \mbox{relation\_rng\_restriction}(a,\mbox{relation\_dom\_restriction}(c,b)) )\qquad\mbox{fof}(\mbox{t140\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU863$\wedge$5.p} TPS problem from FINITE-FINITE1-EQUIV






$\mbox{cA}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA\_type},\mbox{type})$

$\mbox{cFINITE}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cFINITE\_type},\mbox{type})$

$\mbox{cNAT}{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cNAT\_type},\mbox{type})$

$\mbox{cSUCC}{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cSUCC\_type},\mbox{type})$

$\mbox{cZERO}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cZERO\_type},\mbox{type})$

$\mbox{cZERO} = ( \lambda \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cZERO\_def},\mbox{definition})$

$\mbox{cSUCC} = ( \lambda \mbox{xn}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o},\mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx}\mbox{ and }\mbox{xn}@ \lambda \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xt}\ne \mbox{xx}\mbox{ and }\mbox{xp}@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cSUCC\_def},\mbox{definition})$

$\mbox{cNAT} = ( \lambda \mbox{xn}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;\forall \mbox{xp}{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{cZERO}\mbox{ and }\forall \mbox{xx}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cSUCC}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xn} ) ) )\qquad\mbox{thf}(\mbox{cNAT\_def},\mbox{definition})$

$\mbox{cFINITE} = ( \lambda \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xn}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \mbox{cNAT}@ \mbox{xn}\mbox{ and }\mbox{xn}@ \mbox{xp} ) )\qquad\mbox{thf}(\mbox{cFINITE\_def},\mbox{definition})$

$\forall \mbox{xw}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{cA} ) ) \ \Rightarrow \ ( \mbox{cFINITE}@ \mbox{cA} )\qquad\mbox{thf}(\mbox{cTHM537\_pme},\mbox{conjecture})$

\medskip\textbf{SEU936$\wedge$5.p} TPS problem FN-THM-3






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$c{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\forall f{:}\;a\to b,g{:}\;b\to c{:}\;( \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( g@ ( f@ \mbox{xx} ) ) = ( g@ ( f@ \mbox{xy} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cFN\_THM\_3\_pme},\mbox{conjecture})$

\medskip\textbf{SEU478$\wedge$1.p} A terminating relation is normalizing






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{term}@ r ) \ \Rightarrow \ ( \mbox{norm}@ r ) )\qquad\mbox{thf}(\mbox{terminating\_implies\_normalizing},\mbox{conjecture})$

\medskip\textbf{SEU875$\wedge$5.p} TPS problem from SET-TOPOLOGY-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$t{:}\;a\qquad\mbox{thf}(t,\mbox{type})$

$\mbox{cA}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xx}\mbox{ and }\mbox{xx}@ t ) \ \Rightarrow \ ( \mbox{cA}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xx}\mbox{ and }\mbox{xx}@ t ) \ \Rightarrow \ \exists \mbox{xe}{:}\;a\to \mbox{\$o}{:}\;( \forall x{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( x@ \lambda \mbox{xy}{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xx}_0{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{xx}_0 ) \ \Rightarrow \ \forall \mbox{xt}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xt}_0 ) \ \Rightarrow \ ( x@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{xx}_0@ \mbox{xz}\mbox{ or }\mbox{xt}_0 = \mbox{xz} ) ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xe} ) )\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xx}@ \mbox{xx}_0 ) )\mbox{ and }\forall \mbox{xy}{:}\;a\to \mbox{\$o}{:}\;( ( \mbox{cA}@ \mbox{xy}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( \mbox{xe}@ \mbox{xx}_0 ) \ \Rightarrow \ ( \mbox{xy}@ \mbox{xx}_0 ) ) ) \ \Rightarrow \ ( \mbox{cA}@ \mbox{xy}\mbox{ and }\mbox{xy}@ t ) ) ) )\qquad\mbox{thf}(\mbox{cDOMLEMMA5\_pme},\mbox{conjecture})$

\medskip\textbf{SEU843$\wedge$5.p} TPS problem GAZING-THM23






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o}{:}\;( ( \lambda \mbox{xz}{:}\;a{:}\;( s@ \mbox{xz}\mbox{ or }t@ \mbox{xz} ) ) = s \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( t@ \mbox{xx} ) \ \Rightarrow \ ( s@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM23\_pme},\mbox{conjecture})$

\medskip\textbf{SEU945$\wedge$5.p} TPS problem THM3






$\exists \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xf}@ \mbox{xx} ) = ( \mbox{xf}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM3\_pme},\mbox{conjecture})$

\medskip\textbf{SEU812$\wedge$2.p} Transitive Sets

(! X:i.transitiveset X $\to$ (! A:i.in A X $\to$ subset A X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{transitiveset}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{transitiveset\_type},\mbox{type})$

$\mbox{transitiveset} = ( \lambda a{:}\;\mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ a ) \ \Rightarrow \ ( \subseteq @ x@ a ) ) )\qquad\mbox{thf}(\mbox{transitiveset},\mbox{definition})$

$\forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{transitiveset}@ x ) \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ x ) \ \Rightarrow \ ( \subseteq @ a@ x ) ) )\qquad\mbox{thf}(\mbox{transitivesetOp}_1,\mbox{conjecture})$

\medskip\textbf{SEU954$\wedge$5.p} TPS problem from FUNCTION-THMS






$f{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(f,\mbox{type})$

$g{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(g,\mbox{type})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( g@ \mbox{xx} ) = ( g@ \mbox{xy} ) \ \Rightarrow \ ( f@ \mbox{xx} ) = ( f@ \mbox{xy} ) ) \ \Rightarrow \ \exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ ( g@ \mbox{xx} ) ) ) = f\qquad\mbox{thf}(\mbox{cTHM588\_pme},\mbox{conjecture})$

\medskip\textbf{SEU530$\wedge$2.p} Preliminary Notions - Equality Laws

(! x:i.! y:i.in x (setadjoin y emptyset) $\to$ in y
 (setadjoin x emptyset))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{eqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{eqinunit\_type},\mbox{type})$

$\mbox{eqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{eqinunit},\mbox{definition})$

$\mbox{uniqinunit} \ \Rightarrow \ ( \mbox{eqinunit} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{singletonsswitch},\mbox{conjecture})$

\medskip\textbf{SEU149+1.p} MPTP bushy problem t8\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t8\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU951$\wedge$5.p} TPS problem THM574

Challenge from Dana Scott stemming from injective Cantor Theorem.




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$h{:}\;( a\to \mbox{\$o} )\to b\qquad\mbox{thf}(h,\mbox{type})$

$\forall \mbox{xx}{:}\;a\to \mbox{\$o},\mbox{xy}{:}\;a\to \mbox{\$o}{:}\;( ( h@ \mbox{xx} ) = ( h@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xg}{:}\;b\to a\to \mbox{\$o}{:}\;\forall y{:}\;a\to \mbox{\$o}{:}\;\exists x{:}\;b{:}\;( \mbox{xg}@ x ) = y\qquad\mbox{thf}(\mbox{cTHM574\_pme},\mbox{conjecture})$

\medskip\textbf{SEU506$\wedge$2.p} Preliminary Notions - Basic Laws of Logic

(! A:i.(! x:i.~(in x A)) $\to$ A = emptyset)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ ( \mbox{setext} \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{in}@ \mbox{xx}@ a \ \Rightarrow \ a = \mbox{emptyset} ) )\qquad\mbox{thf}(\mbox{emptyI},\mbox{conjecture})$

\medskip\textbf{SEU546$\wedge$2.p} Preliminary Notions - Equivalence Laws

(! phi:i$>$o.(? x:i.! y:i.phi y $<$$\to$ y = x) $\to$ exu ($\wedge$ x:i.phi x))




$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu},\mbox{definition})$

$\mbox{exuI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{exuI1\_type},\mbox{type})$

$\mbox{exuI}_1 = ( \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{exuI}_1,\mbox{definition})$

$\mbox{exuI}_1 \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} )\iff \mbox{xy} = \mbox{xx} ) \ \Rightarrow \ ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{exuI}_2,\mbox{conjecture})$

\medskip\textbf{SEU786$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ in (kpair x y) (binunion R S) $\to$
 in (kpair x y) R | in (kpair x y) S))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionE\_type},\mbox{type})$

$\mbox{binunionE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a\mbox{ or }\mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binunionE},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{binunionE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r\mbox{ or }\mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionE},\mbox{conjecture})$

\medskip\textbf{SEU832$\wedge$5.p} TPS problem GAZING-THM31






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( s@ \mbox{xx}\mbox{ and }s@ \mbox{xx} ) ) = s\qquad\mbox{thf}(\mbox{cGAZING\_THM31\_pme},\mbox{conjecture})$

\medskip\textbf{SEU134+1.p} MPTP bushy problem t37\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{set\_difference}(a,b) = \mbox{empty\_set}\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t37\_xboole}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{set\_difference}(a,b) = \mbox{empty\_set}\iff a\subseteq b )\qquad\mbox{fof}(\mbox{l32\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SEU855$\wedge$5.p} TPS problem GAZING-THM33






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o},u{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( t@ \mbox{xx} ) \ \Rightarrow \ ( u@ \mbox{xx} ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx} ) )\iff ( \lambda \mbox{xx}{:}\;a{:}\;( s@ \mbox{xx}\mbox{ and }u@ \mbox{xx}\mbox{ and }\neg\,t@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM33\_pme},\mbox{conjecture})$

\medskip\textbf{SEU653$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.! z:i.! u:i.setadjoin (setadjoin x emptyset)
 (setadjoin (setadjoin x (setadjoin y emptyset)) emptyset) =
 setadjoin (setadjoin z emptyset) (setadjoin (setadjoin z
 (setadjoin u emptyset)) emptyset) $\to$ y = u)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{secondinupair}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{secondinupair\_type},\mbox{type})$

$\mbox{secondinupair} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{secondinupair},\mbox{definition})$

$\mbox{upairset2E}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{upairset2E\_type},\mbox{type})$

$\mbox{upairset2E} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) \ \Rightarrow \ ( \mbox{xz} = \mbox{xx}\mbox{ or }\mbox{xz} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{upairset2E},\mbox{definition})$

$\mbox{setukpairinjL}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjL2\_type},\mbox{type})$

$\mbox{setukpairinjL}_2 = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xu}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) )\qquad\mbox{thf}(\mbox{setukpairinjL}_2,\mbox{definition})$

$\mbox{setukpairinjR}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjR1\_type},\mbox{type})$

$\mbox{setukpairinjR}_1 = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xu}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) \ \Rightarrow \ ( \mbox{xz} = \mbox{xu} \ \Rightarrow \ \mbox{xy} = \mbox{xu} ) ) )\qquad\mbox{thf}(\mbox{setukpairinjR}_1,\mbox{definition})$

$\mbox{upairequniteq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{upairequniteq\_type},\mbox{type})$

$\mbox{upairequniteq} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{upairequniteq},\mbox{definition})$

$\mbox{secondinupair} \ \Rightarrow \ ( \mbox{upairset2E} \ \Rightarrow \ ( \mbox{setukpairinjL}_2 \ \Rightarrow \ ( \mbox{setukpairinjR}_1 \ \Rightarrow \ ( \mbox{upairequniteq} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xu}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xy} = \mbox{xu} ) ) ) ) )\qquad\mbox{thf}(\mbox{setukpairinjR}_2,\mbox{conjecture})$

\medskip\textbf{SEU976$\wedge$5.p} TPS problem from COINDUCTIVE-PU-ALG-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cR}{:}\;a\to a\qquad\mbox{thf}(\mbox{cR},\mbox{type})$

$\mbox{cP}{:}\;a\to a\to a\qquad\mbox{thf}(\mbox{cP},\mbox{type})$

$\mbox{cL}{:}\;a\to a\qquad\mbox{thf}(\mbox{cL},\mbox{type})$

$\mbox{cZ}{:}\;a\qquad\mbox{thf}(\mbox{cZ},\mbox{type})$

$( ( \mbox{cL}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }( \mbox{cR}@ \mbox{cZ} ) = \mbox{cZ}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cL}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx}\mbox{ and }\forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( \mbox{cR}@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xt}{:}\;a{:}\;( \mbox{xt}\ne \mbox{cZ}\iff \mbox{xt} = ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xt} ) ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xt} = \mbox{cZ}\iff \mbox{xu} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) \ \Rightarrow \ \forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ \mbox{xt} = \mbox{xu} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a,\mbox{xy}{:}\;a{:}\;( ( \forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{cZ}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( x@ \mbox{xx}_0 ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xx}_0@ \mbox{cZ} ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xx} ) )\mbox{ and }\forall x{:}\;a\to \mbox{\$o}{:}\;( ( x@ \mbox{cZ}\mbox{ and }\forall \mbox{xx}_0{:}\;a{:}\;( ( x@ \mbox{xx}_0 ) \ \Rightarrow \ ( x@ ( \mbox{cP}@ \mbox{xx}_0@ \mbox{cZ} ) ) ) ) \ \Rightarrow \ ( x@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ \mbox{xx}@ \mbox{xy} )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xu} = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) )\mbox{ or }\exists x{:}\;a\to \mbox{\$o}{:}\;( x@ ( \mbox{cP}@ \mbox{xy}@ \mbox{xx} )\mbox{ and }\forall \mbox{xt}{:}\;a,\mbox{xu}{:}\;a{:}\;( ( x@ ( \mbox{cP}@ \mbox{xt}@ \mbox{xu} ) ) \ \Rightarrow \ ( ( \mbox{xu} = \mbox{cZ} \ \Rightarrow \ \mbox{xt} = \mbox{cZ} )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cL}@ \mbox{xt} )@ ( \mbox{cL}@ \mbox{xu} ) )\mbox{ and }x@ ( \mbox{cP}@ ( \mbox{cR}@ \mbox{xt} )@ ( \mbox{cR}@ \mbox{xu} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cPU\_LEM3B\_pme},\mbox{conjecture})$

\medskip\textbf{SEU849$\wedge$5.p} TPS problem GAZING-THM39






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o}{:}\;( ( \lambda \mbox{xx}{:}\;a{:}\;( s@ \mbox{xx}\mbox{ and }t@ \mbox{xx} ) ) = s\iff \forall \mbox{xx}{:}\;a{:}\;( ( s@ \mbox{xx} ) \ \Rightarrow \ ( t@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM39\_pme},\mbox{conjecture})$

\medskip\textbf{SEU984$\wedge$5.p} TPS problem from FINITE-SETS-CHECKERBOARD






$\mbox{cCKB\_FIN}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_FIN\_type},\mbox{type})$

$\mbox{cCKB\_INF}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_INF\_type},\mbox{type})$

$\mbox{cCKB\_INJ}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_INJ\_type},\mbox{type})$

$\mbox{cCKB\_XPL}{:}\;( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB\_XPL\_type},\mbox{type})$

$\mbox{cCKB\_INJ} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall \mbox{xx}_1{:}\;\mbox{\$i},\mbox{xy}_1{:}\;\mbox{\$i},\mbox{xx}_2{:}\;\mbox{\$i},\mbox{xy}_2{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( ( \mbox{xh}@ \mbox{xx}_1@ \mbox{xy}_1@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xh}@ \mbox{xx}_2@ \mbox{xy}_2@ \mbox{xu}@ \mbox{xv} ) \ \Rightarrow \ ( \mbox{xx}_1 = \mbox{xx}_2\mbox{ and }\mbox{xy}_1 = \mbox{xy}_2 ) ) )\qquad\mbox{thf}(\mbox{cCKB\_INJ\_def},\mbox{definition})$

$\mbox{cCKB\_XPL} = ( \lambda \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ \mbox{xm}@ \mbox{xn}\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xk}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \exists \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;( \mbox{xh}@ \mbox{xx}@ \mbox{xy}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\mbox{xk}@ \mbox{xu}@ \mbox{xv}\mbox{ and }\neg\,\mbox{xu} = \mbox{xm}\mbox{ and }\mbox{xv} = \mbox{xn} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB\_XPL\_def},\mbox{definition})$

$\mbox{cCKB\_INF} = ( \lambda \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xh}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xm}{:}\;\mbox{\$i},\mbox{xn}{:}\;\mbox{\$i}{:}\;( \mbox{cCKB\_INJ}@ \mbox{xh}\mbox{ and }\mbox{cCKB\_XPL}@ \mbox{xh}@ \mbox{xk}@ \mbox{xm}@ \mbox{xn} ) )\qquad\mbox{thf}(\mbox{cCKB\_INF\_def},\mbox{definition})$

$\mbox{cCKB\_FIN} = ( \lambda \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\mbox{cCKB\_INF}@ \mbox{xk} )\qquad\mbox{thf}(\mbox{cCKB\_FIN\_def},\mbox{definition})$

$\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xw}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xp} ) ) \ \Rightarrow \ \forall \mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xw}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xq} ) ) \ \Rightarrow \ \forall \mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xk}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx}\mbox{ and }\mbox{xq}@ \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{cCKB\_FIN}@ \mbox{xk} ) ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_L70000\_pme},\mbox{conjecture})$

\medskip\textbf{SEU200+1.p} MPTP bushy problem t118\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_rng\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k8\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{fc6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_rng}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_rng}(a)) ) )\qquad\mbox{fof}(\mbox{fc8\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_rng\_restriction}(a,b)\subseteq b )\qquad\mbox{fof}(\mbox{t117\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_rng}(\mbox{relation\_rng\_restriction}(a,b))\subseteq \mbox{relation\_rng}(b) )\qquad\mbox{fof}(\mbox{t118\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{relation}(b) \ \Rightarrow \ ( a\subseteq b \ \Rightarrow \ ( \mbox{relation\_dom}(a)\subseteq \mbox{relation\_dom}(b)\mbox{ and }\mbox{relation\_rng}(a)\subseteq \mbox{relation\_rng}(b) ) ) ) )\qquad\mbox{fof}(\mbox{t25\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU719$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! x:i.in x A $\to$ in x X $\to$
 ~(in x (setminus A X))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{setminusER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) ) )\qquad\mbox{thf}(\mbox{complementTI}_1,\mbox{conjecture})$

\medskip\textbf{SEU241+1.p} MPTP bushy problem l3\_wellord1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{antisymmetric}(a)\iff \mbox{is\_antisymmetric\_in}(a,\mbox{relation\_field}(a)) ) )\qquad\mbox{fof}(\mbox{d12\_relat}_2,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( \mbox{is\_antisymmetric\_in}(a,b)\iff \forall c,d{:}\;( ( \mbox{in}(c,b)\mbox{ and }\mbox{in}(d,b)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(c,d),a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) \ \Rightarrow \ c = d ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_2,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \mbox{relation\_field}(a) = \mbox{set\_union}_2(\mbox{relation\_dom}(a),\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{d6\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \mbox{antisymmetric}(a)\iff \forall b,c{:}\;( ( \mbox{in}(\mbox{ordered\_pair}(b,c),a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(c,b),a) ) \ \Rightarrow \ b = c ) ) )\qquad\mbox{fof}(\mbox{l3\_wellord}_1,\mbox{conjecture})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\mbox{set\_union}_2(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t1\_boole},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(\mbox{ordered\_pair}(a,b),c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_field}(c))\mbox{ and }\mbox{in}(b,\mbox{relation\_field}(c)) ) ) )\qquad\mbox{fof}(\mbox{t30\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU133+1.p} MPTP bushy problem t36\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_difference}(a,b)\subseteq a\qquad\mbox{fof}(\mbox{t36\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU473$\wedge$1.p} The transitive reflexive symmetric closure operator is monotonic






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\mbox{mono}@ \mbox{trsc}\qquad\mbox{thf}(\mbox{transitive\_reflexive\_symmetric\_closure\_op\_is\_monotonic},\mbox{conjecture})$

\medskip\textbf{SEU686$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i.func A B f $\to$ (! x:i.in x A $\to$
 (! y:i.in y B $\to$ in (kpair x y) f $\to$ ap A B f x = y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{app}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{app\_type},\mbox{type})$

$\mbox{app} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} )@ b ) ) ) )\qquad\mbox{thf}(\mbox{app},\mbox{definition})$

$\mbox{ex1E}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ex1E2\_type},\mbox{type})$

$\mbox{ex1E}_2 = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{ex}_1@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ex1E}_2,\mbox{definition})$

$\mbox{funcGraphProp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcGraphProp1\_type},\mbox{type})$

$\mbox{funcGraphProp}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) )@ \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{funcGraphProp}_1,\mbox{definition})$

$\mbox{app} \ \Rightarrow \ ( \mbox{ex1E}_2 \ \Rightarrow \ ( \mbox{funcGraphProp}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{funcGraphProp}_2,\mbox{conjecture})$

\medskip\textbf{SEU955$\wedge$5.p} TPS problem from FUNCTION-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$\exists \mbox{xc}{:}\;( a\to \mbox{\$o} )\to a{:}\;\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xt}{:}\;a{:}\;( x@ \mbox{xt} ) \ \Rightarrow \ ( x@ ( \mbox{xc}@ x ) ) ) \ \Rightarrow \ ( \exists \mbox{xg}{:}\;a\to b{:}\;\forall y{:}\;b{:}\;\exists x{:}\;a{:}\;( \mbox{xg}@ x ) = y\mbox{ or }\exists \mbox{xf}{:}\;b\to a{:}\;\forall y{:}\;a{:}\;\exists x{:}\;b{:}\;( \mbox{xf}@ x ) = y )\qquad\mbox{thf}(\mbox{cTHM609\_pme},\mbox{conjecture})$

\medskip\textbf{SEU588$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.subset B (binunion A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIR\_type},\mbox{type})$

$\mbox{binunionIR} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIR},\mbox{definition})$

$\mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{binunionIR} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \subseteq @ b@ ( \mbox{binunion}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{binunionRsub},\mbox{conjecture})$

\medskip\textbf{SEU902$\wedge$5.p} TPS problem THM143

A lemma for the Injective Cantor Theorem X5309.




$d{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(d,\mbox{type})$

$\forall \mbox{xh}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}{:}\;( ( \forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xq}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xh}@ \mbox{xp} ) = ( \mbox{xh}@ \mbox{xq} ) \ \Rightarrow \ \mbox{xp} = \mbox{xq} )\mbox{ and }d = ( \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;\exists \mbox{xt}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \neg\,\mbox{xt}@ ( \mbox{xh}@ \mbox{xt} )\mbox{ and }\mbox{xz} = ( \mbox{xh}@ \mbox{xt} ) ) ) ) \ \Rightarrow \ \neg\,d@ ( \mbox{xh}@ d ) )\qquad\mbox{thf}(\mbox{cTHM143\_pme},\mbox{conjecture})$

\medskip\textbf{SEU797$\wedge$2.p} More about Functions - Images of Functions

(! A:i.! f:i$>$i.! x:i.(? y:i.in y A \& x = f y) $\to$
 in x (image1 A ($\wedge$ y:i.f y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{image}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{image1\_type},\mbox{type})$

$\mbox{image1Equiv}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{image1Equiv\_type},\mbox{type})$

$\mbox{image1Equiv} = ( \forall a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{image}_1@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) ) )\iff \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xx} = ( \mbox{xf}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{image1Equiv},\mbox{definition})$

$\mbox{image1Equiv} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xx} = ( \mbox{xf}@ \mbox{xy} ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{image}_1@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{image1I},\mbox{conjecture})$

\medskip\textbf{SEU965$\wedge$5.p} TPS problem from FUNCTION-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xf}{:}\;a\to a,\mbox{xg}{:}\;a\to a,\mbox{xh}{:}\;a\to a{:}\;( ( \forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xg}@ ( \mbox{xh}@ \mbox{xx} ) ) ) )\mbox{ and }\forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xh} ) ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xh}@ ( \mbox{xg}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM94\_pme},\mbox{conjecture})$

\medskip\textbf{SEU119+1.p} MPTP bushy problem t3\_xboole\_0






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_intersection}_2(a,b) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{d7\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\neg\,\mbox{disjoint}(a,b)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b)\mbox{ and }\neg\,\exists c{:}\;( \mbox{in}(c,a)\mbox{ and }\mbox{in}(c,b) )\mbox{ and }\mbox{disjoint}(a,b) )\qquad\mbox{fof}(\mbox{t3\_xboole}_0,\mbox{conjecture})$

\medskip\textbf{SEU755$\wedge$2.p} Typed Set Theory - Laws for Typed Sets - DeMorgan Laws

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! x:i.in x A $\to$ in x (setminus A X) $\to$ in x (setminus A Y) $\to$
 in x (setminus A (binunion X Y)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{binunionTEcontra}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionTEcontra\_type},\mbox{type})$

$\mbox{binunionTEcontra} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ x \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ y \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ x@ y ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionTEcontra},\mbox{definition})$

$\mbox{setminusI} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ ( \mbox{binunionTEcontra} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ y ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{demorgan2b}_2,\mbox{conjecture})$

\medskip\textbf{SEU541$\wedge$2.p} Preliminary Notions - Dependent Connective Laws

(! phi:o.phi $\to$ set2prop (prop2set phi))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{prop2set}{:}\;\mbox{\$o}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{prop2set\_type},\mbox{type})$

$\mbox{prop2setI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{prop2setI\_type},\mbox{type})$

$\mbox{prop2setI} = ( \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( \mbox{xphi} \ \Rightarrow \ ( \mbox{in}@ \mbox{emptyset}@ ( \mbox{prop2set}@ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{prop2setI},\mbox{definition})$

$\mbox{set2prop}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{set2prop\_type},\mbox{type})$

$\mbox{set2prop} = ( \lambda a{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{emptyset}@ a ) )\qquad\mbox{thf}(\mbox{set2prop},\mbox{definition})$

$\mbox{prop2setI} \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( \mbox{xphi} \ \Rightarrow \ ( \mbox{set2prop}@ ( \mbox{prop2set}@ \mbox{xphi} ) ) )\qquad\mbox{thf}(\mbox{prop2set2propI},\mbox{conjecture})$

\medskip\textbf{SEU882$\wedge$5.p} TPS problem THM139

Every object is in the range of some function.




$\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;\exists \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = \mbox{xy}\qquad\mbox{thf}(\mbox{cTHM139\_pme},\mbox{conjecture})$

\medskip\textbf{SEU248+1.p} MPTP bushy problem l29\_wellord1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \forall c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_rng\_restriction}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff ( \mbox{in}(e,a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,e),b) ) ) ) ) )\qquad\mbox{fof}(\mbox{d12\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_dom}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(c,d),a) ) ) )\qquad\mbox{fof}(\mbox{d4\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_rng\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k8\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \mbox{relation}(b)\mbox{ and }\mbox{function}(b) ) \ \Rightarrow \ ( \mbox{relation}(\mbox{relation\_rng\_restriction}(a,b))\mbox{ and }\mbox{function}(\mbox{relation\_rng\_restriction}(a,b)) ) )\qquad\mbox{fof}(\mbox{fc5\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation\_dom}(\mbox{relation\_rng\_restriction}(a,b))\subseteq \mbox{relation\_dom}(b) )\qquad\mbox{fof}(\mbox{l29\_wellord}_1,\mbox{conjecture})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU483$\wedge$1.p} A symmetric relation is non-terminating






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \exists x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( r@ x@ y ) \ \Rightarrow \ ( ( \mbox{symm}@ r ) \ \Rightarrow \ \neg\,\mbox{term}@ r ) )\qquad\mbox{thf}(\mbox{symmetric\_implies\_non\_terminating},\mbox{conjecture})$

\medskip\textbf{SEU859$\wedge$5.p} TPS problem THM164

Direct consequence of the definition of FINITE1.




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xr}{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( \forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}_0{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}_0{:}\;a\to \mbox{\$o},\mbox{xx}_0{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr}_0 ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}_0@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx}_0 ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xr} ) ) \ \Rightarrow \ \forall \mbox{xw}{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}_0{:}\;a{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}_0{:}\;a\to \mbox{\$o},\mbox{xx}_0{:}\;a{:}\;( ( \mbox{xw}@ \mbox{xr}_0 ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}_0@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx}_0 ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;a{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cTHM164\_pme},\mbox{conjecture})$

\medskip\textbf{SEU519$\wedge$2.p} Preliminary Notions - Power Sets and Unions

(! A:i.in emptyset (powerset A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ ( \mbox{powersetI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{emptyset}@ ( \mbox{powerset}@ a ) ) )\qquad\mbox{thf}(\mbox{emptyinPowerset},\mbox{conjecture})$

\medskip\textbf{SEU842$\wedge$5.p} TPS problem GAZING-THM25






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall s{:}\;a\to \mbox{\$o},t{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( s@ \mbox{xx}\mbox{ and }t@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( t@ \mbox{xx}\mbox{ and }s@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cGAZING\_THM25\_pme},\mbox{conjecture})$

\medskip\textbf{SEU873$\wedge$5.p} TPS problem from FINITE-SET-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\mbox{cC}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cC},\mbox{type})$

$\mbox{cB}{:}\;a\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cB},\mbox{type})$

$( \forall p{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \forall e{:}\;a\to \mbox{\$o}{:}\;( \neg\,\exists \mbox{xt}{:}\;a{:}\;( e@ \mbox{xt} ) \ \Rightarrow \ ( p@ e ) )\mbox{ and }\forall y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a,z{:}\;a\to \mbox{\$o}{:}\;( ( p@ y\mbox{ and }\forall \mbox{xx}_{20}{:}\;a{:}\;( ( z@ \mbox{xx}_{20} )\iff ( y@ \mbox{xx}_{20}\mbox{ or }\mbox{xx}_{20} = \mbox{xx} ) ) ) \ \Rightarrow \ ( p@ z ) ) ) \ \Rightarrow \ ( p@ \mbox{cB} ) )\mbox{ and }\forall p{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \forall e{:}\;a\to \mbox{\$o}{:}\;( \neg\,\exists \mbox{xt}{:}\;a{:}\;( e@ \mbox{xt} ) \ \Rightarrow \ ( p@ e ) )\mbox{ and }\forall y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a,z{:}\;a\to \mbox{\$o}{:}\;( ( p@ y\mbox{ and }\forall \mbox{xx}_{21}{:}\;a{:}\;( ( z@ \mbox{xx}_{21} )\iff ( y@ \mbox{xx}_{21}\mbox{ or }\mbox{xx}_{21} = \mbox{xx} ) ) ) \ \Rightarrow \ ( p@ z ) ) ) \ \Rightarrow \ ( p@ \mbox{cC} ) ) ) \ \Rightarrow \ \forall p{:}\;( a\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \forall e{:}\;a\to \mbox{\$o}{:}\;( \neg\,\exists \mbox{xt}{:}\;a{:}\;( e@ \mbox{xt} ) \ \Rightarrow \ ( p@ e ) )\mbox{ and }\forall y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a,z{:}\;a\to \mbox{\$o}{:}\;( ( p@ y\mbox{ and }\forall \mbox{xx}_{22}{:}\;a{:}\;( ( z@ \mbox{xx}_{22} )\iff ( y@ \mbox{xx}_{22}\mbox{ or }\mbox{xx}_{22} = \mbox{xx} ) ) ) \ \Rightarrow \ ( p@ z ) ) ) \ \Rightarrow \ ( p@ \lambda \mbox{xz}{:}\;a{:}\;( \mbox{cB}@ \mbox{xz}\mbox{ or }\mbox{cC}@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cTHM549\_pme},\mbox{conjecture})$

\medskip\textbf{SEU969$\wedge$5.p} TPS problem from CHECKERBOARD-RELNS






$c_1{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{c1\_type},\mbox{type})$

$s{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{s\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_type},\mbox{type})$

$\mbox{cCKB6\_NUM}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_type},\mbox{type})$

$\mbox{cCKB6\_BLACK} = ( \lambda \mbox{xu}{:}\;\mbox{\$i},\mbox{xv}{:}\;\mbox{\$i}{:}\;\forall \mbox{xw}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xw}@ c_1@ c_1\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xj}@ \mbox{xk} ) \ \Rightarrow \ ( \mbox{xw}@ ( s@ ( s@ \mbox{xj} ) )@ \mbox{xk}\mbox{ and }\mbox{xw}@ ( s@ \mbox{xj} )@ ( s@ \mbox{xk} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{xu}@ \mbox{xv} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_BLACK\_def},\mbox{definition})$

$\mbox{cCKB6\_NUM} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\forall \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{xp}@ c_1\mbox{ and }\forall \mbox{xw}{:}\;\mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xw} ) \ \Rightarrow \ ( \mbox{xp}@ ( s@ \mbox{xw} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cCKB6\_NUM\_def},\mbox{definition})$

$\forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{cCKB6\_BLACK}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ ( \mbox{cCKB6\_NUM}@ \mbox{xx}\mbox{ and }\mbox{cCKB6\_NUM}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{cCKB6\_L6000\_pme},\mbox{conjecture})$

\medskip\textbf{SEU562$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.(! x:i.in x A $\to$ in x B) $\to$ subset A B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\subseteq  = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{subset},\mbox{definition})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{conjecture})$

\medskip\textbf{SEU607$\wedge$2.p} Preliminary Notions - Operations on Sets - Set Difference

(! A:i.! B:i.! x:i.~(in x A) $\to$ ~(in x (setminus A B)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusEL\_type},\mbox{type})$

$\mbox{setminusEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{setminusEL},\mbox{definition})$

$\mbox{setminusEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( \neg\,\mbox{in}@ \mbox{xx}@ a \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) )\qquad\mbox{thf}(\mbox{setminusILneg},\mbox{conjecture})$

\medskip\textbf{SEU482$\wedge$1.p} A normalizing relation is not necessarily terminating






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\exists y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;y\ne z\qquad\mbox{thf}(\mbox{two\_individuals},\mbox{hypothesis})$

$\neg\,\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{norm}@ r ) \ \Rightarrow \ ( \mbox{term}@ r ) )\qquad\mbox{thf}(\mbox{normalizing\_does\_not\_imply\_terminating},\mbox{conjecture})$

\medskip\textbf{SEU637$\wedge$2.p} Ordered Pairs - Singletons

(! A:i.! phi:i$>$o.(! x:i.in x A $\to$ (! y:i.in y A $\to$ phi x $\to$
 phi y $\to$ x = y)) $\to$ (? x:i.in x A \& phi x) $\to$ singleton
 (dsetconstr A ($\wedge$ x:i.phi x)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{eqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{eqinunit\_type},\mbox{type})$

$\mbox{eqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{eqinunit},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{setext} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ ( \mbox{eqinunit} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) ) ) \ \Rightarrow \ ( \exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }\mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{singletonprop},\mbox{conjecture})$

\medskip\textbf{SEU827$\wedge$1.p} About sets 2






$\mbox{leibeq}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{leibeq\_type},\mbox{type})$

$\mbox{leibeq} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall p{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( p@ x ) \ \Rightarrow \ ( p@ y ) ) )\qquad\mbox{thf}(\mbox{leibeq},\mbox{definition})$

$u{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{u\_type},\mbox{type})$

$u = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},u{:}\;\mbox{\$i}{:}\;( x@ u\mbox{ or }y@ u ) )\qquad\mbox{thf}(u,\mbox{definition})$

$n{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{n\_type},\mbox{type})$

$n = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},u{:}\;\mbox{\$i}{:}\;( x@ u\mbox{ and }y@ u ) )\qquad\mbox{thf}(n,\mbox{definition})$

$a{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$c{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{c\_type},\mbox{type})$

$\mbox{leibeq}@ ( u@ a@ ( n@ b@ c ) )@ ( n@ ( u@ a@ b )@ ( u@ a@ c ) )\qquad\mbox{thf}(\mbox{conj},\mbox{conjecture})$

\medskip\textbf{SEU713$\wedge$2.p} Typed Set Theory - Types of Set Operators

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 in (setminus X Y) (powerset A)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\mbox{powersetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetE\_type},\mbox{type})$

$\mbox{powersetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetE},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusEL\_type},\mbox{type})$

$\mbox{setminusEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{setminusEL},\mbox{definition})$

$\mbox{powersetI} \ \Rightarrow \ ( \mbox{powersetE} \ \Rightarrow \ ( \mbox{setminusEL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ x@ y )@ ( \mbox{powerset}@ a ) ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusT\_lem},\mbox{conjecture})$

\medskip\textbf{SEU123+1.p} MPTP bushy problem t3\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{t2\_xboole}_1,\mbox{axiom})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t3\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU938$\wedge$5.p} TPS problem THM196

It is not true that if [k COMPOSE j] is an iterate of j, provided
 we assume extensionality and the existence of the described
 function h (which can be proved if we have distinct elements a
 and b and descriptions).




$a{:}\;\mbox{\$i}\qquad\mbox{thf}(a,\mbox{type})$

$b{:}\;\mbox{\$i}\qquad\mbox{thf}(b,\mbox{type})$

$h{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(h,\mbox{type})$

$( ( h@ a ) = a\mbox{ and }( h@ b )\ne a\mbox{ and }\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xx} ) = ( \mbox{xg}@ \mbox{xx} ) \ \Rightarrow \ \mbox{xf} = \mbox{xg} ) ) \ \Rightarrow \ \neg\,\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xk}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xj}\mbox{ and }\forall \mbox{xj}_6{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj}_6 ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xj}@ ( \mbox{xj}_6@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xk}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xj}\mbox{ and }\forall \mbox{xj}_7{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj}_7 ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xj}@ ( \mbox{xj}_7@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xk} ) ) )\qquad\mbox{thf}(\mbox{cTHM196\_pme},\mbox{conjecture})$

\medskip\textbf{SEU147+3.p} Basic properties of sets, theorem 1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{powerset}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c\subseteq a ) )\qquad\mbox{fof}(\mbox{d1\_zfmisc}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\mbox{powerset}(\mbox{empty\_set}) = \mbox{singleton}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{t1\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t3\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SEU656$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.ksnd (kpair x y) = y)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{iskpair}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{iskpair\_type},\mbox{type})$

$\mbox{iskpair} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setunion}@ a )\mbox{ and }a = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{iskpair},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{kpairp}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{kpairp\_type},\mbox{type})$

$\mbox{kpairp} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{iskpair}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{kpairp},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{theprop}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{theprop\_type},\mbox{type})$

$\mbox{theprop} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{singleton}@ x ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setunion}@ x )@ x ) ) )\qquad\mbox{thf}(\mbox{theprop},\mbox{definition})$

$\mbox{kfst}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kfst\_type},\mbox{type})$

$\mbox{setukpairinjR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjR\_type},\mbox{type})$

$\mbox{setukpairinjR} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ \mbox{xy} = \mbox{xu} ) )\qquad\mbox{thf}(\mbox{setukpairinjR},\mbox{definition})$

$\mbox{ksndsingleton}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ksndsingleton\_type},\mbox{type})$

$\mbox{ksndsingleton} = ( \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{iskpair}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ ( \mbox{setunion}@ \mbox{xu} )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xu} = ( \mbox{kpair}@ ( \mbox{kfst}@ \mbox{xu} )@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{ksndsingleton},\mbox{definition})$

$\mbox{ksnd}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ksnd\_type},\mbox{type})$

$\mbox{ksnd} = ( \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{dsetconstr}@ ( \mbox{setunion}@ \mbox{xu} )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xu} = ( \mbox{kpair}@ ( \mbox{kfst}@ \mbox{xu} )@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ksnd},\mbox{definition})$

$\mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{kpairp} \ \Rightarrow \ ( \mbox{theprop} \ \Rightarrow \ ( \mbox{setukpairinjR} \ \Rightarrow \ ( \mbox{ksndsingleton} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{ksnd}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{ksndpairEq},\mbox{conjecture})$

\medskip\textbf{SEU162+1.p} MPTP bushy problem t65\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{disjoint}(\mbox{singleton}(a),b)\mbox{ and }\mbox{in}(a,b)\qquad\mbox{fof}(\mbox{l25\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{in}(a,b) \ \Rightarrow \ \mbox{disjoint}(\mbox{singleton}(a),b) )\qquad\mbox{fof}(\mbox{l28\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b) \ \Rightarrow \ \mbox{disjoint}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{set\_difference}(a,\mbox{singleton}(b)) = a\iff \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{t65\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_difference}(a,b) = a )\qquad\mbox{fof}(\mbox{t83\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SEU803$\wedge$2.p} More about Functions - Surjective Functions

(! A:i.! B:i.! f:i$>$i.(! x:i.in x A $\to$ in (f x) B) $\to$
 (! g:i.in g (funcSet B A) $\to$ (! x:i.in x A $\to$
 ap B A g (f x) = x) $\to$ surjective B A g))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{funcSet}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{funcSet\_type},\mbox{type})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{surjective}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{surjective\_type},\mbox{type})$

$\mbox{surjective} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xy} ) = \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{surjective},\mbox{definition})$

$\forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{xf}@ \mbox{xx} )@ b ) ) \ \Rightarrow \ \forall \mbox{xg}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xg}@ ( \mbox{funcSet}@ b@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ b@ a@ \mbox{xg}@ ( \mbox{xf}@ \mbox{xx} ) ) = \mbox{xx} ) \ \Rightarrow \ ( \mbox{surjective}@ b@ a@ \mbox{xg} ) ) ) )\qquad\mbox{thf}(\mbox{leftInvIsSurj},\mbox{conjecture})$

\medskip\textbf{SEU749$\wedge$2.p} Typed Set Theory - Laws for Typed Sets - DeMorgan Laws

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 in (setminus A (binunion X Y)) (powerset (setminus A X))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{binunionT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionT\_lem\_type},\mbox{type})$

$\mbox{binunionT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{binunion}@ x@ y )@ ( \mbox{powerset}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionT\_lem},\mbox{definition})$

$\mbox{complementT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{complementT\_lem\_type},\mbox{type})$

$\mbox{complementT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{complementT\_lem},\mbox{definition})$

$\mbox{powersetTI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetTI1\_type},\mbox{type})$

$\mbox{powersetTI}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ x ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ y ) ) ) \ \Rightarrow \ ( \mbox{in}@ x@ ( \mbox{powerset}@ y ) ) ) ) ) )\qquad\mbox{thf}(\mbox{powersetTI}_1,\mbox{definition})$

$\mbox{demorgan2a}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{demorgan2a1\_type},\mbox{type})$

$\mbox{demorgan2a}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ x ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{demorgan2a}_1,\mbox{definition})$

$\mbox{binunionT\_lem} \ \Rightarrow \ ( \mbox{complementT\_lem} \ \Rightarrow \ ( \mbox{powersetTI}_1 \ \Rightarrow \ ( \mbox{demorgan2a}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) )@ ( \mbox{powerset}@ ( \mbox{setminus}@ a@ x ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{complementUnionInPowersetComplement},\mbox{conjecture})$

\medskip\textbf{SEU131+1.p} MPTP bushy problem l32\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{set\_difference}(a,b) = \mbox{empty\_set}\iff a\subseteq b )\qquad\mbox{fof}(\mbox{l32\_xboole}_1,\mbox{conjecture})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(a,\mbox{empty\_set}) = a\qquad\mbox{fof}(\mbox{t3\_boole},\mbox{axiom})$

$\forall a{:}\;\mbox{set\_difference}(\mbox{empty\_set},a) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t4\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU557$\wedge$2.p} A simple congruence property of descr

(forall phi:i$>$o.forall psi:i$>$o.(forall x:i.forall y:i.x = y $\to$
 (phi x $<$$\to$ psi y)) $\to$ exu (lambda x:i.phi x) $\to$
 exu (lambda x:i.psi x) $\to$ descr (lambda x:i.phi x) =
 descr (lambda x:i.psi x))




$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exuEu}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{exuEu\_type},\mbox{type})$

$\mbox{descr}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{descr\_type},\mbox{type})$

$\mbox{descrp}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{descrp\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu\_def},\mbox{definition})$

$\mbox{exuEu} = ( \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{exuEu\_def},\mbox{definition})$

$\mbox{descrp} = ( \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{xphi}@ ( \mbox{descr}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{descrp\_def},\mbox{definition})$

$\mbox{descrp} \ \Rightarrow \ ( \mbox{exuEu} \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xpsi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} )\iff ( \mbox{xpsi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ ( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xpsi}@ \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{descr}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) = ( \mbox{descr}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xpsi}@ \mbox{xx} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{descr\_\_Cong},\mbox{conjecture})$

\medskip\textbf{SEU796$\wedge$2.p} More about Functions - Images of Functions

(! A:i.! f:i$>$i.! x:i.in x (image1 A ($\wedge$ y:i.f y)) $\to$
 (? y:i.in y A \& x = f y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{image}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{image1\_type},\mbox{type})$

$\mbox{image1Equiv}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{image1Equiv\_type},\mbox{type})$

$\mbox{image1Equiv} = ( \forall a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{image}_1@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) ) )\iff \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xx} = ( \mbox{xf}@ \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{image1Equiv},\mbox{definition})$

$\mbox{image1Equiv} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{image}_1@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\mbox{xx} = ( \mbox{xf}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{image1E},\mbox{conjecture})$

\medskip\textbf{SEU549$\wedge$2.p} Preliminary Notions - Equality Laws

(! phi:i$>$o.exu ($\wedge$ x:i.phi x) $\to$ (! x:i.! y:i.phi x $\to$ phi y $\to$
 x = y))




$\mbox{exu}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{exu\_type},\mbox{type})$

$\mbox{exu} = ( \lambda \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx}\mbox{ and }\forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exu},\mbox{definition})$

$\forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{exu}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{exuE3u},\mbox{conjecture})$

\medskip\textbf{SEU573$\wedge$2.p} Preliminary Notions - Relations on Sets - Subsets

(! A:i.! B:i.subset A B $\to$ in A (powerset B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{powersetI} \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( \mbox{in}@ a@ ( \mbox{powerset}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subset2powerset},\mbox{conjecture})$

\medskip\textbf{SEU705$\wedge$2.p} Conditionals

(! A:i.! phi:o.! x:i.in x A $\to$ (! y:i.in y A $\to$
 in (if A phi x y) A))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{sepSubset}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{sepSubset\_type},\mbox{type})$

$\mbox{sepSubset} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \subseteq @ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) )@ a ) )\qquad\mbox{thf}(\mbox{sepSubset},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{theprop}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{theprop\_type},\mbox{type})$

$\mbox{theprop} = ( \forall x{:}\;\mbox{\$i}{:}\;( ( \mbox{singleton}@ x ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{setunion}@ x )@ x ) ) )\qquad\mbox{thf}(\mbox{theprop},\mbox{definition})$

$\mbox{ifSingleton}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ifSingleton\_type},\mbox{type})$

$\mbox{ifSingleton} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ifSingleton},\mbox{definition})$

$\mbox{if}{:}\;\mbox{\$i}\to \mbox{\$o}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{if\_type},\mbox{type})$

$\mbox{if} = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setunion}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xx} )\mbox{ or }( \neg\,\mbox{xphi}\mbox{ and }\mbox{xz} = \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{if},\mbox{definition})$

$\mbox{subsetE} \ \Rightarrow \ ( \mbox{sepSubset} \ \Rightarrow \ ( \mbox{theprop} \ \Rightarrow \ ( \mbox{ifSingleton} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{if}@ a@ \mbox{xphi}@ \mbox{xx}@ \mbox{xy} )@ a ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ifp},\mbox{conjecture})$

\medskip\textbf{SEU709$\wedge$2.p} Conditionals

(! A:i.! phi:o.! x:i.in x A $\to$ (! y:i.in y A $\to$
 in (if A phi x y) (setadjoin x (setadjoin y emptyset))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\mbox{secondinupair}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{secondinupair\_type},\mbox{type})$

$\mbox{secondinupair} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{secondinupair},\mbox{definition})$

$\mbox{if}{:}\;\mbox{\$i}\to \mbox{\$o}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{if\_type},\mbox{type})$

$\mbox{iftrue}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{iftrue\_type},\mbox{type})$

$\mbox{iftrue} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{xphi} \ \Rightarrow \ ( \mbox{if}@ a@ \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) = \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{iftrue},\mbox{definition})$

$\mbox{iffalse}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{iffalse\_type},\mbox{type})$

$\mbox{iffalse} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \neg\,\mbox{xphi} \ \Rightarrow \ ( \mbox{if}@ a@ \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) = \mbox{xy} ) ) ) )\qquad\mbox{thf}(\mbox{iffalse},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{in\_\_Cong} \ \Rightarrow \ ( \mbox{secondinupair} \ \Rightarrow \ ( \mbox{iftrue} \ \Rightarrow \ ( \mbox{iffalse} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{if}@ a@ \mbox{xphi}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{iftrueorfalse},\mbox{conjecture})$

\medskip\textbf{SEU597$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.subset (binintersect A B) B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectER\_type},\mbox{type})$

$\mbox{binintersectER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersectER},\mbox{definition})$

$\mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{binintersectER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \subseteq @ ( \mbox{binintersect}@ a@ b )@ b ) )\qquad\mbox{thf}(\mbox{binintersectRsub},\mbox{conjecture})$

\medskip\textbf{SEU642$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! x:i.! y:i.! z:i.in (setadjoin z emptyset) (setadjoin
 (setadjoin x emptyset) (setadjoin (setadjoin x
 (setadjoin y emptyset)) emptyset)) $\to$ x = z)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{upairset2E}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{upairset2E\_type},\mbox{type})$

$\mbox{upairset2E} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xz}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) ) \ \Rightarrow \ ( \mbox{xz} = \mbox{xx}\mbox{ or }\mbox{xz} = \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{upairset2E},\mbox{definition})$

$\mbox{singletonsuniq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{singletonsuniq\_type},\mbox{type})$

$\mbox{singletonsuniq} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) = ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{singletonsuniq},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ ( \mbox{upairset2E} \ \Rightarrow \ ( \mbox{singletonsuniq} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{setadjoin}@ \mbox{xz}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xz} ) ) ) )\qquad\mbox{thf}(\mbox{setukpairinjL}_1,\mbox{conjecture})$

\medskip\textbf{SEU655$\wedge$2.p} Ordered Pairs - Properties of Pairs

(! u:i.iskpair u $\to$ singleton (dsetconstr (setunion u)
 ($\wedge$ x:i.u = kpair (kfst u) x)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{iskpair}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{iskpair\_type},\mbox{type})$

$\mbox{iskpair} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setunion}@ a )\mbox{ and }a = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{iskpair},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{ex1I}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ex1I\_type},\mbox{type})$

$\mbox{ex1I} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xy} = \mbox{xx} ) ) \ \Rightarrow \ ( \mbox{ex}_1@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ex1I},\mbox{definition})$

$\mbox{kfst}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kfst\_type},\mbox{type})$

$\mbox{kfstpairEq}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{kfstpairEq\_type},\mbox{type})$

$\mbox{kfstpairEq} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{kfst}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) = \mbox{xx} )\qquad\mbox{thf}(\mbox{kfstpairEq},\mbox{definition})$

$\mbox{setukpairinjR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setukpairinjR\_type},\mbox{type})$

$\mbox{setukpairinjR} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i},\mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) = ( \mbox{kpair}@ \mbox{xz}@ \mbox{xu} ) \ \Rightarrow \ \mbox{xy} = \mbox{xu} ) )\qquad\mbox{thf}(\mbox{setukpairinjR},\mbox{definition})$

$\mbox{ex1I} \ \Rightarrow \ ( \mbox{kfstpairEq} \ \Rightarrow \ ( \mbox{setukpairinjR} \ \Rightarrow \ \forall \mbox{xu}{:}\;\mbox{\$i}{:}\;( ( \mbox{iskpair}@ \mbox{xu} ) \ \Rightarrow \ ( \mbox{singleton}@ ( \mbox{dsetconstr}@ ( \mbox{setunion}@ \mbox{xu} )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{xu} = ( \mbox{kpair}@ ( \mbox{kfst}@ \mbox{xu} )@ \mbox{xx} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ksndsingleton},\mbox{conjecture})$

\medskip\textbf{SEU186+1.p} MPTP bushy problem t56\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a)\iff \forall b{:}\;\neg\,\mbox{in}(b,a)\mbox{ and }\forall c,d{:}\;b\ne \mbox{ordered\_pair}(c,d) )\qquad\mbox{fof}(\mbox{d1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ ( \forall b,c{:}\;\neg\,\mbox{in}(\mbox{ordered\_pair}(b,c),a) \ \Rightarrow \ a = \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{t56\_relat}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU524$\wedge$2.p} Preliminary Notions - Power Sets and Unions

(! A:i.! x:i.in x A $\to$ in x (powerset (setunion A)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{powersetI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{powersetI\_type},\mbox{type})$

$\mbox{powersetI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ b@ ( \mbox{powerset}@ a ) ) ) )\qquad\mbox{thf}(\mbox{powersetI},\mbox{definition})$

$\mbox{setunionI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionI\_type},\mbox{type})$

$\mbox{setunionI} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{setunionI},\mbox{definition})$

$\mbox{powersetI} \ \Rightarrow \ ( \mbox{setunionI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{powerset}@ ( \mbox{setunion}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{subPowSU},\mbox{conjecture})$

\medskip\textbf{SEU590$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.! C:i.subset C A $\to$ subset C B $\to$ subset C
 (binintersect A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{subsetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetE\_type},\mbox{type})$

$\mbox{subsetE} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) ) )\qquad\mbox{thf}(\mbox{subsetE},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersect} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersect},\mbox{definition})$

$\mbox{binintersectI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectI\_type},\mbox{type})$

$\mbox{binintersectI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectI},\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{subsetE} \ \Rightarrow \ ( \mbox{binintersectI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( ( \subseteq @ c@ a ) \ \Rightarrow \ ( ( \subseteq @ c@ b ) \ \Rightarrow \ ( \subseteq @ c@ ( \mbox{binintersect}@ a@ b ) ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectSubset}_5,\mbox{conjecture})$

\medskip\textbf{SEU489$\wedge$1.p} Church-Rosser property implies confluence






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{cr}@ r ) \ \Rightarrow \ ( \mbox{confl}@ r ) )\qquad\mbox{thf}(\mbox{church\_rosser\_implies\_confluence},\mbox{conjecture})$

\medskip\textbf{SEU963$\wedge$5.p} TPS problem from FUNCTION-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xf}{:}\;a\to a,\mbox{xg}{:}\;a\to a,\mbox{xh}{:}\;a\to a{:}\;( ( \forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xg}@ ( \mbox{xh}@ \mbox{xx} ) ) ) )\mbox{ and }\forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xh} ) ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg} ) ) )\qquad\mbox{thf}(\mbox{cTHM93\_pme},\mbox{conjecture})$

\medskip\textbf{SEU683$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! phi:i$>$o.ex1 A ($\wedge$ x:i.phi x) $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ phi x $\to$ phi y $\to$ x = y)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{uniqinunit}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{uniqinunit\_type},\mbox{type})$

$\mbox{uniqinunit} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) )\qquad\mbox{thf}(\mbox{uniqinunit},\mbox{definition})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{uniqinunit} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{ex}_1@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ex1E}_2,\mbox{conjecture})$

\medskip\textbf{SEU156+3.p} Basic properties of sets, theorem 33






$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c,d{:}\;\neg\,\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(c,d)\mbox{ and }a\ne c\mbox{ and }a\ne d\qquad\mbox{fof}(\mbox{t10\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{ordered\_pair}(a,b) = \mbox{ordered\_pair}(c,d) \ \Rightarrow \ ( a = c\mbox{ and }b = d ) )\qquad\mbox{fof}(\mbox{t33\_zfmisc}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{unordered\_pair}(a,a) = \mbox{singleton}(a)\qquad\mbox{fof}(\mbox{t69\_enumset}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq \mbox{singleton}(b) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t6\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t8\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{t9\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU862$\wedge$5.p} TPS problem from FINITE-FINITE1-EQUIV






$\mbox{cA}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA\_type},\mbox{type})$

$\mbox{cFINITE}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cFINITE\_type},\mbox{type})$

$\mbox{cNAT}{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cNAT\_type},\mbox{type})$

$\mbox{cSUCC}{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cSUCC\_type},\mbox{type})$

$\mbox{cZERO}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cZERO\_type},\mbox{type})$

$\mbox{cZERO} = ( \lambda \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cZERO\_def},\mbox{definition})$

$\mbox{cSUCC} = ( \lambda \mbox{xn}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o},\mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xp}@ \mbox{xx}\mbox{ and }\mbox{xn}@ \lambda \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xt}\ne \mbox{xx}\mbox{ and }\mbox{xp}@ \mbox{xt} ) ) )\qquad\mbox{thf}(\mbox{cSUCC\_def},\mbox{definition})$

$\mbox{cNAT} = ( \lambda \mbox{xn}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;\forall \mbox{xp}{:}\;( ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{cZERO}\mbox{ and }\forall \mbox{xx}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xp}@ ( \mbox{cSUCC}@ \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xn} ) ) )\qquad\mbox{thf}(\mbox{cNAT\_def},\mbox{definition})$

$\mbox{cFINITE} = ( \lambda \mbox{xp}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists \mbox{xn}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( \mbox{cNAT}@ \mbox{xn}\mbox{ and }\mbox{xn}@ \mbox{xp} ) )\qquad\mbox{thf}(\mbox{cFINITE\_def},\mbox{definition})$

$( \mbox{cFINITE}@ \mbox{cA} ) \ \Rightarrow \ \forall \mbox{xw}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}{:}\;( ( \mbox{xw}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\mbox{\$false}\mbox{ and }\forall \mbox{xr}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xw}@ \mbox{xr} ) \ \Rightarrow \ ( \mbox{xw}@ \lambda \mbox{xt}{:}\;\mbox{\$i}{:}\;( \mbox{xr}@ \mbox{xt}\mbox{ or }\mbox{xt} = \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \mbox{xw}@ \mbox{cA} ) )\qquad\mbox{thf}(\mbox{cTHM538\_pme},\mbox{conjecture})$

\medskip\textbf{SEU128+1.p} MPTP bushy problem t19\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }a\subseteq c ) \ \Rightarrow \ a\subseteq \mbox{set\_intersection}_2(b,c) )\qquad\mbox{fof}(\mbox{t19\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU160+3.p} Basic properties of sets, theorem 39






$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq \mbox{singleton}(b)\iff ( a = \mbox{empty\_set}\mbox{ or }a = \mbox{singleton}(b) ) )\qquad\mbox{fof}(\mbox{t39\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( a\subseteq \mbox{singleton}(b)\iff ( a = \mbox{empty\_set}\mbox{ or }a = \mbox{singleton}(b) ) )\qquad\mbox{fof}(\mbox{l4\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU895$\wedge$5.p} TPS problem X5202






$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$f{:}\;b\to a\qquad\mbox{thf}(f,\mbox{type})$

$y{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(y,\mbox{type})$

$x{:}\;b\to \mbox{\$o}\qquad\mbox{thf}(x,\mbox{type})$

$( \lambda \mbox{xz}{:}\;a{:}\;\exists \mbox{xt}{:}\;b{:}\;( ( x@ \mbox{xt}\mbox{ or }y@ \mbox{xt} )\mbox{ and }\mbox{xz} = ( f@ \mbox{xt} ) ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( \exists \mbox{xt}{:}\;b{:}\;( x@ \mbox{xt}\mbox{ and }\mbox{xz} = ( f@ \mbox{xt} ) )\mbox{ or }\exists \mbox{xt}{:}\;b{:}\;( y@ \mbox{xt}\mbox{ and }\mbox{xz} = ( f@ \mbox{xt} ) ) ) )\qquad\mbox{thf}(\mbox{cX5202\_pme},\mbox{conjecture})$

\medskip\textbf{SEU988$\wedge$5.p} TPS problem from FUNS-AND-RELNS-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$f{:}\;a\to b\to \mbox{\$o}\qquad\mbox{thf}(f,\mbox{type})$

$\forall y{:}\;b\to \mbox{\$o}{:}\;\exists x{:}\;a{:}\;( f@ x ) = y \ \Rightarrow \ \exists \mbox{xg}{:}\;( a\to \mbox{\$o} )\to b\to \mbox{\$o},\mbox{xh}{:}\;( b\to \mbox{\$o} )\to a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xx}_0{:}\;a{:}\;( \lambda \mbox{xy}{:}\;a{:}\;( f@ \mbox{xx}_0 ) = ( f@ \mbox{xy} ) ) = \mbox{xx} \ \Rightarrow \ \mbox{\$true} )\mbox{ and }\forall \mbox{xx}{:}\;b\to \mbox{\$o}{:}\;( \mbox{\$true} \ \Rightarrow \ \exists \mbox{xx}_0{:}\;a{:}\;( \lambda \mbox{xy}{:}\;a{:}\;( f@ \mbox{xx}_0 ) = ( f@ \mbox{xy} ) ) = ( \mbox{xh}@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xy}{:}\;b\to \mbox{\$o}{:}\;( \mbox{xg}@ ( \mbox{xh}@ \mbox{xy} ) ) = \mbox{xy}\mbox{ and }\forall \mbox{xx}{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xx}_0{:}\;a{:}\;( \lambda \mbox{xy}{:}\;a{:}\;( f@ \mbox{xx}_0 ) = ( f@ \mbox{xy} ) ) = \mbox{xx} \ \Rightarrow \ ( \mbox{xh}@ ( \mbox{xg}@ \mbox{xx} ) ) = \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cTHM529\_pme},\mbox{conjecture})$

\medskip\textbf{SEU968$\wedge$5.p} TPS problem from FUNCTION-THMS






$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xg}{:}\;a\to a{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xg}@ \mbox{xx} ) ) = \mbox{xg} \ \Rightarrow \ \forall \mbox{xf}{:}\;a\to a,\mbox{xg}_1{:}\;a\to a,\mbox{xg}_2{:}\;a\to a{:}\;( ( \forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg}_1 ) )\mbox{ and }\forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg}_2 ) ) ) \ \Rightarrow \ \forall \mbox{xp}{:}\;( a\to a )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;a{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;a\to a{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;a{:}\;( \mbox{xg}_1@ ( \mbox{xg}_2@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM135C\_pme},\mbox{conjecture})$

\medskip\textbf{SEU453$\wedge$1.p} The reflexive closure of a binary relation is reflexive






$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{refl}@ ( \mbox{rc}@ r ) )\qquad\mbox{thf}(\mbox{reflexive\_closure\_is\_reflexive},\mbox{conjecture})$

\medskip\textbf{SEU780$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ in (kpair x y) (breln1compset A R S) $\to$
 (? z:i.in z A \& (in (kpair x z) R \& in (kpair z y) S))))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{dpsetconstr}{:}\;\mbox{\$i}\to \mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dpsetconstr\_type},\mbox{type})$

$\mbox{dpsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dpsetconstrER\_type},\mbox{type})$

$\mbox{dpsetconstrER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{dpsetconstr}@ a@ b@ \lambda \mbox{xz}{:}\;\mbox{\$i},\mbox{xu}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xz}@ \mbox{xu} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{dpsetconstrER},\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{breln1compset}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{breln1compset\_type},\mbox{type})$

$\mbox{breln1compset} = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i},s{:}\;\mbox{\$i}{:}\;( \mbox{dpsetconstr}@ a@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xz}@ a\mbox{ and }\mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xz} )@ r\mbox{ and }\mbox{in}@ ( \mbox{kpair}@ \mbox{xz}@ \mbox{xy} )@ s ) ) )\qquad\mbox{thf}(\mbox{breln1compset},\mbox{definition})$

$\mbox{dpsetconstrER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{breln1compset}@ a@ r@ s ) ) \ \Rightarrow \ \exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xz}@ a\mbox{ and }\mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xz} )@ r\mbox{ and }\mbox{in}@ ( \mbox{kpair}@ \mbox{xz}@ \mbox{xy} )@ s ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1compE},\mbox{conjecture})$

\medskip\textbf{SEU687$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! f:i.func A B f $\to$ (! g:i.func A B g $\to$
 (! x:i.in x A $\to$ ap A B f x = ap A B g x) $\to$ (! x:i.in x A $\to$
 (! y:i.in y B $\to$ in (kpair x y) g $\to$ in (kpair x y) f))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{singleton}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{singleton\_type},\mbox{type})$

$\mbox{singleton} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ a\mbox{ and }a = ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

$\mbox{ex}_1{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{ex1\_type},\mbox{type})$

$\mbox{ex}_1 = ( \lambda a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{singleton}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{ex}_1,\mbox{definition})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{func}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{func\_type},\mbox{type})$

$\mbox{func} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ b@ r\mbox{ and }\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ex}_1@ b@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) ) ) ) )\qquad\mbox{thf}(\mbox{func},\mbox{definition})$

$\mbox{ap}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{ap\_type},\mbox{type})$

$\mbox{funcGraphProp}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcGraphProp1\_type},\mbox{type})$

$\mbox{funcGraphProp}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) )@ \mbox{xf} ) ) ) )\qquad\mbox{thf}(\mbox{funcGraphProp}_1,\mbox{definition})$

$\mbox{funcGraphProp}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{funcGraphProp2\_type},\mbox{type})$

$\mbox{funcGraphProp}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = \mbox{xy} ) ) ) ) )\qquad\mbox{thf}(\mbox{funcGraphProp}_2,\mbox{definition})$

$\mbox{funcGraphProp}_1 \ \Rightarrow \ ( \mbox{funcGraphProp}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xf}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xf} ) \ \Rightarrow \ \forall \mbox{xg}{:}\;\mbox{\$i}{:}\;( ( \mbox{func}@ a@ b@ \mbox{xg} ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{ap}@ a@ b@ \mbox{xf}@ \mbox{xx} ) = ( \mbox{ap}@ a@ b@ \mbox{xg}@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xg} ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ \mbox{xf} ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{funcextLem},\mbox{conjecture})$

\medskip\textbf{SEU620$\wedge$2.p} Ordered Pairs - Kuratowski Pairs

(! x:i.! y:i.iskpair (kpair x y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{iskpair}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{iskpair\_type},\mbox{type})$

$\mbox{iskpair} = ( \lambda a{:}\;\mbox{\$i}{:}\;\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a )\mbox{ and }\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ ( \mbox{setunion}@ a )\mbox{ and }a = ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{iskpair},\mbox{definition})$

$\mbox{kpairiskpair}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{kpairiskpair\_type},\mbox{type})$

$\mbox{kpairiskpair} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{iskpair}@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) )\qquad\mbox{thf}(\mbox{kpairiskpair},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{kpairiskpair} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{iskpair}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) )\qquad\mbox{thf}(\mbox{kpairp},\mbox{conjecture})$

\medskip\textbf{SEU610$\wedge$2.p} Preliminary Notions - Operations on Sets - Set Difference

(! A:i.! B:i.setminus A B = emptyset $\to$ subset A B)




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptysetE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetE\_type},\mbox{type})$

$\mbox{emptysetE} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} ) )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{definition})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{emptysetE} \ \Rightarrow \ ( \mbox{in\_\_Cong} \ \Rightarrow \ ( \mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{setminusI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{setminus}@ a@ b ) = \mbox{emptyset} \ \Rightarrow \ ( \subseteq @ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusSubset}_1,\mbox{conjecture})$

\medskip\textbf{SEU633$\wedge$2.p} Ordered Pairs - Singletons

(! A:i.! x:i.in x (setunion A) $\to$ (? X:i.in X A \& in x X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{setunionE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionE\_type},\mbox{type})$

$\mbox{setunionE} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( \forall b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ \mbox{xphi} ) ) \ \Rightarrow \ \mbox{xphi} ) ) )\qquad\mbox{thf}(\mbox{setunionE},\mbox{definition})$

$\mbox{setunionE} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) \ \Rightarrow \ \exists x{:}\;\mbox{\$i}{:}\;( \mbox{in}@ x@ a\mbox{ and }\mbox{in}@ \mbox{xx}@ x ) )\qquad\mbox{thf}(\mbox{setunionE}_2,\mbox{conjecture})$

\medskip\textbf{SEU558$\wedge$2.p} A simple congruence property of dsetconstr

(forall A:i.forall B:i.A = B $\to$ (forall phi:i$>$o.forall psi:i$>$o.(
 forall x:i.in x A $\to$ (forall y:i.in y B $\to$ x = y $\to$ (phi x $<$$\to$
 psi y))) $\to$ dsetconstr A (lambda x:i.phi x) = dsetconstr B
 (lambda x:i.psi x)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{dsetconstrEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrEL\_type},\mbox{type})$

$\mbox{dsetconstrEL} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{dsetconstrEL},\mbox{definition})$

$\mbox{dsetconstrER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrER\_type},\mbox{type})$

$\mbox{dsetconstrER} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{dsetconstrER},\mbox{definition})$

$\mbox{setext}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setext\_type},\mbox{type})$

$\mbox{setext} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setext},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{dsetconstrEL} \ \Rightarrow \ ( \mbox{dsetconstrER} \ \Rightarrow \ ( \mbox{setext} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xpsi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} )\iff ( \mbox{xpsi}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xx} ) ) = ( \mbox{dsetconstr}@ b@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xpsi}@ \mbox{xx} ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstr\_\_Cong},\mbox{conjecture})$

\medskip\textbf{SEU768$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! phi:i$>$o.(! x:i.in x A $\to$
 (! y:i.in y A $\to$ in (kpair x y) R $\to$ phi (kpair x y))) $\to$
 (! x:i.in x R $\to$ phi x)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{brelnall}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{brelnall2\_type},\mbox{type})$

$\mbox{brelnall}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ r ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{xphi}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ r ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{brelnall}_2,\mbox{definition})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{breln}_1 = ( \lambda a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( \mbox{breln}@ a@ a@ r ) )\qquad\mbox{thf}(\mbox{breln}_1,\mbox{definition})$

$\mbox{brelnall}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{xphi}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ r ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{breln1all}_2,\mbox{conjecture})$

\medskip\textbf{SEU475$\wedge$1.p} Not swapping symmetric closure and transitive closure

Taking the symmetric closure of the transitive closure is NOT the
 same as taking the transitive closure of the symmetric closure.




$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\exists x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( x\ne y\mbox{ and }x\ne z\mbox{ and }y\ne z )\qquad\mbox{thf}(\mbox{three\_individuals},\mbox{hypothesis})$

$\neg\,\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{sc}@ ( \mbox{tc}@ r ) ) = ( \mbox{tc}@ ( \mbox{sc}@ r ) )\qquad\mbox{thf}(\mbox{composing\_symmetric\_closure\_and\_transitive\_closure},\mbox{conjecture})$

\medskip\textbf{SEU161+3.p} Basic properties of sets, theorem 46






$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_union}_2(\mbox{singleton}(a),b) = b )\qquad\mbox{fof}(\mbox{t46\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_union}_2(\mbox{singleton}(a),b) = b )\qquad\mbox{fof}(\mbox{l23\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SEU197+1.p} MPTP bushy problem t115\_relat\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \forall c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( c = \mbox{relation\_rng\_restriction}(a,b)\iff \forall d,e{:}\;( \mbox{in}(\mbox{ordered\_pair}(d,e),c)\iff ( \mbox{in}(e,a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(d,e),b) ) ) ) ) )\qquad\mbox{fof}(\mbox{d12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;\mbox{in}(\mbox{ordered\_pair}(d,c),a) ) ) )\qquad\mbox{fof}(\mbox{d5\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{relation}(b) \ \Rightarrow \ \mbox{relation}(\mbox{relation\_rng\_restriction}(a,b)) )\qquad\mbox{fof}(\mbox{dt\_k8\_relat}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_m1\_subset}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{fc6\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_rng}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_rng}(a)) ) )\qquad\mbox{fof}(\mbox{fc8\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{relation}(c) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_rng}(\mbox{relation\_rng\_restriction}(b,c)))\iff ( \mbox{in}(a,b)\mbox{ and }\mbox{in}(a,\mbox{relation\_rng}(c)) ) ) )\qquad\mbox{fof}(\mbox{t115\_relat}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU601$\wedge$2.p} Preliminary Notions - Operations on Sets - Set Difference

(! A:i.! B:i.! x:i.in x A $\to$ ~(in x B) $\to$ in x (setminus A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminus} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ a@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminus},\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{conjecture})$

\medskip\textbf{SEU617$\wedge$2.p} Ordered Pairs - Kuratowski Pairs

(! x:i.! y:i.in x (setunion (setadjoin (setadjoin x emptyset)
 (setadjoin (setadjoin x (setadjoin y emptyset)) emptyset))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{setunion}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setunion\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{setunionI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setunionI\_type},\mbox{type})$

$\mbox{setunionI} = ( \forall a{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ b@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{setunionI},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{setunionI} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setunion}@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) ) ) )\qquad\mbox{thf}(\mbox{setukpairIL},\mbox{conjecture})$

\medskip\textbf{SEU130+1.p} MPTP bushy problem t28\_xboole\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k3\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b)\subseteq a\qquad\mbox{fof}(\mbox{t17\_xboole}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_intersection}_2(a,b) = a )\qquad\mbox{fof}(\mbox{t28\_xboole}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{set\_intersection}_2(a,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_boole},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SEU728$\wedge$2.p} Typed Set Theory - Laws for Typed Sets

(! A:i.! X:i.in X (powerset A) $\to$ (! x:i.in x A $\to$
 in x (setminus A (setminus A X)) $\to$ in x X))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{setminusI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusI\_type},\mbox{type})$

$\mbox{setminusI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \neg\,\mbox{in}@ \mbox{xx}@ b \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{setminusI},\mbox{definition})$

$\mbox{setminusER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setminusER\_type},\mbox{type})$

$\mbox{setminusER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ b ) ) \ \Rightarrow \ \neg\,\mbox{in}@ \mbox{xx}@ b ) )\qquad\mbox{thf}(\mbox{setminusER},\mbox{definition})$

$\mbox{setminusI} \ \Rightarrow \ ( \mbox{setminusER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ ( \mbox{setminus}@ a@ ( \mbox{setminus}@ a@ x ) ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ x ) ) ) ) )\qquad\mbox{thf}(\mbox{doubleComplementE}_1,\mbox{conjecture})$

\medskip\textbf{SEU689$\wedge$2.p} Functions - Extensionality and Beta Reduction

(! A:i.! B:i.! R:i.breln A B R $\to$ (! S:i.breln A B S $\to$
 (! x:i.in x A $\to$ (! y:i.in y B $\to$ in (kpair x y) R $\to$
 in (kpair x y) S)) $\to$ subset R S))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{breln}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln\_type},\mbox{type})$

$\mbox{breln} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \subseteq @ c@ ( \mbox{cartprod}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{breln},\mbox{definition})$

$\mbox{brelnall}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{brelnall1\_type},\mbox{type})$

$\mbox{brelnall}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ r ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{xphi}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} ) ) ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ r ) \ \Rightarrow \ ( \mbox{xphi}@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{brelnall}_1,\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{brelnall}_1 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}@ a@ b@ s ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ r ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) ) ) ) \ \Rightarrow \ ( \subseteq @ r@ s ) ) ) ) )\qquad\mbox{thf}(\mbox{subbreln},\mbox{conjecture})$

\medskip\textbf{SEU903$\wedge$5.p} TPS problem THM131






$g{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{g\_type},\mbox{type})$

$b{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{b\_type},\mbox{type})$

$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall \mbox{xh}_1{:}\;g\to b,\mbox{xh}_2{:}\;b\to a,\mbox{xs}_1{:}\;g\to \mbox{\$o},\mbox{xf}_1{:}\;g\to g,\mbox{xs}_2{:}\;b\to \mbox{\$o},\mbox{xf}_2{:}\;b\to b,\mbox{xs}_3{:}\;a\to \mbox{\$o},\mbox{xf}_3{:}\;a\to a{:}\;( ( \forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_1@ ( \mbox{xf}_1@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xf}_2@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx} ) ) = ( \mbox{xf}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_2@ ( \mbox{xf}_2@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_2@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;b{:}\;( ( \mbox{xs}_2@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xh}_2@ ( \mbox{xf}_2@ \mbox{xx} ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ \mbox{xx} ) ) ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_1@ ( \mbox{xf}_1@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( \mbox{xs}_3@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xf}_3@ \mbox{xx} ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xs}_3@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) ) )\mbox{ and }\forall \mbox{xx}{:}\;g{:}\;( ( \mbox{xs}_1@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{xh}_2@ ( \mbox{xh}_1@ ( \mbox{xf}_1@ \mbox{xx} ) ) ) = ( \mbox{xf}_3@ ( \mbox{xh}_2@ ( \mbox{xh}_1@ \mbox{xx} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{cTHM131\_pme},\mbox{conjecture})$

\medskip\textbf{SEU587$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.subset A (binunion A B))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI2\_type},\mbox{type})$

$\mbox{subsetI}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_2,\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIL\_type},\mbox{type})$

$\mbox{binunionIL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIL},\mbox{definition})$

$\mbox{subsetI}_2 \ \Rightarrow \ ( \mbox{binunionIL} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \subseteq @ a@ ( \mbox{binunion}@ a@ b ) ) )\qquad\mbox{thf}(\mbox{binunionLsub},\mbox{conjecture})$

\medskip\textbf{SEU146+1.p} MPTP bushy problem l4\_zfmisc\_1






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( a = b\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d10\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_tarski},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_xboole}_0,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq b\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l2\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ ( \mbox{in}(c,a)\mbox{ or }a\subseteq \mbox{set\_difference}(b,\mbox{singleton}(c)) ) )\qquad\mbox{fof}(\mbox{l3\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq \mbox{singleton}(b)\iff ( a = \mbox{empty\_set}\mbox{ or }a = \mbox{singleton}(b) ) )\qquad\mbox{fof}(\mbox{l4\_zfmisc}_1,\mbox{conjecture})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{t2\_xboole}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{set\_difference}(a,b) = \mbox{empty\_set}\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t37\_xboole}_1,\mbox{axiom})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t3\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SEU944$\wedge$5.p} TPS problem THM15C






$\forall \mbox{xf}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \exists \mbox{xg}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \forall \mbox{xp}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}{:}\;( ( \mbox{xp}@ \lambda \mbox{xu}{:}\;\mbox{\$i}{:}\;\mbox{xu}\mbox{ and }\forall \mbox{xj}{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{xp}@ \mbox{xj} ) \ \Rightarrow \ ( \mbox{xp}@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ ( \mbox{xj}@ \mbox{xx} ) ) ) ) ) \ \Rightarrow \ ( \mbox{xp}@ \mbox{xg} ) )\mbox{ and }\exists \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{xg}@ \mbox{xx} ) = \mbox{xx}\mbox{ and }\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{xg}@ \mbox{xz} ) = \mbox{xz} \ \Rightarrow \ \mbox{xz} = \mbox{xx} ) ) ) \ \Rightarrow \ \exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xf}@ \mbox{xy} ) = \mbox{xy} )\qquad\mbox{thf}(\mbox{cTHM15C\_pme},\mbox{conjecture})$

\medskip\textbf{SEU811$\wedge$2.p} Omega and Peano

(! x:i.in x omega $\to$ (! y:i.in y omega $\to$ omegaS x = omegaS y $\to$
 x = y))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{omega}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{omega\_type},\mbox{type})$

$\mbox{setadjoinIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinIL\_type},\mbox{type})$

$\mbox{setadjoinIL} = ( \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{setadjoinIL},\mbox{definition})$

$\mbox{setadjoinE}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setadjoinE\_type},\mbox{type})$

$\mbox{setadjoinE} = ( \forall \mbox{xx}{:}\;\mbox{\$i},a{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ ( \mbox{setadjoin}@ \mbox{xx}@ a ) ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( \mbox{xy} = \mbox{xx} \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{setadjoinE},\mbox{definition})$

$\mbox{in\_\_Cong}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{in\_\_Cong\_type},\mbox{type})$

$\mbox{in\_\_Cong} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( a = b \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xx} = \mbox{xy} \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ a )\iff ( \mbox{in}@ \mbox{xy}@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{in\_\_Cong},\mbox{definition})$

$\mbox{notinself}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{notinself2\_type},\mbox{type})$

$\mbox{notinself}_2 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ a@ b ) \ \Rightarrow \ \neg\,\mbox{in}@ b@ a ) )\qquad\mbox{thf}(\mbox{notinself}_2,\mbox{definition})$

$\mbox{omegaS}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{omegaS\_type},\mbox{type})$

$\mbox{omegaS} = ( \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ \mbox{xx}@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{omegaS},\mbox{definition})$

$\mbox{setadjoinIL} \ \Rightarrow \ ( \mbox{setadjoinE} \ \Rightarrow \ ( \mbox{in\_\_Cong} \ \Rightarrow \ ( \mbox{notinself}_2 \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{omega} ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ \mbox{omega} ) \ \Rightarrow \ ( ( \mbox{omegaS}@ \mbox{xx} ) = ( \mbox{omegaS}@ \mbox{xy} ) \ \Rightarrow \ \mbox{xx} = \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{peanoSinj},\mbox{conjecture})$

\medskip\textbf{SEU600$\wedge$2.p} Preliminary Notions - Ops on Sets - Unions and Intersections

(! A:i.! B:i.! C:i.binintersect A (binunion B C) = binunion
 (binintersect A B) (binintersect A C))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\subseteq {:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{subset\_type},\mbox{type})$

$\mbox{subsetI}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{subsetI1\_type},\mbox{type})$

$\mbox{subsetI}_1 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) \ \Rightarrow \ ( \subseteq @ a@ b ) ) )\qquad\mbox{thf}(\mbox{subsetI}_1,\mbox{definition})$

$\mbox{setextsub}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{setextsub\_type},\mbox{type})$

$\mbox{setextsub} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( ( \subseteq @ a@ b ) \ \Rightarrow \ ( ( \subseteq @ b@ a ) \ \Rightarrow \ a = b ) ) )\qquad\mbox{thf}(\mbox{setextsub},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIL\_type},\mbox{type})$

$\mbox{binunionIL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIL},\mbox{definition})$

$\mbox{binunionIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIR\_type},\mbox{type})$

$\mbox{binunionIR} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIR},\mbox{definition})$

$\mbox{binunionEcases}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionEcases\_type},\mbox{type})$

$\mbox{binunionEcases} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$o}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ ( ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ \mbox{xphi} ) \ \Rightarrow \ \mbox{xphi} ) ) ) )\qquad\mbox{thf}(\mbox{binunionEcases},\mbox{definition})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{binintersectI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectI\_type},\mbox{type})$

$\mbox{binintersectI} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{binintersectI},\mbox{definition})$

$\mbox{binintersectEL}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectEL\_type},\mbox{type})$

$\mbox{binintersectEL} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ a ) ) )\qquad\mbox{thf}(\mbox{binintersectEL},\mbox{definition})$

$\mbox{binintersectER}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binintersectER\_type},\mbox{type})$

$\mbox{binintersectER} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ ( \mbox{binintersect}@ a@ b ) ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ b ) ) )\qquad\mbox{thf}(\mbox{binintersectER},\mbox{definition})$

$\mbox{subsetI}_1 \ \Rightarrow \ ( \mbox{setextsub} \ \Rightarrow \ ( \mbox{binunionIL} \ \Rightarrow \ ( \mbox{binunionIR} \ \Rightarrow \ ( \mbox{binunionEcases} \ \Rightarrow \ ( \mbox{binintersectI} \ \Rightarrow \ ( \mbox{binintersectEL} \ \Rightarrow \ ( \mbox{binintersectER} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},c{:}\;\mbox{\$i}{:}\;( \mbox{binintersect}@ a@ ( \mbox{binunion}@ b@ c ) ) = ( \mbox{binunion}@ ( \mbox{binintersect}@ a@ b )@ ( \mbox{binintersect}@ a@ c ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{bs114d},\mbox{conjecture})$

\medskip\textbf{SEU149+3.p} Basic properties of sets, theorem 8






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t8\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SEU630$\wedge$2.p} Ordered Pairs - Cartesian Products

(! A:i.! B:i.! x:i.in x A $\to$ (! y:i.in y B $\to$ in (kpair x y)
 (cartprod A B)))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{setadjoin}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setadjoin\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{dsetconstr}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\qquad\mbox{thf}(\mbox{dsetconstr\_type},\mbox{type})$

$\mbox{dsetconstrI}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{dsetconstrI\_type},\mbox{type})$

$\mbox{dsetconstrI} = ( \forall a{:}\;\mbox{\$i},\mbox{xphi}{:}\;\mbox{\$i}\to \mbox{\$o},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ ( ( \mbox{xphi}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{dsetconstr}@ a@ \lambda \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{xphi}@ \mbox{xy} ) ) ) ) ) )\qquad\mbox{thf}(\mbox{dsetconstrI},\mbox{definition})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{kpair} = ( \lambda \mbox{xx}{:}\;\mbox{\$i},\mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ \mbox{emptyset} )@ ( \mbox{setadjoin}@ ( \mbox{setadjoin}@ \mbox{xx}@ ( \mbox{setadjoin}@ \mbox{xy}@ \mbox{emptyset} ) )@ \mbox{emptyset} ) ) )\qquad\mbox{thf}(\mbox{kpair},\mbox{definition})$

$\mbox{cartprod}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{cartprod\_type},\mbox{type})$

$\mbox{cartprod} = ( \lambda a{:}\;\mbox{\$i},b{:}\;\mbox{\$i}{:}\;( \mbox{dsetconstr}@ ( \mbox{powerset}@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) )@ \lambda \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xy}@ a\mbox{ and }\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{in}@ \mbox{xz}@ b\mbox{ and }\mbox{xx} = ( \mbox{kpair}@ \mbox{xy}@ \mbox{xz} ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprod},\mbox{definition})$

$\mbox{ubforcartprodlem}_3{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{ubforcartprodlem3\_type},\mbox{type})$

$\mbox{ubforcartprodlem}_3 = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{powerset}@ ( \mbox{powerset}@ ( \mbox{binunion}@ a@ b ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{ubforcartprodlem}_3,\mbox{definition})$

$\mbox{dsetconstrI} \ \Rightarrow \ ( \mbox{ubforcartprodlem}_3 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ b ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{cartprod}@ a@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{cartprodpairin},\mbox{conjecture})$

\medskip\textbf{SEU784$\wedge$2.p} Binary Relations on a Set

(! A:i.! R:i.breln1 A R $\to$ (! S:i.breln1 A S $\to$ (! x:i.in x A $\to$
 (! y:i.in y A $\to$ in (kpair x y) S $\to$
 in (kpair x y) (binunion R S)))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binunionIR}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionIR\_type},\mbox{type})$

$\mbox{binunionIR} = ( \forall a{:}\;\mbox{\$i},b{:}\;\mbox{\$i},\mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ b ) \ \Rightarrow \ ( \mbox{in}@ \mbox{xx}@ ( \mbox{binunion}@ a@ b ) ) ) )\qquad\mbox{thf}(\mbox{binunionIR},\mbox{definition})$

$\mbox{kpair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{kpair\_type},\mbox{type})$

$\mbox{breln}_1{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{breln1\_type},\mbox{type})$

$\mbox{binunionIR} \ \Rightarrow \ \forall a{:}\;\mbox{\$i},r{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ r ) \ \Rightarrow \ \forall s{:}\;\mbox{\$i}{:}\;( ( \mbox{breln}_1@ a@ s ) \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ a ) \ \Rightarrow \ \forall \mbox{xy}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xy}@ a ) \ \Rightarrow \ ( ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ s ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{kpair}@ \mbox{xx}@ \mbox{xy} )@ ( \mbox{binunion}@ r@ s ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{breln1unionIR},\mbox{conjecture})$

\medskip\textbf{SEU277+1.p} MPTP bushy problem s1\_xboole\_0\_\_e1\_8\_1\_1\_\_relat\_1






$\forall a,b,c{:}\;( ( \mbox{relation}(b)\mbox{ and }\mbox{relation}(c)\mbox{ and }\mbox{function}(c) ) \ \Rightarrow \ \exists d{:}\;\forall e{:}\;( \mbox{in}(e,d)\iff ( \mbox{in}(e,\mbox{cartesian\_product}_2(a,a))\mbox{ and }\exists f,g{:}\;( e = \mbox{ordered\_pair}(f,g)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(\mbox{apply}(c,f),\mbox{apply}(c,g)),b) ) ) ) )\qquad\mbox{fof}(\mbox{s1\_xboole\_0\_\_e6\_21\_\_wellord2\_}_1,\mbox{conjecture})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) )\qquad\mbox{fof}(\mbox{rc3\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{ordinal}(a) \ \Rightarrow \ ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) )\qquad\mbox{fof}(\mbox{cc1\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a) ) \ \Rightarrow \ \mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{cc2\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc1\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc2\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) )\qquad\mbox{fof}(\mbox{rc3\_ordinal}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc2\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{empty}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a)\mbox{ and }\mbox{one\_to\_one}(a) ) )\qquad\mbox{fof}(\mbox{cc2\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{epsilon\_transitive}(a)\mbox{ and }\mbox{epsilon\_connected}(a)\mbox{ and }\mbox{ordinal}(a) ) )\qquad\mbox{fof}(\mbox{cc3\_ordinal}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k1\_funct}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k2\_zfmisc}_1,\mbox{axiom})$

$\mbox{\$true}\qquad\mbox{fof}(\mbox{dt\_k4\_tarski},\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{relation}(b)\mbox{ and }\mbox{relation}(c)\mbox{ and }\mbox{function}(c) ) \ \Rightarrow \ ( \forall d,e,f{:}\;( ( d = e\mbox{ and }\exists g,h{:}\;( e = \mbox{ordered\_pair}(g,h)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(\mbox{apply}(c,g),\mbox{apply}(c,h)),b) )\mbox{ and }d = f\mbox{ and }\exists i,j{:}\;( f = \mbox{ordered\_pair}(i,j)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(\mbox{apply}(c,i),\mbox{apply}(c,j)),b) ) ) \ \Rightarrow \ e = f ) \ \Rightarrow \ \exists d{:}\;\forall e{:}\;( \mbox{in}(e,d)\iff \exists f{:}\;( \mbox{in}(f,\mbox{cartesian\_product}_2(a,a))\mbox{ and }f = e\mbox{ and }\exists k,l{:}\;( e = \mbox{ordered\_pair}(k,l)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(\mbox{apply}(c,k),\mbox{apply}(c,l)),b) ) ) ) ) )\qquad\mbox{fof}(\mbox{s1\_tarski\_\_e6\_21\_\_wellord2\_}_1,\mbox{axiom})$

\medskip\textbf{SEU476$\wedge$1.p} Swaping transitive closure and reflexive closure

Taking the transitive closure of the reflexive closure is the same
 as taking the reflexive closure of the transitive closure.




$\mbox{include}(\mbox{'Axioms/SET009\string^0.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{tc}@ ( \mbox{rc}@ r ) ) = ( \mbox{rc}@ ( \mbox{tc}@ r ) )\qquad\mbox{thf}(\mbox{composing\_transitive\_closure\_and\_reflexive\_closure},\mbox{conjecture})$

\medskip\textbf{SEU764$\wedge$2.p} Typed Set Theory - First Wizard of Oz Examples - WoZ1 Problems

(! A:i.! X:i.in X (powerset A) $\to$ (! Y:i.in Y (powerset A) $\to$
 (! Z:i.in Z (powerset A) $\to$ (! W:i.in W (powerset A) $\to$
 setminus A (binintersect (binunion X Y) (binunion Z W)) =
 binunion (binintersect (setminus A X) (setminus A Y))
 (binintersect (setminus A Z) (setminus A W))))))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{powerset}{:}\;\mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{powerset\_type},\mbox{type})$

$\mbox{binunion}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binunion\_type},\mbox{type})$

$\mbox{binintersect}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{binintersect\_type},\mbox{type})$

$\mbox{setminus}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{setminus\_type},\mbox{type})$

$\mbox{binunionT\_lem}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{binunionT\_lem\_type},\mbox{type})$

$\mbox{binunionT\_lem} = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{in}@ ( \mbox{binunion}@ x@ y )@ ( \mbox{powerset}@ a ) ) ) ) )\qquad\mbox{thf}(\mbox{binunionT\_lem},\mbox{definition})$

$\mbox{demorgan}_1{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{demorgan1\_type},\mbox{type})$

$\mbox{demorgan}_1 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{setminus}@ a@ ( \mbox{binintersect}@ x@ y ) ) = ( \mbox{binunion}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{setminus}@ a@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{demorgan}_1,\mbox{definition})$

$\mbox{demorgan}_2{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{demorgan2\_type},\mbox{type})$

$\mbox{demorgan}_2 = ( \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{setminus}@ a@ ( \mbox{binunion}@ x@ y ) ) = ( \mbox{binintersect}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{setminus}@ a@ y ) ) ) ) )\qquad\mbox{thf}(\mbox{demorgan}_2,\mbox{definition})$

$\mbox{binunionT\_lem} \ \Rightarrow \ ( \mbox{demorgan}_1 \ \Rightarrow \ ( \mbox{demorgan}_2 \ \Rightarrow \ \forall a{:}\;\mbox{\$i},x{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ x@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall y{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ y@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall z{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ z@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ \forall w{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ w@ ( \mbox{powerset}@ a ) ) \ \Rightarrow \ ( \mbox{setminus}@ a@ ( \mbox{binintersect}@ ( \mbox{binunion}@ x@ y )@ ( \mbox{binunion}@ z@ w ) ) ) = ( \mbox{binunion}@ ( \mbox{binintersect}@ ( \mbox{setminus}@ a@ x )@ ( \mbox{setminus}@ a@ y ) )@ ( \mbox{binintersect}@ ( \mbox{setminus}@ a@ z )@ ( \mbox{setminus}@ a@ w ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{woz1}_2,\mbox{conjecture})$

\medskip\textbf{SEU501$\wedge$2.p} Preliminary Notions - Basic Laws of Logic

(! x:i.in x emptyset $\to$ (! phi:o.phi))




$\mbox{in}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{in\_type},\mbox{type})$

$\mbox{emptyset}{:}\;\mbox{\$i}\qquad\mbox{thf}(\mbox{emptyset\_type},\mbox{type})$

$\mbox{emptysetAx}{:}\;\mbox{\$o}\qquad\mbox{thf}(\mbox{emptysetAx\_type},\mbox{type})$

$\mbox{emptysetAx} = ( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\neg\,\mbox{in}@ \mbox{xx}@ \mbox{emptyset} )\qquad\mbox{thf}(\mbox{emptysetAx},\mbox{definition})$

$\mbox{emptysetAx} \ \Rightarrow \ \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{in}@ \mbox{xx}@ \mbox{emptyset} ) \ \Rightarrow \ \forall \mbox{xphi}{:}\;\mbox{\$o}{:}\;\mbox{xphi} )\qquad\mbox{thf}(\mbox{emptysetE},\mbox{conjecture})$

\medskip
\end{document}