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    \textbf{SET002-0.ax} Set theory axioms






$\neg\,x\in \mbox{empty\_set}\qquad\mbox{cnf}(\mbox{empty\_set},\mbox{axiom})$

$( \mbox{element}\in \mbox{subset}\mbox{ and }\mbox{subset}\subseteq \mbox{superset} ) \ \Rightarrow \ \mbox{element}\in \mbox{superset}\qquad\mbox{cnf}(\mbox{membership\_in\_subsets},\mbox{axiom})$

$\mbox{subset}\subseteq \mbox{superset}\mbox{ or }\mbox{member\_of\_1\_not\_of}_2(\mbox{subset},\mbox{superset})\in \mbox{subset}\qquad\mbox{cnf}(\mbox{subsets\_axiom}_1,\mbox{axiom})$

$\mbox{member\_of\_1\_not\_of}_2(\mbox{subset},\mbox{superset})\in \mbox{superset} \ \Rightarrow \ \mbox{subset}\subseteq \mbox{superset}\qquad\mbox{cnf}(\mbox{subsets\_axiom}_2,\mbox{axiom})$

$x\in \mbox{xs}\mbox{ or }x\in \mbox{xs}'\qquad\mbox{cnf}(\mbox{member\_of\_set\_or\_complement},\mbox{axiom})$

$x\in \mbox{xs} \ \Rightarrow \ \neg\,x\in \mbox{xs}'\qquad\mbox{cnf}(\mbox{not\_member\_of\_set\_and\_complement},\mbox{axiom})$

$x\in \mbox{xs} \ \Rightarrow \ x\in \mbox{union}(\mbox{xs},\mbox{ys})\qquad\mbox{cnf}(\mbox{member\_of\_set1\_is\_member\_of\_union},\mbox{axiom})$

$x\in \mbox{ys} \ \Rightarrow \ x\in \mbox{union}(\mbox{xs},\mbox{ys})\qquad\mbox{cnf}(\mbox{member\_of\_set2\_is\_member\_of\_union},\mbox{axiom})$

$x\in \mbox{union}(\mbox{xs},\mbox{ys}) \ \Rightarrow \ ( x\in \mbox{xs}\mbox{ or }x\in \mbox{ys} )\qquad\mbox{cnf}(\mbox{member\_of\_union\_is\_member\_of\_one\_set},\mbox{axiom})$

$( x\in \mbox{xs}\mbox{ and }x\in \mbox{ys} ) \ \Rightarrow \ x\in \mbox{intersection}(\mbox{xs},\mbox{ys})\qquad\mbox{cnf}(\mbox{member\_of\_both\_is\_member\_of\_intersection},\mbox{axiom})$

$x\in \mbox{intersection}(\mbox{xs},\mbox{ys}) \ \Rightarrow \ x\in \mbox{xs}\qquad\mbox{cnf}(\mbox{member\_of\_intersection\_is\_member\_of\_set}_1,\mbox{axiom})$

$x\in \mbox{intersection}(\mbox{xs},\mbox{ys}) \ \Rightarrow \ x\in \mbox{ys}\qquad\mbox{cnf}(\mbox{member\_of\_intersection\_is\_member\_of\_set}_2,\mbox{axiom})$

$\mbox{equal\_sets}(\mbox{subset},\mbox{superset}) \ \Rightarrow \ \mbox{subset}\subseteq \mbox{superset}\qquad\mbox{cnf}(\mbox{set\_equal\_sets\_are\_subsets}_1,\mbox{axiom})$

$\mbox{equal\_sets}(\mbox{superset},\mbox{subset}) \ \Rightarrow \ \mbox{subset}\subseteq \mbox{superset}\qquad\mbox{cnf}(\mbox{set\_equal\_sets\_are\_subsets}_2,\mbox{axiom})$

$( \mbox{set}_1\subseteq \mbox{set}_2\mbox{ and }\mbox{set}_2\subseteq \mbox{set}_1 ) \ \Rightarrow \ \mbox{equal\_sets}(\mbox{set}_2,\mbox{set}_1)\qquad\mbox{cnf}(\mbox{subsets\_are\_set\_equal\_sets},\mbox{axiom})$

$\mbox{equal\_sets}(\mbox{xs},\mbox{xs})\qquad\mbox{cnf}(\mbox{reflexivity\_for\_set\_equal},\mbox{axiom})$

$\mbox{equal\_sets}(\mbox{xs},\mbox{ys}) \ \Rightarrow \ \mbox{equal\_sets}(\mbox{ys},\mbox{xs})\qquad\mbox{cnf}(\mbox{symmetry\_for\_set\_equal},\mbox{axiom})$

$( \mbox{equal\_sets}(\mbox{xs},\mbox{ys})\mbox{ and }\mbox{equal\_sets}(\mbox{ys},\mbox{zs}) ) \ \Rightarrow \ \mbox{equal\_sets}(\mbox{xs},\mbox{zs})\qquad\mbox{cnf}(\mbox{transitivity\_for\_set\_equal},\mbox{axiom})$

$\mbox{equal\_elements}(x,x)\qquad\mbox{cnf}(\mbox{reflexivity\_for\_equal\_elements},\mbox{axiom})$

$\mbox{equal\_elements}(x,y) \ \Rightarrow \ \mbox{equal\_elements}(y,x)\qquad\mbox{cnf}(\mbox{symmetry\_for\_equal\_elements},\mbox{axiom})$

$( \mbox{equal\_elements}(x,y)\mbox{ and }\mbox{equal\_elements}(y,z) ) \ \Rightarrow \ \mbox{equal\_elements}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_for\_equal\_elements},\mbox{axiom})$

\medskip\textbf{SET001-0.ax} Membership and subsets






$( \mbox{element}\in \mbox{subset}\mbox{ and }\mbox{subset}\subseteq \mbox{superset} ) \ \Rightarrow \ \mbox{element}\in \mbox{superset}\qquad\mbox{cnf}(\mbox{membership\_in\_subsets},\mbox{axiom})$

$\mbox{subset}\subseteq \mbox{superset}\mbox{ or }\mbox{member\_of\_1\_not\_of}_2(\mbox{subset},\mbox{superset})\in \mbox{subset}\qquad\mbox{cnf}(\mbox{subsets\_axiom}_1,\mbox{axiom})$

$\mbox{member\_of\_1\_not\_of}_2(\mbox{subset},\mbox{superset})\in \mbox{superset} \ \Rightarrow \ \mbox{subset}\subseteq \mbox{superset}\qquad\mbox{cnf}(\mbox{subsets\_axiom}_2,\mbox{axiom})$

$\mbox{equal\_sets}(\mbox{subset},\mbox{superset}) \ \Rightarrow \ \mbox{subset}\subseteq \mbox{superset}\qquad\mbox{cnf}(\mbox{set\_equal\_sets\_are\_subsets}_1,\mbox{axiom})$

$\mbox{equal\_sets}(\mbox{superset},\mbox{subset}) \ \Rightarrow \ \mbox{subset}\subseteq \mbox{superset}\qquad\mbox{cnf}(\mbox{set\_equal\_sets\_are\_subsets}_2,\mbox{axiom})$

$( \mbox{set}_1\subseteq \mbox{set}_2\mbox{ and }\mbox{set}_2\subseteq \mbox{set}_1 ) \ \Rightarrow \ \mbox{equal\_sets}(\mbox{set}_2,\mbox{set}_1)\qquad\mbox{cnf}(\mbox{subsets\_are\_set\_equal\_sets},\mbox{axiom})$

\medskip\textbf{SET006+2.ax} Equivalence relation axioms for the SET006+0 set theory axioms






$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \neg\,\exists x{:}\;( x\in a\mbox{ and }x\in b ) )\qquad\mbox{fof}(\mbox{disjoint},\mbox{axiom})$

$\forall a,e{:}\;( \mbox{partition}(a,e)\iff ( \forall x{:}\;( x\in a \ \Rightarrow \ x\subseteq e )\mbox{ and }\forall x{:}\;( x\in e \ \Rightarrow \ \exists y{:}\;( y\in a\mbox{ and }x\in y ) )\mbox{ and }\forall x,y{:}\;( ( x\in a\mbox{ and }y\in a ) \ \Rightarrow \ ( \exists z{:}\;( z\in x\mbox{ and }z\in y ) \ \Rightarrow \ x = y ) ) ) )\qquad\mbox{fof}(\mbox{partition},\mbox{axiom})$

$\forall a,r{:}\;( \mbox{equivalence}(r,a)\iff ( \forall x{:}\;( x\in a \ \Rightarrow \ \mbox{apply}(r,x,x) )\mbox{ and }\forall x,y{:}\;( ( x\in a\mbox{ and }y\in a ) \ \Rightarrow \ ( \mbox{apply}(r,x,y) \ \Rightarrow \ \mbox{apply}(r,y,x) ) )\mbox{ and }\forall x,y,z{:}\;( ( x\in a\mbox{ and }y\in a\mbox{ and }z\in a ) \ \Rightarrow \ ( ( \mbox{apply}(r,x,y)\mbox{ and }\mbox{apply}(r,y,z) ) \ \Rightarrow \ \mbox{apply}(r,x,z) ) ) ) )\qquad\mbox{fof}(\mbox{equivalence},\mbox{axiom})$

$\forall r,e,a,x{:}\;( x\in \mbox{equivalence\_class}(a,e,r)\iff ( x\in e\mbox{ and }\mbox{apply}(r,a,x) ) )\qquad\mbox{fof}(\mbox{equivalence\_class},\mbox{axiom})$

$\forall r,e{:}\;( \mbox{pre\_order}(r,e)\iff ( \forall x{:}\;( x\in e \ \Rightarrow \ \mbox{apply}(r,x,x) )\mbox{ and }\forall x,y,z{:}\;( ( x\in e\mbox{ and }y\in e\mbox{ and }z\in e ) \ \Rightarrow \ ( ( \mbox{apply}(r,x,y)\mbox{ and }\mbox{apply}(r,y,z) ) \ \Rightarrow \ \mbox{apply}(r,x,z) ) ) ) )\qquad\mbox{fof}(\mbox{pre\_order},\mbox{axiom})$

\medskip\textbf{SET006+0.ax} Naive set theory based on Goedel's set theory






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

$\forall a,b{:}\;( \mbox{equal\_set}(a,b)\iff ( a\subseteq b\mbox{ and }b\subseteq a ) )\qquad\mbox{fof}(\mbox{equal\_set},\mbox{axiom})$

$\forall x,a{:}\;( x\in \mbox{power\_set}(a)\iff x\subseteq a )\qquad\mbox{fof}(\mbox{power\_set},\mbox{axiom})$

$\forall x,a,b{:}\;( x\in \mbox{intersection}(a,b)\iff ( x\in a\mbox{ and }x\in b ) )\qquad\mbox{fof}(\mbox{intersection},\mbox{axiom})$

$\forall x,a,b{:}\;( x\in \mbox{union}(a,b)\iff ( x\in a\mbox{ or }x\in b ) )\qquad\mbox{fof}(\mbox{union},\mbox{axiom})$

$\forall x{:}\;\neg\,x\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set},\mbox{axiom})$

$\forall b,a,e{:}\;( b\in ( e\setminus a )\iff ( b\in e\mbox{ and }\neg\,b\in a ) )\qquad\mbox{fof}(\mbox{difference},\mbox{axiom})$

$\forall x,a{:}\;( x\in \mbox{singleton}(a)\iff x = a )\qquad\mbox{fof}(\mbox{singleton},\mbox{axiom})$

$\forall x,a,b{:}\;( x\in \mbox{unordered\_pair}(a,b)\iff ( x = a\mbox{ or }x = b ) )\qquad\mbox{fof}(\mbox{unordered\_pair},\mbox{axiom})$

$\forall x,a{:}\;( x\in \mbox{\mbox{sum}}(a)\iff \exists y{:}\;( y\in a\mbox{ and }x\in y ) )\qquad\mbox{fof}(\mbox{sum},\mbox{axiom})$

$\forall x,a{:}\;( x\in \mbox{\mbox{product}}(a)\iff \forall y{:}\;( y\in a \ \Rightarrow \ x\in y ) )\qquad\mbox{fof}(\mbox{product},\mbox{axiom})$

\medskip\textbf{SET001-3.ax} Membership and difference






$( \mbox{set}_1\setminus \mbox{set}_2{=}\mbox{difference}\mbox{ and }\mbox{element}\in \mbox{difference} ) \ \Rightarrow \ \mbox{element}\in \mbox{set}_1\qquad\mbox{cnf}(\mbox{member\_of\_difference},\mbox{axiom})$

$( \mbox{element}\in \mbox{set}_1\mbox{ and }\mbox{element}\in \mbox{set}_2 ) \ \Rightarrow \ \neg\,\mbox{a\_set}\setminus \mbox{set}_1{=}\mbox{set}_2\qquad\mbox{cnf}(\mbox{not\_member\_of\_difference},\mbox{axiom})$

$( \mbox{element}\in \mbox{set}_1\mbox{ and }\mbox{set}_1\setminus \mbox{set}_2{=}\mbox{difference} ) \ \Rightarrow \ ( \mbox{element}\in \mbox{difference}\mbox{ or }\mbox{element}\in \mbox{set}_2 )\qquad\mbox{cnf}(\mbox{member\_of\_difference\_or\_set}_2,\mbox{axiom})$

$\mbox{set}_1\setminus \mbox{set}_2{=}\mbox{difference}\mbox{ or }k(\mbox{set}_1,\mbox{set}_2,\mbox{difference})\in \mbox{set}_1\mbox{ or }k(\mbox{set}_1,\mbox{set}_2,\mbox{difference})\in \mbox{difference}\qquad\mbox{cnf}(\mbox{difference\_axiom}_2,\mbox{axiom})$

$k(\mbox{set}_1,\mbox{set}_2,\mbox{difference})\in \mbox{set}_2 \ \Rightarrow \ ( k(\mbox{set}_1,\mbox{set}_2,\mbox{difference})\in \mbox{difference}\mbox{ or }\mbox{set}_1\setminus \mbox{set}_2{=}\mbox{difference} )\qquad\mbox{cnf}(\mbox{difference\_axiom}_1,\mbox{axiom})$

$( k(\mbox{set}_1,\mbox{set}_2,\mbox{difference})\in \mbox{difference}\mbox{ and }k(\mbox{set}_1,\mbox{set}_2,\mbox{difference})\in \mbox{set}_1 ) \ \Rightarrow \ ( k(\mbox{set}_1,\mbox{set}_2,\mbox{difference})\in \mbox{set}_2\mbox{ or }\mbox{set}_1\setminus \mbox{set}_2{=}\mbox{difference} )\qquad\mbox{cnf}(\mbox{difference\_axiom}_3,\mbox{axiom})$

\medskip\textbf{SET001-1.ax} Membership and union






$( \mbox{union}(\mbox{set}_1,\mbox{set}_2,\mbox{union})\mbox{ and }\mbox{element}\in \mbox{union} ) \ \Rightarrow \ ( \mbox{element}\in \mbox{set}_1\mbox{ or }\mbox{element}\in \mbox{set}_2 )\qquad\mbox{cnf}(\mbox{member\_of\_union\_is\_member\_of\_one\_set},\mbox{axiom})$

$( \mbox{union}(\mbox{set}_1,\mbox{set}_2,\mbox{union})\mbox{ and }\mbox{element}\in \mbox{set}_1 ) \ \Rightarrow \ \mbox{element}\in \mbox{union}\qquad\mbox{cnf}(\mbox{member\_of\_set1\_is\_member\_of\_union},\mbox{axiom})$

$( \mbox{union}(\mbox{set}_1,\mbox{set}_2,\mbox{union})\mbox{ and }\mbox{element}\in \mbox{set}_2 ) \ \Rightarrow \ \mbox{element}\in \mbox{union}\qquad\mbox{cnf}(\mbox{member\_of\_set2\_is\_member\_of\_union},\mbox{axiom})$

$\mbox{union}(\mbox{set}_1,\mbox{set}_2,\mbox{union})\mbox{ or }g(\mbox{set}_1,\mbox{set}_2,\mbox{union})\in \mbox{set}_1\mbox{ or }g(\mbox{set}_1,\mbox{set}_2,\mbox{union})\in \mbox{set}_2\mbox{ or }g(\mbox{set}_1,\mbox{set}_2,\mbox{union})\in \mbox{union}\qquad\mbox{cnf}(\mbox{union\_axiom}_1,\mbox{axiom})$

$( g(\mbox{set}_1,\mbox{set}_2,\mbox{union})\in \mbox{set}_1\mbox{ and }g(\mbox{set}_1,\mbox{set}_2,\mbox{union})\in \mbox{union} ) \ \Rightarrow \ \mbox{union}(\mbox{set}_1,\mbox{set}_2,\mbox{union})\qquad\mbox{cnf}(\mbox{union\_axiom}_2,\mbox{axiom})$

$( g(\mbox{set}_1,\mbox{set}_2,\mbox{union})\in \mbox{set}_2\mbox{ and }g(\mbox{set}_1,\mbox{set}_2,\mbox{union})\in \mbox{union} ) \ \Rightarrow \ \mbox{union}(\mbox{set}_1,\mbox{set}_2,\mbox{union})\qquad\mbox{cnf}(\mbox{union\_axiom}_3,\mbox{axiom})$

\medskip\textbf{SET004-1.ax} Set theory (Boolean algebra) axioms based on NBG set theory






$\mbox{subclass}(\mbox{compose\_class}(x),\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{compose\_class\_definition}_1,\mbox{axiom})$

$\mbox{ordered\_pair}(y,z)\in \mbox{compose\_class}(x) \ \Rightarrow \ x\circ y = z\qquad\mbox{cnf}(\mbox{compose\_class\_definition}_2,\mbox{axiom})$

$( \mbox{ordered\_pair}(y,z)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\mbox{ and }x\circ y = z ) \ \Rightarrow \ \mbox{ordered\_pair}(y,z)\in \mbox{compose\_class}(x)\qquad\mbox{cnf}(\mbox{compose\_class\_definition}_3,\mbox{axiom})$

$\mbox{subclass}(\mbox{composition\_function},\mbox{cross\_product}(\mbox{universal\_class},\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})))\qquad\mbox{cnf}(\mbox{definition\_of\_composition\_function}_1,\mbox{axiom})$

$\mbox{ordered\_pair}(x,\mbox{ordered\_pair}(y,z))\in \mbox{composition\_function} \ \Rightarrow \ x\circ y = z\qquad\mbox{cnf}(\mbox{definition\_of\_composition\_function}_2,\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}) \ \Rightarrow \ \mbox{ordered\_pair}(x,\mbox{ordered\_pair}(y,x\circ y))\in \mbox{composition\_function}\qquad\mbox{cnf}(\mbox{definition\_of\_composition\_function}_3,\mbox{axiom})$

$\mbox{subclass}(\mbox{domain\_relation},\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{definition\_of\_domain\_relation}_1,\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{domain\_relation} \ \Rightarrow \ \mbox{domain\_of}(x) = y\qquad\mbox{cnf}(\mbox{definition\_of\_domain\_relation}_2,\mbox{axiom})$

$x\in \mbox{universal\_class} \ \Rightarrow \ \mbox{ordered\_pair}(x,\mbox{domain\_of}(x))\in \mbox{domain\_relation}\qquad\mbox{cnf}(\mbox{definition\_of\_domain\_relation}_3,\mbox{axiom})$

$\mbox{first}(\mbox{not\_subclass\_element}(x\circ x',\mbox{identity\_relation})) = \mbox{single\_valued}_1(x)\qquad\mbox{cnf}(\mbox{single\_valued\_term\_defn}_1,\mbox{axiom})$

$\mbox{second}(\mbox{not\_subclass\_element}(x\circ x',\mbox{identity\_relation})) = \mbox{single\_valued}_2(x)\qquad\mbox{cnf}(\mbox{single\_valued\_term\_defn}_2,\mbox{axiom})$

$1_{\text{dom}(x)} = \mbox{single\_valued}_3(x)\qquad\mbox{cnf}(\mbox{single\_valued\_term\_defn}_3,\mbox{axiom})$

$\mbox{intersection}(( \mbox{element\_relation}\circ \mbox{identity\_relation}' )',\mbox{element\_relation}) = \mbox{singleton\_relation}\qquad\mbox{cnf}(\mbox{compose\_can\_define\_singleton},\mbox{axiom})$

$\mbox{subclass}(\mbox{application\_function},\mbox{cross\_product}(\mbox{universal\_class},\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})))\qquad\mbox{cnf}(\mbox{application\_function\_defn}_1,\mbox{axiom})$

$\mbox{ordered\_pair}(x,\mbox{ordered\_pair}(y,z))\in \mbox{application\_function} \ \Rightarrow \ y\in \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{application\_function\_defn}_2,\mbox{axiom})$

$\mbox{ordered\_pair}(x,\mbox{ordered\_pair}(y,z))\in \mbox{application\_function} \ \Rightarrow \ \mbox{apply}(x,y) = z\qquad\mbox{cnf}(\mbox{application\_function\_defn}_3,\mbox{axiom})$

$( \mbox{ordered\_pair}(x,\mbox{ordered\_pair}(y,z))\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\mbox{ and }y\in \mbox{domain\_of}(x) ) \ \Rightarrow \ \mbox{ordered\_pair}(x,\mbox{ordered\_pair}(y,\mbox{apply}(x,y)))\in \mbox{application\_function}\qquad\mbox{cnf}(\mbox{application\_function\_defn}_4,\mbox{axiom})$

$\mbox{maps}(\mbox{xf},x,y) \ \Rightarrow \ \mbox{function}(\mbox{xf})\qquad\mbox{cnf}(\mbox{maps}_1,\mbox{axiom})$

$\mbox{maps}(\mbox{xf},x,y) \ \Rightarrow \ \mbox{domain\_of}(\mbox{xf}) = x\qquad\mbox{cnf}(\mbox{maps}_2,\mbox{axiom})$

$\mbox{maps}(\mbox{xf},x,y) \ \Rightarrow \ \mbox{subclass}(\mbox{range\_of}(\mbox{xf}),y)\qquad\mbox{cnf}(\mbox{maps}_3,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }\mbox{subclass}(\mbox{range\_of}(\mbox{xf}),y) ) \ \Rightarrow \ \mbox{maps}(\mbox{xf},\mbox{domain\_of}(\mbox{xf}),y)\qquad\mbox{cnf}(\mbox{maps}_4,\mbox{axiom})$

\medskip\textbf{SET006+3.ax} Order relation (Naive set theory)






$\forall r,e{:}\;( \mbox{order}(r,e)\iff ( \forall x{:}\;( x\in e \ \Rightarrow \ \mbox{apply}(r,x,x) )\mbox{ and }\forall x,y{:}\;( ( x\in e\mbox{ and }y\in e ) \ \Rightarrow \ ( ( \mbox{apply}(r,x,y)\mbox{ and }\mbox{apply}(r,y,x) ) \ \Rightarrow \ x = y ) )\mbox{ and }\forall x,y,z{:}\;( ( x\in e\mbox{ and }y\in e\mbox{ and }z\in e ) \ \Rightarrow \ ( ( \mbox{apply}(r,x,y)\mbox{ and }\mbox{apply}(r,y,z) ) \ \Rightarrow \ \mbox{apply}(r,x,z) ) ) ) )\qquad\mbox{fof}(\mbox{order},\mbox{axiom})$

$\forall r,e{:}\;( \mbox{total\_order}(r,e)\iff ( \mbox{order}(r,e)\mbox{ and }\forall x,y{:}\;( ( x\in e\mbox{ and }y\in e ) \ \Rightarrow \ ( \mbox{apply}(r,x,y)\mbox{ or }\mbox{apply}(r,y,x) ) ) ) )\qquad\mbox{fof}(\mbox{total\_order},\mbox{axiom})$

$\forall r,e,m{:}\;( b\iff \forall x{:}\;( x\in e \ \Rightarrow \ \mbox{apply}(r,x,m) ) )\qquad\mbox{fof}(\mbox{upper\_bound},\mbox{axiom})$

$\forall r,e,m{:}\;( a\iff \forall x{:}\;( x\in e \ \Rightarrow \ \mbox{apply}(r,m,x) ) )\qquad\mbox{fof}(\mbox{lower\_bound},\mbox{axiom})$

$\forall r,e,m{:}\;( \mbox{greatest}(m,r,e)\iff ( m\in e\mbox{ and }\forall x{:}\;( x\in e \ \Rightarrow \ \mbox{apply}(r,x,m) ) ) )\qquad\mbox{fof}(\mbox{greatest},\mbox{axiom})$

$\forall r,e,m{:}\;( \mbox{least}(m,r,e)\iff ( m\in e\mbox{ and }\forall x{:}\;( x\in e \ \Rightarrow \ \mbox{apply}(r,m,x) ) ) )\qquad\mbox{fof}(\mbox{least},\mbox{axiom})$

$\forall r,e,m{:}\;( \mbox{max}(m,r,e)\iff ( m\in e\mbox{ and }\forall x{:}\;( ( x\in e\mbox{ and }\mbox{apply}(r,m,x) ) \ \Rightarrow \ m = x ) ) )\qquad\mbox{fof}(\mbox{max},\mbox{axiom})$

$\forall r,e,m{:}\;( \mbox{min}(m,r,e)\iff ( m\in e\mbox{ and }\forall x{:}\;( ( x\in e\mbox{ and }\mbox{apply}(r,x,m) ) \ \Rightarrow \ m = x ) ) )\qquad\mbox{fof}(\mbox{min},\mbox{axiom})$

$\forall a,x,r,e{:}\;( \mbox{least\_upper\_bound}(a,x,r,e)\iff ( a\in x\mbox{ and }b\mbox{ and }\forall m{:}\;( ( m\in e\mbox{ and }b ) \ \Rightarrow \ \mbox{apply}(r,a,m) ) ) )\qquad\mbox{fof}(\mbox{least\_upper\_bound},\mbox{axiom})$

$\forall a,x,r,e{:}\;( \mbox{greatest\_lower\_bound}(a,x,r,e)\iff ( a\in x\mbox{ and }a\mbox{ and }\forall m{:}\;( ( m\in e\mbox{ and }a ) \ \Rightarrow \ \mbox{apply}(r,m,a) ) ) )\qquad\mbox{fof}(\mbox{greatest\_lower\_bound},\mbox{axiom})$

\medskip\textbf{SET001-2.ax} Membership and intersection






$( \mbox{intersection}(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\mbox{ and }\mbox{element}\in \mbox{intersection} ) \ \Rightarrow \ \mbox{element}\in \mbox{set}_1\qquad\mbox{cnf}(\mbox{member\_of\_intersection\_is\_member\_of\_set}_1,\mbox{axiom})$

$( \mbox{intersection}(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\mbox{ and }\mbox{element}\in \mbox{intersection} ) \ \Rightarrow \ \mbox{element}\in \mbox{set}_2\qquad\mbox{cnf}(\mbox{member\_of\_intersection\_is\_member\_of\_set}_2,\mbox{axiom})$

$( \mbox{intersection}(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\mbox{ and }\mbox{element}\in \mbox{set}_2\mbox{ and }\mbox{element}\in \mbox{set}_1 ) \ \Rightarrow \ \mbox{element}\in \mbox{intersection}\qquad\mbox{cnf}(\mbox{member\_of\_both\_is\_member\_of\_intersection},\mbox{axiom})$

$h(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\in \mbox{intersection}\mbox{ or }\mbox{intersection}(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\mbox{ or }h(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\in \mbox{set}_1\qquad\mbox{cnf}(\mbox{intersection\_axiom}_1,\mbox{axiom})$

$h(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\in \mbox{intersection}\mbox{ or }\mbox{intersection}(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\mbox{ or }h(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\in \mbox{set}_2\qquad\mbox{cnf}(\mbox{intersection\_axiom}_2,\mbox{axiom})$

$( h(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\in \mbox{intersection}\mbox{ and }h(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\in \mbox{set}_2\mbox{ and }h(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\in \mbox{set}_1 ) \ \Rightarrow \ \mbox{intersection}(\mbox{set}_1,\mbox{set}_2,\mbox{intersection})\qquad\mbox{cnf}(\mbox{intersection\_axiom}_3,\mbox{axiom})$

\medskip\textbf{SET006+4.ax} Ordinal numbers for the SET006+0 set theory axioms






$\forall a{:}\;( a\in \mbox{on}\iff ( \mbox{set}(a)\mbox{ and }\mbox{strict\_well\_order}(\mbox{member\_predicate},a)\mbox{ and }\forall x{:}\;( x\in a \ \Rightarrow \ x\subseteq a ) ) )\qquad\mbox{fof}(\mbox{ordinal\_number},\mbox{axiom})$

$\forall r,e{:}\;( \mbox{strict\_well\_order}(r,e)\iff ( \mbox{strict\_order}(r,e)\mbox{ and }\forall a{:}\;( ( a\subseteq e\mbox{ and }\exists x{:}\;x\in a ) \ \Rightarrow \ \exists y{:}\;\mbox{least}(y,r,a) ) ) )\qquad\mbox{fof}(\mbox{strict\_well\_order},\mbox{axiom})$

$\forall r,e,m{:}\;( \mbox{least}(m,r,e)\iff ( m\in e\mbox{ and }\forall x{:}\;( x\in e \ \Rightarrow \ ( m = x\mbox{ or }\mbox{apply}(r,m,x) ) ) ) )\qquad\mbox{fof}(\mbox{least},\mbox{axiom})$

$\forall x,y{:}\;( \mbox{apply}(\mbox{member\_predicate},x,y)\iff x\in y )\qquad\mbox{fof}(\mbox{rel\_member},\mbox{axiom})$

$\forall r,e{:}\;( \mbox{strict\_order}(r,e)\iff ( \forall x,y{:}\;( ( x\in e\mbox{ and }y\in e ) \ \Rightarrow \ \neg\,\mbox{apply}(r,x,y)\mbox{ and }\mbox{apply}(r,y,x) )\mbox{ and }\forall x,y,z{:}\;( ( x\in e\mbox{ and }y\in e\mbox{ and }z\in e ) \ \Rightarrow \ ( ( \mbox{apply}(r,x,y)\mbox{ and }\mbox{apply}(r,y,z) ) \ \Rightarrow \ \mbox{apply}(r,x,z) ) ) ) )\qquad\mbox{fof}(\mbox{strict\_order},\mbox{axiom})$

$\forall x{:}\;( \mbox{set}(x) \ \Rightarrow \ \forall y{:}\;( y\in x \ \Rightarrow \ \mbox{set}(y) ) )\qquad\mbox{fof}(\mbox{set\_member},\mbox{axiom})$

$\forall x,r,a,y{:}\;( y\in \mbox{initial\_segment}(x,r,a)\iff ( y\in a\mbox{ and }\mbox{apply}(r,y,x) ) )\qquad\mbox{fof}(\mbox{initial\_segment},\mbox{axiom})$

$\forall a,x{:}\;( x\in \mbox{suc}(a)\iff x\in \mbox{union}(a,\mbox{singleton}(a)) )\qquad\mbox{fof}(\mbox{successor},\mbox{axiom})$

\medskip\textbf{SET008$\wedge$1.ax} Definitions for functions






$\mbox{fun\_image}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{fun\_image\_decl},\mbox{type})$

$\mbox{fun\_image} = ( \lambda f{:}\;\mbox{\$i}\to \mbox{\$i},a{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}{:}\;\exists x{:}\;\mbox{\$i}{:}\;( a@ x\mbox{ and }y = ( f@ x ) ) )\qquad\mbox{thf}(\mbox{fun\_image},\mbox{definition})$

$\mbox{fun\_composition}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$i}\to \mbox{\$i}\qquad\mbox{thf}(\mbox{fun\_composition\_decl},\mbox{type})$

$\mbox{fun\_composition} = ( \lambda f{:}\;\mbox{\$i}\to \mbox{\$i},g{:}\;\mbox{\$i}\to \mbox{\$i},x{:}\;\mbox{\$i}{:}\;( g@ ( f@ x ) ) )\qquad\mbox{thf}(\mbox{fun\_composition},\mbox{definition})$

$\mbox{fun\_inv\_image}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{fun\_inv\_image\_decl},\mbox{type})$

$\mbox{fun\_inv\_image} = ( \lambda f{:}\;\mbox{\$i}\to \mbox{\$i},b{:}\;\mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( b@ y\mbox{ and }y = ( f@ x ) ) )\qquad\mbox{thf}(\mbox{fun\_inv\_image},\mbox{definition})$

$\mbox{fun\_injective}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{fun\_injective\_decl},\mbox{type})$

$\mbox{fun\_injective} = ( \lambda f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( f@ x ) = ( f@ y ) \ \Rightarrow \ x = y ) )\qquad\mbox{thf}(\mbox{fun\_injective},\mbox{definition})$

$\mbox{fun\_surjective}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{fun\_surjective\_decl},\mbox{type})$

$\mbox{fun\_surjective} = ( \lambda f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;\forall y{:}\;\mbox{\$i}{:}\;\exists x{:}\;\mbox{\$i}{:}\;y = ( f@ x ) )\qquad\mbox{thf}(\mbox{fun\_surjective},\mbox{definition})$

$\mbox{fun\_bijective}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{fun\_bijective\_decl},\mbox{type})$

$\mbox{fun\_bijective} = ( \lambda f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{fun\_injective}@ f\mbox{ and }\mbox{fun\_surjective}@ f ) )\qquad\mbox{thf}(\mbox{fun\_bijective},\mbox{definition})$

$\mbox{fun\_decreasing}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{fun\_decreasing\_decl},\mbox{type})$

$\mbox{fun\_decreasing} = ( \lambda f{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{sMALLER}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{sMALLER}@ x@ y ) \ \Rightarrow \ ( \mbox{sMALLER}@ ( f@ y )@ ( f@ x ) ) ) )\qquad\mbox{thf}(\mbox{fun\_decreasing},\mbox{definition})$

$\mbox{fun\_increasing}{:}\;( \mbox{\$i}\to \mbox{\$i} )\to ( \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{fun\_increasing\_decl},\mbox{type})$

$\mbox{fun\_increasing} = ( \lambda f{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{sMALLER}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( ( \mbox{sMALLER}@ x@ y ) \ \Rightarrow \ ( \mbox{sMALLER}@ ( f@ x )@ ( f@ y ) ) ) )\qquad\mbox{thf}(\mbox{fun\_increasing},\mbox{definition})$

\medskip\textbf{SET008$\wedge$0.ax} Basic set theory definitions






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

$\mbox{in} = ( \lambda x{:}\;\mbox{\$i},m{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( m@ x ) )\qquad\mbox{thf}(\mbox{in},\mbox{definition})$

$\mbox{is\_a}{:}\;\mbox{\$i}\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$o}\qquad\mbox{thf}(\mbox{is\_a\_decl},\mbox{type})$

$\mbox{is\_a} = ( \lambda x{:}\;\mbox{\$i},m{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( m@ x ) )\qquad\mbox{thf}(\mbox{is\_a},\mbox{definition})$

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

$\mbox{emptyset} = ( \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$false} )\qquad\mbox{thf}(\mbox{emptyset},\mbox{definition})$

$\mbox{unord\_pair}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{unord\_pair\_decl},\mbox{type})$

$\mbox{unord\_pair} = ( \lambda x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},u{:}\;\mbox{\$i}{:}\;( u = x\mbox{ or }u = y ) )\qquad\mbox{thf}(\mbox{unord\_pair},\mbox{definition})$

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

$\mbox{singleton} = ( \lambda x{:}\;\mbox{\$i},u{:}\;\mbox{\$i}{:}\;u = x )\qquad\mbox{thf}(\mbox{singleton},\mbox{definition})$

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

$\mbox{union} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},u{:}\;\mbox{\$i}{:}\;( x@ u\mbox{ or }y@ u ) )\qquad\mbox{thf}(\mbox{union},\mbox{definition})$

$\mbox{excl\_union}{:}\;( \mbox{\$i}\to \mbox{\$o} )\to ( \mbox{\$i}\to \mbox{\$o} )\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{excl\_union\_decl},\mbox{type})$

$\mbox{excl\_union} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},u{:}\;\mbox{\$i}{:}\;( ( x@ u\mbox{ and }\neg\,y@ u )\mbox{ or }( \neg\,x@ u\mbox{ and }y@ u ) ) )\qquad\mbox{thf}(\mbox{excl\_union},\mbox{definition})$

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

$\mbox{intersection} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},u{:}\;\mbox{\$i}{:}\;( x@ u\mbox{ and }y@ u ) )\qquad\mbox{thf}(\mbox{intersection},\mbox{definition})$

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

$\mbox{setminus} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},u{:}\;\mbox{\$i}{:}\;( x@ u\mbox{ and }\neg\,y@ u ) )\qquad\mbox{thf}(\mbox{setminus},\mbox{definition})$

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

$\mbox{complement} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},u{:}\;\mbox{\$i}{:}\;\neg\,x@ u )\qquad\mbox{thf}(\mbox{complement},\mbox{definition})$

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

$\mbox{disjoint} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{intersection}@ x@ y ) = \mbox{emptyset} )\qquad\mbox{thf}(\mbox{disjoint},\mbox{definition})$

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

$\subseteq  = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\forall u{:}\;\mbox{\$i}{:}\;( ( x@ u ) \ \Rightarrow \ ( y@ u ) ) )\qquad\mbox{thf}(\mbox{subset},\mbox{definition})$

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

$\mbox{meets} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists u{:}\;\mbox{\$i}{:}\;( x@ u\mbox{ and }y@ u ) )\qquad\mbox{thf}(\mbox{meets},\mbox{definition})$

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

$\mbox{misses} = ( \lambda x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\exists u{:}\;\mbox{\$i}{:}\;( x@ u\mbox{ and }y@ u ) )\qquad\mbox{thf}(\mbox{misses},\mbox{definition})$

\medskip\textbf{SET545-6.p} Application special case 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_application\_special\_case1}_1,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{universal\_class},x)\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_application\_special\_case1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET942+1.p} subset(A,B) =$>$ subset(union(A),union(B))






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

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

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

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

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

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

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{union}(a)\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{t95\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET243-6.p} Restriction property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{restrict}(\mbox{xf},x_1,y_1),x_2,y_2)\ne \mbox{restrict}(\mbox{xf},\mbox{intersection}(x_1,x_2),\mbox{intersection}(y_1,y_2))\qquad\mbox{cnf}(\mbox{prove\_restriction\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET764$\wedge$4.p} The inverse image of empty set is empty






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^1.ax'})$

$\forall f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{fun\_inv\_image}@ f@ \mbox{emptyset} ) = \mbox{emptyset}\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET566-6.p} Compatible function property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{compatible}(\mbox{xh}_1,\mbox{xf}_1,\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{compatible}(\mbox{xh}_2,\mbox{xf}_2,\mbox{xf}_3)\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property4}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{compatible}(\mbox{xh}_2\circ \mbox{xh}_1,\mbox{xf}_1,\mbox{xf}_3)\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property4}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET771+4.p} Equality of images defines a equivalence relation






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall f,a,b,r{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\forall x_1,x_2{:}\;( ( x_1\in a\mbox{ and }x_2\in a ) \ \Rightarrow \ ( \mbox{apply}(r,x_1,x_2)\iff \exists y{:}\;( y\in b\mbox{ and }\mbox{apply}(f,x_1,y)\mbox{ and }\mbox{apply}(f,x_2,y) ) ) ) ) \ \Rightarrow \ \mbox{equivalence}(r,a) )\qquad\mbox{fof}(\mbox{thIII}_{07},\mbox{conjecture})$

\medskip\textbf{SET196$\wedge$5.p} TPS problem BOOL-PROP-37

Trybulec's 37th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) \ \Rightarrow \ ( x@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_37\_pme},\mbox{conjecture})$

\medskip\textbf{SET367-6.p} Power class not in null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{power\_class}(x) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_power\_class\_not\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET582+3.p} If x not in X iff x in Y iff x in Z, then X = Y sym\ Z

If for every x : x is not in X iff x is in Y iff x is in Z,
 then X is the symmetric difference of Y and Z.




$\forall b,c{:}\;( \forall d{:}\;( d\in b\iff d\in c ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{member\_equal},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( \forall e{:}\;( \neg\,e\in b\iff ( e\in c\iff e\in d ) ) \ \Rightarrow \ b = \mbox{symmetric\_difference}(c,d) )\qquad\mbox{fof}(\mbox{prove\_th}_{25},\mbox{conjecture})$

\medskip\textbf{SET872+1.p} subset(singleton(A),unordered\_pair(A,B))






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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

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

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

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

$\forall a,b{:}\;\mbox{singleton}(a)\subseteq \mbox{unordered\_pair}(a,b)\qquad\mbox{fof}(\mbox{t12\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET912+1.p} ( in(A,B) \& in(C,B) ) =$>$ intsctn(uno\_pair(A,C),B) = uno\_pair(A,C)






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

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

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

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

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

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

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

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_intersection}_2(a,b) = a )\qquad\mbox{fof}(\mbox{t28\_xboole}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{unordered\_pair}(a,b)\subseteq c\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,c) ) )\qquad\mbox{fof}(\mbox{t38\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{in}(c,b) ) \ \Rightarrow \ \mbox{set\_intersection}_2(\mbox{unordered\_pair}(a,c),b) = \mbox{unordered\_pair}(a,c) )\qquad\mbox{fof}(\mbox{t53\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET802+4.p} M is the least of X iff it is a member and a GLB






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e{:}\;( \mbox{order}(r,e) \ \Rightarrow \ \forall x{:}\;( x\subseteq e \ \Rightarrow \ \forall m{:}\;( \mbox{least}(m,r,x)\iff ( m\in x\mbox{ and }\mbox{greatest\_lower\_bound}(m,x,r,e) ) ) ) )\qquad\mbox{fof}(\mbox{thIV}_{14},\mbox{conjecture})$

\medskip\textbf{SET558-6.p} Compatible functions alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{operation}(\mbox{xf}_1)\qquad\mbox{cnf}(\mbox{prove\_compatible\_functions\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$\mbox{compatible}(\mbox{xh},\mbox{xf}_1,\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_compatible\_functions\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

$\mbox{cross\_product}(\mbox{domain\_of}(\mbox{xh}),\mbox{domain\_of}(\mbox{xh}))\ne \mbox{domain\_of}(\mbox{xf}_1)\qquad\mbox{cnf}(\mbox{prove\_compatible\_functions\_alternate\_defn1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET837-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_X} = \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_X} = \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{v\_X} = \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_x},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{v\_X} = \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_x},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_x},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_X}\ne \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ ( \mbox{v\_X} = \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET385-6.p} Corollary 2 to relation property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$( \mbox{element\_relation}' )'\ne \mbox{element\_relation}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_relation\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET617$\wedge$5.p} TPS problem BOOL-PROP-92

Trybulec's 92nd Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o}{:}\;( ( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,\mbox{\$false} )\mbox{ or }( \mbox{\$false}\mbox{ and }\neg\,x@ \mbox{xz} ) ) ) = x\mbox{ and }( \lambda \mbox{xz}{:}\;a{:}\;( ( \mbox{\$false}\mbox{ and }\neg\,x@ \mbox{xz} )\mbox{ or }( x@ \mbox{xz}\mbox{ and }\neg\,\mbox{\$false} ) ) ) = x )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_92\_pme},\mbox{conjecture})$

\medskip\textbf{SET769+4.p} Equality of equivalence classes 2

Two equivalence classes are equal if and only if they are not
 : disjoint.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall e,r,a,b{:}\;( ( \mbox{equivalence}(r,e)\mbox{ and }a\in e\mbox{ and }b\in e ) \ \Rightarrow \ ( \mbox{equal\_set}(\mbox{equivalence\_class}(a,e,r),\mbox{equivalence\_class}(b,e,r))\iff \neg\,\mbox{disjoint}(\mbox{equivalence\_class}(a,e,r),\mbox{equivalence\_class}(b,e,r)) ) )\qquad\mbox{fof}(\mbox{thIII}_{05},\mbox{conjecture})$

\medskip\textbf{SET504-6.p} Corollary 1 to universal class not set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,\mbox{universal\_class})\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_universal\_class\_not\_set}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET841-2.p} Problem about Zorn's lemma






$\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_refl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_n},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_n} = \mbox{v\_m} )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_subsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET889+1.p} powerset(singleton(A)) = unordered\_pair(empty\_set,singleton(A))






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

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

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

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

$\forall a,b{:}\;( a\subseteq \mbox{singleton}(b)\iff ( a = \mbox{empty\_set}\mbox{ or }a = \mbox{singleton}(b) ) )\qquad\mbox{fof}(\mbox{l4\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a{:}\;\mbox{powerset}(\mbox{singleton}(a)) = \mbox{unordered\_pair}(\mbox{empty\_set},\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{t30\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET517-6.p} If member of X is X then X is not a singleton of a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{singleton}(\mbox{member\_of}(x)) = x\qquad\mbox{cnf}(\mbox{prove\_not\_singleton\_of\_set}_1,\mbox{negated\_conjecture})$

$\mbox{member\_of}(x) = x\qquad\mbox{cnf}(\mbox{prove\_not\_singleton\_of\_set}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET201+3.p} Intersection is monotonic

If X is a subset of Y and Z is a subset of V, then the
 intersection of X and Z is a subset of the intersection of Y
 and V.




$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d,e{:}\;( ( b\subseteq c\mbox{ and }d\subseteq e ) \ \Rightarrow \ \mbox{intersection}(b,d)\subseteq \mbox{intersection}(c,e) )\qquad\mbox{fof}(\mbox{prove\_th}_{41},\mbox{conjecture})$

\medskip\textbf{SET404-6.p} Monotonicity of composition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{xr},\mbox{yr})\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_composition1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{zr}\circ \mbox{xr},\mbox{zr}\circ \mbox{yr})\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_composition1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET738+4.p} Problem on composition of mappings 3

Consider three mappings F from A to B,G from B to C,H from
 C to A. If HoGoF and FoHoG are injective and GoFoH surjective,
 then H is one-to-one.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,c,a)\mbox{ and }\mbox{injective}(\mbox{compose\_function}(h,\mbox{compose\_function}(g,f,a,b,c),a,c,a),a,a)\mbox{ and }\mbox{injective}(\mbox{compose\_function}(f,\mbox{compose\_function}(h,g,b,c,a),b,a,b),b,b)\mbox{ and }\mbox{surjective}(\mbox{compose\_function}(g,\mbox{compose\_function}(f,h,c,a,b),c,b,c),c,c) ) \ \Rightarrow \ \mbox{one\_to\_one}(h,c,a) )\qquad\mbox{fof}(\mbox{thII}_{29},\mbox{conjecture})$

\medskip\textbf{SET500-6.p} Diagonalization property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{range\_of}(\mbox{intersection}(\mbox{xr},\mbox{identity\_relation}))'\ne \mbox{diagonalise}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET320-6.p} Corollary 4 to range property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(\mbox{range\_of}(\mbox{restrict}(x_1,y_1,z_1)),\mbox{range\_of}(\mbox{restrict}(x_2,y_2,z_2))),\mbox{cross\_product}(z_1,z_2))\qquad\mbox{cnf}(\mbox{prove\_corollary\_4\_to\_range\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET056-7.p} Expanded equality definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_equality1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(x,y)\in x\qquad\mbox{cnf}(\mbox{prove\_equality1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{prove\_equality1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET611$\wedge$5.p} TPS problem BOOL-PROP-84

Trybulec's 84th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} )\iff ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) ) = x )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_84\_pme},\mbox{conjecture})$

\medskip\textbf{SET606$\wedge$3.p} X \ X $\wedge$ Y = X \ Y

The difference of X and the intersection of X and Y is the
 difference of X and Y.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{setminus}@ x@ ( \mbox{intersection}@ x@ y ) ) = ( \mbox{setminus}@ x@ y )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET858-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) = \mbox{v\_m} ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_2,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_3,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_U},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_U} = \mbox{v\_m} )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma2}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_m},\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET261-6.p} Domain of null class is the null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(\mbox{null\_class})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_domain\_of\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET014+4.p} Union of subsets is a subset

If A and B are contained in C then the union of A and B is also.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,x,y{:}\;( ( x\subseteq a\mbox{ and }y\subseteq a )\iff \mbox{union}(x,y)\subseteq a )\qquad\mbox{fof}(\mbox{thI}_{45},\mbox{conjecture})$

\medskip\textbf{SET160-6.p} Commutativity of union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,y)\ne \mbox{union}(y,x)\qquad\mbox{cnf}(\mbox{prove\_commutativity\_of\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET827-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_V},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_V},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_V},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET112-6.p} 2nd is the ordered pair, second condition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(\mbox{first}(x),\mbox{second}(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_1st\_and\_2nd\_5}_1,\mbox{negated\_conjecture})$

$\mbox{second}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_1st\_and\_2nd\_5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET058-7.p} Expanded equality definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x\qquad\mbox{cnf}(\mbox{prove\_equality3}_1,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_equality3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(x,y)\in x\qquad\mbox{cnf}(\mbox{prove\_equality3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET321-6.p} Range property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{range\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_range\_property5}_1,\mbox{negated\_conjecture})$

$\neg\,1_{\text{dom}(z)}\in \mbox{domain\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_range\_property5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET689+4.p} Property of subset

If A is a subset of B,B a subset of C and C a subset of A,
 then A is equal to C.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c\mbox{ and }c\subseteq a ) \ \Rightarrow \ \mbox{equal\_set}(a,c) )\qquad\mbox{fof}(\mbox{thI}_{05},\mbox{conjecture})$

\medskip\textbf{SET605+3.p} The difference of X and the union of X and Y is the empty set






$\forall b,c{:}\;b\subseteq \mbox{union}(b,c)\qquad\mbox{fof}(\mbox{subset\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b\setminus c = \mbox{empty\_set}\iff b\subseteq c )\qquad\mbox{fof}(\mbox{difference\_empty\_set},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;b\setminus \mbox{union}(b,c) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{prove\_th}_{76},\mbox{conjecture})$

\medskip\textbf{SET882+1.p} A != B =$>$ diff(unordered\_pair(A,B),singleton(B)) = singleton(A)






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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c) = \mbox{singleton}(a)\iff ( \neg\,\mbox{in}(a,c)\mbox{ and }( \mbox{in}(b,c)\mbox{ or }a = b ) ) )\qquad\mbox{fof}(\mbox{l39\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b{:}\;( a\ne b \ \Rightarrow \ \mbox{set\_difference}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(b)) = \mbox{singleton}(a) )\qquad\mbox{fof}(\mbox{t23\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET591+3.p} If X (= Y \ X, then X = the empty set

If X is a subset of the difference of Y and X, then X is the
 empty set.




$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq ( c\setminus b ) \ \Rightarrow \ b = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{prove\_th}_{50},\mbox{conjecture})$

\medskip\textbf{SET459-6.p} Domain of identity is the universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(\mbox{identity\_relation})\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_domain\_of\_identity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET316-6.p} Range property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{range\_of}(\mbox{restrict}(x,\mbox{universal\_class},z))\ne \mbox{intersection}(\mbox{range\_of}(x),z)\qquad\mbox{cnf}(\mbox{prove\_range\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET608+3.p} X $\wedge$ Y U (X \ Y) = X

The intersection of X and the union of Y and (the difference of
 X and Y) is X.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(\mbox{intersection}(b,c),b\setminus c) = b\qquad\mbox{fof}(\mbox{prove\_union\_intersection\_difference},\mbox{conjecture})$

\medskip\textbf{SET597$\wedge$5.p} TPS problem BOOL-PROP-56

Trybulec's 56th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( x = ( \lambda \mbox{xz}{:}\;a{:}\;( y@ \mbox{xz}\mbox{ or }z@ \mbox{xz} ) )\iff ( \forall \mbox{xx}{:}\;a{:}\;( ( y@ \mbox{xx} ) \ \Rightarrow \ ( x@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( z@ \mbox{xx} ) \ \Rightarrow \ ( x@ \mbox{xx} ) )\mbox{ and }\forall v{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( y@ \mbox{xx} ) \ \Rightarrow \ ( v@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( z@ \mbox{xx} ) \ \Rightarrow \ ( v@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( v@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_56\_pme},\mbox{conjecture})$

\medskip\textbf{SET693+4.p} Property of union

B is a subset of A if and only if A is equal to the union of
 A and B.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;( \mbox{equal\_set}(a,\mbox{union}(a,b))\iff b\subseteq a )\qquad\mbox{fof}(\mbox{thI}_{20},\mbox{conjecture})$

\medskip\textbf{SET319-6.p} Corollary 3 to range property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(u,\mbox{range\_of}(\mbox{restrict}(x,y,z))),\mbox{cross\_product}(u,\mbox{intersection}(z,\mbox{range\_of}(x))))\qquad\mbox{cnf}(\mbox{prove\_corollary\_3\_to\_range\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET078-6.p} Corollary to every singleton is a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{singleton}(y)\in \mbox{unordered\_pair}(x,\mbox{singleton}(y))\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_singletons\_are\_sets}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET595+3.p} If X (= Y, then Y = X U (Y \ X)

If X is a subset of Y, then Y is the union of X and (the
 difference of Y and X).




$\forall b,c{:}\;( \forall d{:}\;( d\in b\iff d\in c ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{member\_equal},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c \ \Rightarrow \ c = \mbox{union}(b,c\setminus b) )\qquad\mbox{fof}(\mbox{prove\_th}_{54},\mbox{conjecture})$

\medskip\textbf{SET830-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET829-2.p} Problem about set theory






$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET818-2.p} Problem about set theory






$\neg\,\mbox{c\_in}(\mbox{v\_a},\mbox{c\_emptyset},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OemptyE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_U},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{SET391-6.p} Composition property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_composition\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(y,z)\in \mbox{yr}\qquad\mbox{cnf}(\mbox{prove\_composition\_property1}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_composition\_property1}_3,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(y,z)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_composition\_property1}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(x,z)\in \mbox{yr}\circ \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_composition\_property1}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET593$\wedge$5.p} TPS problem BOOL-PROP-52

Trybulec's 52nd Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) ) \ \Rightarrow \ ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_52\_pme},\mbox{conjecture})$

\medskip\textbf{SET222-6.p} Cross product property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{intersection}(\mbox{cross\_product}(w,x),\mbox{cross\_product}(y,z)),\mbox{cross\_product}(w,z))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET951+1.p} Basic properties of sets, theorem 104

~ ( in(A,intersection(cart\_product(B,C),cart\_product(D,E)))
 \& ~ ( A = ordered\_pair(F,G) \& in(F,intersection(B,D))
 \& in(G,set\_intersection2(C,E)) ) )




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

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

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

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

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

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

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

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

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

$\forall a,b,c,d,e{:}\;\neg\,\mbox{in}(a,\mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(b,c),\mbox{cartesian\_product}_2(d,e)))\mbox{ and }\forall f,g{:}\;\neg\,a = \mbox{ordered\_pair}(f,g)\mbox{ and }\mbox{in}(f,\mbox{set\_intersection}_2(b,d))\mbox{ and }\mbox{in}(g,\mbox{set\_intersection}_2(c,e))\qquad\mbox{fof}(\mbox{t104\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c,d{:}\;( \mbox{ordered\_pair}(a,b) = \mbox{ordered\_pair}(c,d) \ \Rightarrow \ ( a = c\mbox{ and }b = d ) )\qquad\mbox{fof}(\mbox{t33\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET017-3.p} Left cancellation for non-ordered pairs






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{b\_little\_set},\mbox{hypothesis})$

$\mbox{non\_ordered\_pair}(c,a) = \mbox{non\_ordered\_pair}(d,b)\qquad\mbox{cnf}(\mbox{equal\_non\_ordered\_pairs},\mbox{hypothesis})$

$a\ne c\qquad\mbox{cnf}(\mbox{prove\_left\_cancellation},\mbox{negated\_conjecture})$

\medskip\textbf{SET138-6.p} Kludge 2 to instantiate variables in unordered triples






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple\_kludge2}_1,\mbox{negated\_conjecture})$

$v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple\_kludge2}_2,\mbox{negated\_conjecture})$

$w\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple\_kludge2}_3,\mbox{negated\_conjecture})$

$\neg\,u\in \mbox{set\_builder}(u,\mbox{set\_builder}(v,\mbox{set\_builder}(w,\mbox{null\_class})))\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple\_kludge2}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET760+4.p} Composition of images 2

If F is surjective, then the image of the inverse image of Y is
 equal to Y.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,y{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{surjective}(f,a,b)\mbox{ and }y\subseteq b ) \ \Rightarrow \ \mbox{equal\_set}(\mbox{image}_3(f,\mbox{inverse\_image}_3(f,y,a),b),y) )\qquad\mbox{fof}(\mbox{thIIa}_{10},\mbox{conjecture})$

\medskip\textbf{SET159-6.p} Associativity of union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{union}(x,y),z)\ne \mbox{union}(x,\mbox{union}(y,z))\qquad\mbox{cnf}(\mbox{prove\_associativity\_of\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET037-6.p} Properties of apply for functions, part 2 of 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_application\_property10}_1,\mbox{negated\_conjecture})$

$x\in \mbox{domain\_of}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_application\_property10}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(x,\mbox{apply}(\mbox{xf},x))\in \mbox{xf}\qquad\mbox{cnf}(\mbox{prove\_application\_property10}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET205-6.p} Cross product with null class 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(x,\mbox{null\_class})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_with\_null\_class1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET416-6.p} Composition with singleton function 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,y\in \mbox{domain\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_composition\_with\_singleton\_function2}_1,\mbox{negated\_conjecture})$

$z\circ \mbox{singleton}(\mbox{ordered\_pair}(x,y))\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_composition\_with\_singleton\_function2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET041-4.p} Properties of apply for composition of functions, 3 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$a\in \mbox{domain\_of}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{member\_of\_domain},\mbox{hypothesis})$

$\mbox{apply}(\mbox{another\_function},\mbox{apply}(\mbox{a\_function},a))\ne \mbox{apply}(\mbox{another\_function}\circ \mbox{a\_function},a)\qquad\mbox{cnf}(\mbox{prove\_apply\_for\_composition}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET235-6.p} Cross product property 10






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property10}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(\mbox{first}(\mbox{not\_subclass\_element}(x,y)),\mbox{second}(\mbox{not\_subclass\_element}(x,y)))\in x\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property10}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property10}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET074-6.p} Corollary to unordered pair axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_unordered\_pair\_axiom2}_1,\mbox{negated\_conjecture})$

$\mbox{unordered\_pair}(x,y) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_unordered\_pair\_axiom2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET817+4.p} The product of a nonempty set of ordinals is an ordinal






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+4.ax'})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{thI}_3,\mbox{axiom})$

$\forall x{:}\;( \mbox{set}(x) \ \Rightarrow \ \mbox{set}(\mbox{\mbox{product}}(x)) )\qquad\mbox{fof}(\mbox{set\_product},\mbox{axiom})$

$\forall a,x{:}\;( x\in a \ \Rightarrow \ \mbox{\mbox{product}}(a)\subseteq x )\qquad\mbox{fof}(\mbox{thI}_{42},\mbox{axiom})$

$\forall a{:}\;( ( a\subseteq \mbox{on}\mbox{ and }\mbox{set}(a)\mbox{ and }\exists x{:}\;x\in a ) \ \Rightarrow \ \mbox{\mbox{product}}(a)\in \mbox{on} )\qquad\mbox{fof}(\mbox{thV}_{21},\mbox{conjecture})$

\medskip\textbf{SET713+4.p} The inverse of a one-to-one mapping is one-to-one






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{one\_to\_one}(f,a,b) ) \ \Rightarrow \ \mbox{one\_to\_one}(\mbox{inverse\_function}(f,a,b),b,a) )\qquad\mbox{fof}(\mbox{thII}_{04},\mbox{conjecture})$

\medskip\textbf{SET509-6.p} Corollary 2 to singleton in unordered pair axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{unordered\_pair}(\mbox{universal\_class},x)\ne \mbox{singleton}(x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_singleton\_in\_unordered\_pair}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET804+4.p} Two different minimal elements implies no least element






$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e{:}\;( \mbox{order}(r,e) \ \Rightarrow \ \forall m_1,m_2{:}\;( ( \mbox{min}(m_1,r,e)\mbox{ and }\mbox{min}(m_2,r,e)\mbox{ and }m_1\ne m_2 ) \ \Rightarrow \ \neg\,\exists m{:}\;\mbox{least}(m,r,e) ) )\qquad\mbox{fof}(\mbox{thIV}_{16},\mbox{conjecture})$

\medskip\textbf{SET535-6.p} Application property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{image}(\mbox{xf},\mbox{singleton}(x))\qquad\mbox{cnf}(\mbox{prove\_application\_property3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(y,\mbox{apply}(\mbox{xf},x))\qquad\mbox{cnf}(\mbox{prove\_application\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET640$\wedge$3.p} A a subset of R (X to Y) =$>$ A a subset of X x Y

If A is a subset of a relation R from X to Y then A is a subset
 of X x Y.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},q{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{sub\_rel}@ r@ q ) \ \Rightarrow \ ( \mbox{sub\_rel}@ r@ ( \mbox{cartesian\_product}@ \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$true}@ \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$true} ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET081+1.p} Only X can belong to {X}






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( y\in \mbox{singleton}(x) \ \Rightarrow \ y = x )\qquad\mbox{fof}(\mbox{only\_member\_in\_singleton},\mbox{conjecture})$

\medskip\textbf{SET521-6.p} Ordered pair determines components 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{first}(\mbox{ordered\_pair}(x,y)) = \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_determines\_components1}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_determines\_components1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET822-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_P}(\mbox{v\_f}(\mbox{v\_b}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_f}(\mbox{v\_U}),\mbox{v\_V},\mbox{tc\_IntDef\_Oint}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U},\mbox{v\_V}),\mbox{v\_V},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_f}(\mbox{v\_U}),\mbox{v\_V},\mbox{tc\_IntDef\_Oint}) \ \Rightarrow \ \neg\,\mbox{v\_P}(\mbox{v\_x}(\mbox{v\_U},\mbox{v\_V}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET844-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Main\_OUnion\_\_least\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_least}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Main\_OUnion\_\_least\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_least}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_OsuccI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oeq\_\_succ\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET370-6.p} Power class property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,y),\mbox{power\_class}(\mbox{power\_class}(\mbox{union}(x,\mbox{power\_class}(y)))))\qquad\mbox{cnf}(\mbox{prove\_power\_class\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET624+3.p} X intersects Y U Z iff X intersects Y or X intersects Z

X intersects the union of Y and Z iff X intersects Y or X
 intersects Z.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( \mbox{intersect}(b,\mbox{union}(c,d))\iff ( \mbox{intersect}(b,c)\mbox{ or }\mbox{intersect}(b,d) ) )\qquad\mbox{fof}(\mbox{prove\_intersect\_with\_union},\mbox{conjecture})$

\medskip\textbf{SET348-6.p} Sum class of universal class is universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{sum\_class}(\mbox{universal\_class})\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_of\_universal\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET194+3.p} X is a subset of the union of X and Y






$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;b\subseteq \mbox{union}(b,c)\qquad\mbox{fof}(\mbox{prove\_subset\_of\_union},\mbox{conjecture})$

\medskip\textbf{SET025-4.p} Ordered pairs are little sets






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\neg\,\mbox{little\_set}(\mbox{ordered\_pair}(a,b))\qquad\mbox{cnf}(\mbox{prove\_ordered\_pairs\_are\_small},\mbox{negated\_conjecture})$

\medskip\textbf{SET113-6.p} 1st is unique if x is not an ordered pair of sets, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{ordered\_pair}(\mbox{first}_1(x),\mbox{second}_1(x))\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_unique\_1st\_and\_2nd\_in\_pair\_of\_non\_sets1}_1,\mbox{negated\_conjecture})$

$\mbox{first}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_unique\_1st\_in\_pair\_of\_non\_sets},\mbox{negated\_conjecture})$

\medskip\textbf{SET252-6.p} Restriction property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{restrict}(x,y,z),x)\qquad\mbox{cnf}(\mbox{prove\_restriction\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET035-4.p} Maps for composition






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{maps}(\mbox{function}_1,a,b)\qquad\mbox{cnf}(\mbox{one\_mapping},\mbox{hypothesis})$

$\mbox{maps}(\mbox{function}_2,c,d)\qquad\mbox{cnf}(\mbox{another\_mapping},\mbox{hypothesis})$

$\neg\,\mbox{maps}(\mbox{function}_2\circ \mbox{function}_1,a,d)\qquad\mbox{cnf}(\mbox{prove\_maps\_for\_composition},\mbox{negated\_conjecture})$

\medskip\textbf{SET095+1.p} Property 2 of singletons






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( x\in y \ \Rightarrow \ \mbox{subclass}(\mbox{singleton}(x),y) )\qquad\mbox{fof}(\mbox{property\_of\_singletons}_2,\mbox{conjecture})$

\medskip\textbf{SET589$\wedge$5.p} TPS problem BOOL-PROP-48

Trybulec's 48th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o},v{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( z@ \mbox{xx} ) \ \Rightarrow \ ( v@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }\neg\,v@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_48\_pme},\mbox{conjecture})$

\medskip\textbf{SET267-6.p} Domain is monotonic 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{domain\_of}(x)',\mbox{domain\_of}(x'))\qquad\mbox{cnf}(\mbox{prove\_domain\_is\_monotonic4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET762+4.p} The image of empty set is empty






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b{:}\;( \mbox{maps}(f,a,b) \ \Rightarrow \ \mbox{equal\_set}(\mbox{image}_2(f,\mbox{empty\_set}),\mbox{empty\_set}) )\qquad\mbox{fof}(\mbox{thIIa}_{12},\mbox{conjecture})$

\medskip\textbf{SET399-6.p} Left compose with null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{null\_class}\circ x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_left\_compose\_with\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET588+3.p} If X (= Y, then X \ Z (= Y \ Z

If X is a subset of Y, then the difference of X and Z is a
 subset of the difference of Y and Z.




$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq c \ \Rightarrow \ ( b\setminus d )\subseteq ( c\setminus d ) )\qquad\mbox{fof}(\mbox{prove\_difference\_subset}_1,\mbox{conjecture})$

\medskip\textbf{SET040-6.p} Properties of apply for composition of functions, 2 of 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_application\_property12}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{apply}(\mbox{yf}\circ \mbox{xf},x),\mbox{apply}(\mbox{yf},\mbox{apply}(\mbox{xf},x)))\qquad\mbox{cnf}(\mbox{prove\_application\_property12}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET959+1.p} Basic properties of sets, theorem 112






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

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

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

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

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

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

$\forall a,b{:}\;( ( \forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\forall d,e{:}\;c\ne \mbox{ordered\_pair}(d,e)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\forall d,e{:}\;c\ne \mbox{ordered\_pair}(d,e)\mbox{ and }\forall c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(c,d),a)\iff \mbox{in}(\mbox{ordered\_pair}(c,d),b) ) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t112\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

\medskip\textbf{SET614$\wedge$3.p} X \ Y \ Z = X \ (Y U Z)

The difference of X and the difference of Y and Z is the
 difference of X and (the union of Y and Z).




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},z{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{setminus}@ ( \mbox{setminus}@ x@ y )@ z ) = ( \mbox{setminus}@ x@ ( \mbox{union}@ y@ z ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET019-3.p} Two sets that contain one another are equal






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$b\subseteq a\qquad\mbox{cnf}(\mbox{a\_contains\_b},\mbox{hypothesis})$

$a\subseteq b\qquad\mbox{cnf}(\mbox{b\_contains\_a},\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{prove\_\_a\_equals\_b},\mbox{negated\_conjecture})$

\medskip\textbf{SET003-1.p} A set is a subset of the union of itself with itself






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-1.ax'})$

$\mbox{union}(a,a,\mbox{aUa})\qquad\mbox{cnf}(\mbox{a\_union\_a\_is\_aUa},\mbox{hypothesis})$

$\neg\,a\subseteq \mbox{aUa}\qquad\mbox{cnf}(\mbox{prove\_a\_is\_a\_subset\_of\_aUa},\mbox{negated\_conjecture})$

\medskip\textbf{SET583$\wedge$7.p} Extensionality






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{qmltpeq}@ x@ x ) )\qquad\mbox{thf}(\mbox{reflexivity},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{qmltpeq}@ x@ y )@ ( \mbox{qmltpeq}@ y@ x ) ) ) )\qquad\mbox{thf}(\mbox{symmetry},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda z{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{qmltpeq}@ x@ y )@ ( \mbox{qmltpeq}@ y@ z ) )@ ( \mbox{qmltpeq}@ x@ z ) ) ) ) )\qquad\mbox{thf}(\mbox{transitivity},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda a{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{qmltpeq}@ a@ b )@ ( \in @ a@ c ) )@ ( \in @ b@ c ) ) ) ) )\qquad\mbox{thf}(\mbox{member\_substitution}_1,\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda a{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{qmltpeq}@ a@ b )@ ( \in @ c@ a ) )@ ( \in @ c@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{member\_substitution}_2,\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda a{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{qmltpeq}@ a@ b )@ ( \subseteq @ a@ c ) )@ ( \subseteq @ b@ c ) ) ) ) )\qquad\mbox{thf}(\mbox{subset\_substitution}_1,\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda a{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{qmltpeq}@ a@ b )@ ( \subseteq @ c@ a ) )@ ( \subseteq @ c@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{subset\_substitution}_2,\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mequiv}@ ( \mbox{qmltpeq}@ b@ c )@ ( \mbox{mand}@ ( \subseteq @ b@ c )@ ( \subseteq @ c@ b ) ) ) ) )\qquad\mbox{thf}(\mbox{equal\_defn},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mequiv}@ ( \subseteq @ b@ c )@ ( \mbox{mforall\_ind}@ \lambda d{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \in @ d@ b )@ ( \in @ d@ c ) ) ) ) ) )\qquad\mbox{thf}(\mbox{subset\_defn},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \subseteq @ b@ b ) )\qquad\mbox{thf}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mand}@ ( \subseteq @ b@ c )@ ( \subseteq @ c@ b ) )@ ( \mbox{qmltpeq}@ b@ c ) ) ) )\qquad\mbox{thf}(\mbox{prove\_extensionality},\mbox{conjecture})$

\medskip\textbf{SET485-6.p} Replacement property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_replacement\_property7}_1,\mbox{negated\_conjecture})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property7}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{restrict}(x,y,\mbox{universal\_class})\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property7}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET011+3.p} X \ (X \ Y) = X $\wedge$ Y

The difference of X and (the difference of X and Y) is the
 intersection of X and Y.




$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;b\setminus ( b\setminus c ) = \mbox{intersection}(b,c)\qquad\mbox{fof}(\mbox{prove\_difference\_difference\_intersection},\mbox{conjecture})$

\medskip\textbf{SET084-6.p} A singleton set depends on its element, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{singleton}(x) = \mbox{singleton}(y)\qquad\mbox{cnf}(\mbox{prove\_singleton\_identified\_by\_element2}_1,\mbox{negated\_conjecture})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_singleton\_identified\_by\_element2}_2,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_singleton\_identified\_by\_element2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET601$\wedge$5.p} TPS problem BOOL-PROP-72

Trybulec's 72nd Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }y@ \mbox{xz} )\mbox{ or }( y@ \mbox{xz}\mbox{ and }z@ \mbox{xz} )\mbox{ or }( z@ \mbox{xz}\mbox{ and }x@ \mbox{xz} ) ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} )\mbox{ and }( y@ \mbox{xx}\mbox{ or }z@ \mbox{xx} )\mbox{ and }( z@ \mbox{xx}\mbox{ or }x@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_72\_pme},\mbox{conjecture})$

\medskip\textbf{SET542-6.p} Corollary to application property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_application\_property9}_1,\mbox{negated\_conjecture})$

$y\in \mbox{image}(\mbox{xf},x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_application\_property9}_2,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{xf},1_{\text{dom}(\mbox{xf})})\ne y\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_application\_property9}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET196+3.p} The intersection of X and Y is a subset of X






$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c)\subseteq b\qquad\mbox{fof}(\mbox{prove\_intersection\_is\_subset},\mbox{conjecture})$

\medskip\textbf{SET069+1.p} If one argument is a proper class, pair contains only the other






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( \neg\,y\in \mbox{universal\_class} \ \Rightarrow \ \mbox{unordered\_pair}(x,y) = \mbox{singleton}(x) )\qquad\mbox{fof}(\mbox{pair\_contains\_only\_other}_2,\mbox{conjecture})$

\medskip\textbf{SET824-2.p} Problem about set theory






$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_c},\mbox{c\_uminus}(\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OComplD\_\_dest}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_U},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{SET935+1.p} union(powset(A),powset(B)) = powset(union(A,B)) =$>$ inc\_comp(A,B)






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

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

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

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

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

$\forall a,b{:}\;( \mbox{inclusion\_comparable}(a,b)\iff ( a\subseteq b\mbox{ or }b\subseteq a ) )\qquad\mbox{fof}(\mbox{d9\_xboole}_0,\mbox{axiom})$

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

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

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

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

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

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

$\forall a,b{:}\;\mbox{inclusion\_comparable}(a,a)\qquad\mbox{fof}(\mbox{reflexivity\_r3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{inclusion\_comparable}(a,b) \ \Rightarrow \ \mbox{inclusion\_comparable}(b,a) )\qquad\mbox{fof}(\mbox{symmetry\_r3\_xboole}_0,\mbox{axiom})$

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

$\forall a,b{:}\;( \mbox{set\_union}_2(\mbox{powerset}(a),\mbox{powerset}(b)) = \mbox{powerset}(\mbox{set\_union}_2(a,b)) \ \Rightarrow \ \mbox{inclusion\_comparable}(a,b) )\qquad\mbox{fof}(\mbox{t82\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET615$\wedge$3.p} (X U Y) \ Z = (X \ Z) U (Y \ Z)

The difference of (the union of X and Y) and Z is the union of
 (the difference of X and Z) and (the difference of Y and Z).




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},z{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{setminus}@ ( \mbox{union}@ x@ y )@ z ) = ( \mbox{union}@ ( \mbox{setminus}@ x@ z )@ ( \mbox{setminus}@ y@ z ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET454-6.p} Identity alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{identity\_relation},\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_identity\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET859-2.p} Problem about Zorn's lemma






$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_n},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET287-6.p} Domain property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_domain\_property8}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{intersection}(y',x))\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_domain\_property8}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET910+1.p} intersection(A,singleton(B)) = singleton(B) =$>$ in(B,A)






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

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

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

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

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

$\forall a,b{:}\;( \mbox{set\_intersection}_2(a,\mbox{singleton}(b)) = \mbox{singleton}(b) \ \Rightarrow \ \mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{t51\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{set\_intersection}_2(a,\mbox{singleton}(b)) = \mbox{singleton}(b) \ \Rightarrow \ \mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{l30\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET139-6.p} Triple reduction 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$\mbox{set\_builder}(x,\mbox{set\_builder}(x,\mbox{set\_builder}(y,\mbox{null\_class})))\ne \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_triple\_reduction1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET282-6.p} Domain of domain relation






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(\mbox{domain\_relation})\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_domain\_of\_domain\_relation}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET108+1.p} 1st and 2nd make the ordered pair






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;\exists u,v{:}\;( ( u\in \mbox{universal\_class}\mbox{ and }v\in \mbox{universal\_class}\mbox{ and }x = \mbox{ordered\_pair}(u,v) )\mbox{ or }( \neg\,\exists y,z{:}\;( y\in \mbox{universal\_class}\mbox{ and }z\in \mbox{universal\_class}\mbox{ and }x = \mbox{ordered\_pair}(y,z) )\mbox{ and }u = x\mbox{ and }v = x ) )\qquad\mbox{fof}(\mbox{existence\_of\_first\_and\_second},\mbox{conjecture})$

\medskip\textbf{SET832-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_V},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_V},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_V},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET476-6.p} Intersection subclass






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{intersection}(x,\mbox{identity\_relation}),\mbox{intersection}(x,x'))\qquad\mbox{cnf}(\mbox{prove\_intersection\_subclass}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET637+3.p} Trybulec's 119th Boolean property of sets






$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{not\_equal}(b,c)\iff b\ne c )\qquad\mbox{fof}(\mbox{not\_equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \mbox{not\_equal}(\mbox{intersection}(b,c),\mbox{empty\_set}) )\qquad\mbox{fof}(\mbox{prove\_th}_{119},\mbox{conjecture})$

\medskip\textbf{SET199-6.p} If Z (= X and Z (= Y, then Z (= X $\wedge$ Y






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(z,x)\qquad\mbox{cnf}(\mbox{prove\_greatest\_lower\_bound}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(z,y)\qquad\mbox{cnf}(\mbox{prove\_greatest\_lower\_bound}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(z,\mbox{intersection}(x,y))\qquad\mbox{cnf}(\mbox{prove\_greatest\_lower\_bound}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET141-6.p} Triple reduction 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$\mbox{set\_builder}(x,\mbox{set\_builder}(y,\mbox{set\_builder}(y,\mbox{null\_class})))\ne \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_triple\_reduction3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET483-6.p} Replacement property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property5}_1,\mbox{negated\_conjecture})$

$\neg\,x'\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET495-6.p} Diagonalization alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{diagonalise}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(z,z)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET066$\wedge$1.p} Unordered pair is commutative






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \mbox{unord\_pair}@ x@ y ) = ( \mbox{unord\_pair}@ y@ x )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET727+4.p} If GoF and FoH are identities, then inverse(F) = H






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,a)\mbox{ and }\mbox{maps}(h,b,a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(g,f,a,b,a),a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(f,h,b,a,b),b) ) \ \Rightarrow \ \mbox{equal\_maps}(\mbox{inverse\_function}(f,a,b),h,b,a) )\qquad\mbox{fof}(\mbox{thII}_{18},\mbox{conjecture})$

\medskip\textbf{SET027-7.p} Transitivity of subset






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_subclass}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(y,z)\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_subclass}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_subclass}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET064+1.p} Uniqueness of null class






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall z{:}\;( z = \mbox{null\_class}\mbox{ or }\exists y{:}\;y\in z )\qquad\mbox{fof}(\mbox{null\_class\_is\_unique},\mbox{conjecture})$

\medskip\textbf{SET958+1.p} Basic properties of sets, theorem 111






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

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

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

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

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

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

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

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

$\forall a,b{:}\;( ( \forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\forall d,e{:}\;c\ne \mbox{ordered\_pair}(d,e)\mbox{ and }\forall c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(c,d),a) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(c,d),b) ) ) \ \Rightarrow \ a\subseteq b )\qquad\mbox{fof}(\mbox{t111\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET628$\wedge$5.p} TPS problem BOOL-PROP-110

Trybulec's 110th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }x@ \mbox{xx} )\iff x\ne ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_110\_pme},\mbox{conjecture})$

\medskip\textbf{SET085-6.p} Unordered pair that is a singleton






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{unordered\_pair}(y,z) = \mbox{singleton}(x)\qquad\mbox{cnf}(\mbox{prove\_singleton\_in\_unordered\_pair3}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_singleton\_in\_unordered\_pair3}_2,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_singleton\_in\_unordered\_pair3}_3,\mbox{negated\_conjecture})$

$x\ne z\qquad\mbox{cnf}(\mbox{prove\_singleton\_in\_unordered\_pair3}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET848-2.p} Problem about Zorn's lemma






$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_minus}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}) )\qquad\mbox{cnf}(\mbox{cls\_Set\_ODiffI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_refl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_chain\_\_lemma4}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_minus}(\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_c},\mbox{t\_a})\ne \mbox{v\_c}\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_not\_\_equals}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}) = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET166-6.p} Members of union 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{union}(y,z)\qquad\mbox{cnf}(\mbox{prove\_members\_of\_union1}_1,\mbox{negated\_conjecture})$

$\neg\,x\in y\qquad\mbox{cnf}(\mbox{prove\_members\_of\_union1}_2,\mbox{negated\_conjecture})$

$\neg\,x\in z\qquad\mbox{cnf}(\mbox{prove\_members\_of\_union1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET001-1.p} Set members are superset members

A member of a set is also a member of that set's supersets.




$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{equal\_sets}(b,\mbox{bb})\qquad\mbox{cnf}(\mbox{b\_equals\_bb},\mbox{hypothesis})$

$\mbox{element\_of\_b}\in b\qquad\mbox{cnf}(\mbox{element\_of\_b},\mbox{hypothesis})$

$\neg\,\mbox{element\_of\_b}\in \mbox{bb}\qquad\mbox{cnf}(\mbox{prove\_element\_of\_bb},\mbox{negated\_conjecture})$

\medskip\textbf{SET587+3.p} X \ Y = the empty set iff X (= Y

The difference of X and Y is the empty set iff X is a subset of
 Y.




$\forall b,c{:}\;( \forall d{:}\;( d\in b\iff d\in c ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{member\_equal},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\setminus c = \mbox{empty\_set}\iff b\subseteq c )\qquad\mbox{fof}(\mbox{prove\_difference\_empty\_set},\mbox{conjecture})$

\medskip\textbf{SET506-6.p} Universal class not null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{universal\_class} = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_universal\_class\_not\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET429-6.p} A subclass of a single-valued class is single-valued






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_subclass}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{single\_valued\_class}(\mbox{intersection}(x,y))\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_subclass}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET034-3.p} The composition of functions is a function






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(x)\subseteq \mbox{domain\_of}(y) \ \Rightarrow \ \mbox{domain\_of}(x) = \mbox{domain\_of}(y\circ x)\qquad\mbox{cnf}(\mbox{domain\_of\_composition},\mbox{axiom})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$\mbox{function}(\mbox{another\_function})\qquad\mbox{cnf}(\mbox{another\_function},\mbox{hypothesis})$

$\neg\,\mbox{function}(\mbox{another\_function}\circ \mbox{a\_function})\qquad\mbox{cnf}(\mbox{prove\_their\_composition\_is\_a\_function},\mbox{negated\_conjecture})$

\medskip\textbf{SET338-6.p} Corollary 3 image property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,x\in \mbox{domain\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_3\_image\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{image}(z,\mbox{singleton}(x))\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_3\_image\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET041-6.p} Properties of apply for composition of functions, 3 of 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_application\_property11}_1,\mbox{negated\_conjecture})$

$x\in \mbox{domain\_of}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_application\_property11}_2,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{yf}\circ \mbox{xf},x)\ne \mbox{apply}(\mbox{yf},\mbox{apply}(\mbox{xf},x))\qquad\mbox{cnf}(\mbox{prove\_application\_property11}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET860-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET171$\wedge$3.p} Union distributes over intersection

The union of X and (the intersection of Y and Z) is the
 intersection of (the union of X and Y) and (the union of X and Z).




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o},b{:}\;\mbox{\$i}\to \mbox{\$o},c{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{union}@ a@ ( \mbox{intersection}@ b@ c ) ) = ( \mbox{intersection}@ ( \mbox{union}@ a@ b )@ ( \mbox{union}@ a@ c ) )\qquad\mbox{thf}(\mbox{union\_distributes\_over\_intersection},\mbox{conjecture})$

\medskip\textbf{SET345-6.p} Sum class alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{sum\_class}(y)\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{range}(\mbox{element\_relation},x,y)\in y\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET453-6.p} Subset relation alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subset\_relation\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_subset\_relation\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(x,y)\in \mbox{subset\_relation}\qquad\mbox{cnf}(\mbox{prove\_subset\_relation\_alternate\_defn3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET547-6.p} Application special case 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{apply}(x,\mbox{universal\_class})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_application\_special\_case3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET255-6.p} Monotonicity of restriction 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(y_1,y_2)\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_restriction2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{restrict}(x,y_1,z),\mbox{restrict}(x,y_2,z))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_restriction2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET012+4.p} Complement is an involution






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,e{:}\;( a\subseteq e \ \Rightarrow \ \mbox{equal\_set}(e\setminus ( e\setminus a ),a) )\qquad\mbox{fof}(\mbox{thI}_{23},\mbox{conjecture})$

\medskip\textbf{SET501-6.p} Diagonalization property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{diagonalise}(\mbox{xr}'\circ \mbox{xs})\ne \mbox{domain\_of}(\mbox{intersection}(\mbox{xr},\mbox{xs}))'\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET717+4.p} The composition of surjective mappings is surjective






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{surjective}(f,a,b)\mbox{ and }\mbox{surjective}(g,b,c) ) \ \Rightarrow \ \mbox{surjective}(\mbox{compose\_function}(g,f,a,b,c),a,c) )\qquad\mbox{fof}(\mbox{thII}_{08},\mbox{conjecture})$

\medskip\textbf{SET532-6.p} Application property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_application\_property2}_1,\mbox{negated\_conjecture})$

$x\in \mbox{domain\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_application\_property2}_2,\mbox{negated\_conjecture})$

$\mbox{image}(z,\mbox{singleton}(x))\ne \mbox{singleton}(\mbox{apply}(z,x))\qquad\mbox{cnf}(\mbox{prove\_application\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET455-6.p} Identity alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{identity\_relation}\qquad\mbox{cnf}(\mbox{prove\_identity\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_identity\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET162-6.p} Null class is identity for union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{null\_class},x)\ne x\qquad\mbox{cnf}(\mbox{prove\_union\_with\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET148+4.p} A set intersection itself is itself






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a{:}\;\mbox{equal\_set}(\mbox{intersection}(a,a),a)\qquad\mbox{fof}(\mbox{thI}_{13},\mbox{conjecture})$

\medskip\textbf{SET227-6.p} Cross product left cancellation 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(u,v) = \mbox{cross\_product}(w,x)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_left\_cancellation2}_1,\mbox{negated\_conjecture})$

$w\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_left\_cancellation2}_2,\mbox{negated\_conjecture})$

$v\ne x\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_left\_cancellation2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET215-6.p} Corollary 6 to cross product product monotonicity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(\mbox{intersection}(x,y),z),\mbox{cross\_product}(y,z))\qquad\mbox{cnf}(\mbox{prove\_corollary\_6\_to\_X\_product\_monotonicity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET612+3.p} X \ (Y U Z) = (X \ Y) $\wedge$ (X \ Z)

The difference of X and (the union of Y and Z) is the
 intersection of (the difference of X and Y) and (the difference
 of X and Z).




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;b\subseteq \mbox{union}(b,c)\qquad\mbox{fof}(\mbox{subset\_of\_union},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }b\subseteq d ) \ \Rightarrow \ b\subseteq \mbox{intersection}(c,d) )\qquad\mbox{fof}(\mbox{intersection\_of\_subsets},\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq c \ \Rightarrow \ ( d\setminus c )\subseteq ( d\setminus b ) )\qquad\mbox{fof}(\mbox{subset\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;b\setminus \mbox{union}(c,d) = \mbox{intersection}(b\setminus c,b\setminus d)\qquad\mbox{fof}(\mbox{prove\_th}_{85},\mbox{conjecture})$

\medskip\textbf{SET916+1.p} ~ ( ~ in(A,B) \& ~ in(C,B) \& ~ disjoint(unordered\_pair(A,C),B) )






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

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

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

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

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

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

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

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

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

$\forall a,b,c{:}\;\neg\,\neg\,\mbox{in}(a,b)\mbox{ and }\neg\,\mbox{in}(c,b)\mbox{ and }\neg\,\mbox{disjoint}(\mbox{unordered\_pair}(a,c),b)\qquad\mbox{fof}(\mbox{t57\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET058-6.p} Expanded equality definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{not\_subclass\_element}(y,x)\in x\qquad\mbox{cnf}(\mbox{prove\_equality3}_1,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_equality3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(x,y)\in x\qquad\mbox{cnf}(\mbox{prove\_equality3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET606+3.p} X \ X $\wedge$ Y = X \ Y

The difference of X and the intersection of X and Y is the
 difference of X and Y.




$\forall b,c{:}\;( \forall d{:}\;( d\in b\iff d\in c ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{member\_equal},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;b\setminus \mbox{intersection}(b,c) = b\setminus c\qquad\mbox{fof}(\mbox{prove\_difference\_into\_intersection},\mbox{conjecture})$

\medskip\textbf{SET861-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OHausdorff\_}_1(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OHausdorff}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_z},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_1,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_subset\_\_chain}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_xa}(\mbox{v\_U}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_xa}(\mbox{v\_U}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_U} = \mbox{v\_xa}(\mbox{v\_U}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_V},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_V},\mbox{v\_x}(\mbox{v\_U}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET010+3.p} X \ Y $\wedge$ Z = (X \ Y) U (X \ Z)

The difference of X and the intersection of Y and Z is the
 union of (the difference of X and Y) and (the difference of X
 and Z).




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }d\subseteq c ) \ \Rightarrow \ \mbox{union}(b,d)\subseteq c )\qquad\mbox{fof}(\mbox{union\_subset},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c)\subseteq b\qquad\mbox{fof}(\mbox{intersection\_is\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq c \ \Rightarrow \ ( d\setminus c )\subseteq ( d\setminus b ) )\qquad\mbox{fof}(\mbox{subset\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;b\setminus \mbox{intersection}(c,d) = \mbox{union}(b\setminus c,b\setminus d)\qquad\mbox{fof}(\mbox{prove\_difference\_and\_intersection\_and\_union},\mbox{conjecture})$

\medskip\textbf{SET155-6.p} De Morgans law 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,y)'\ne \mbox{intersection}(x',y')\qquad\mbox{cnf}(\mbox{prove\_demorgans\_law1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET461-6.p} Domain of restricted identity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(\mbox{restrict}(\mbox{identity\_relation},x,y))\ne \mbox{intersection}(x,y)\qquad\mbox{cnf}(\mbox{prove\_domain\_of\_restricted\_identity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET405-6.p} Monotonicity of composition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{xr},\mbox{yr})\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_composition2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{xr}\circ \mbox{zr},\mbox{yr}\circ \mbox{zr})\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_composition2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET967+1.p} Basic properties of sets, theorem 120






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

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

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

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

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

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

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

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,\mbox{cartesian\_product}_2(b,c))\mbox{ and }\forall d,e{:}\;\mbox{ordered\_pair}(d,e)\ne a\qquad\mbox{fof}(\mbox{t102\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d)) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(b,a),\mbox{cartesian\_product}_2(d,c)) )\qquad\mbox{fof}(\mbox{t107\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( \forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }\forall d,e{:}\;c\ne \mbox{ordered\_pair}(d,e)\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,b)\mbox{ and }\forall d,e{:}\;c\ne \mbox{ordered\_pair}(d,e)\mbox{ and }\forall c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(c,d),a)\iff \mbox{in}(\mbox{ordered\_pair}(c,d),b) ) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t112\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{cartesian\_product}_2(\mbox{set\_union}_2(a,b),c) = \mbox{set\_union}_2(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,c))\mbox{ and }\mbox{cartesian\_product}_2(c,\mbox{set\_union}_2(a,b)) = \mbox{set\_union}_2(\mbox{cartesian\_product}_2(c,a),\mbox{cartesian\_product}_2(c,b)) )\qquad\mbox{fof}(\mbox{t120\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET273-6.p} Corollary to domain property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{xr},\mbox{domain\_of}(\mbox{xr}),y)\ne \mbox{restrict}(\mbox{xr},\mbox{universal\_class},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_domain\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET425-6.p} Single valued class alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,w)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn1}_3,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in z\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn1}_4,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,w)\in z\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn1}_5,\mbox{negated\_conjecture})$

$v\ne w\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn1}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SET037-3.p} Properties of apply for functions, part 2 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(x)\subseteq \mbox{domain\_of}(y) \ \Rightarrow \ \mbox{domain\_of}(x) = \mbox{domain\_of}(y\circ x)\qquad\mbox{cnf}(\mbox{domain\_of\_composition},\mbox{axiom})$

$( \mbox{function}(x)\mbox{ and }\mbox{function}(y) ) \ \Rightarrow \ \mbox{function}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_function},\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},u,v)\mbox{ and }\mbox{maps}(\mbox{xg},v,w) ) \ \Rightarrow \ \mbox{maps}(\mbox{xg}\circ \mbox{xf},u,w)\qquad\mbox{cnf}(\mbox{maps\_for\_composition},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{function}(\mbox{xf})\mbox{ and }\mbox{ordered\_pair}(x,y)\in \mbox{xf} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x) = y\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_1,\mbox{axiom})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$a\in \mbox{domain\_of}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{member\_of\_function\_domain},\mbox{hypothesis})$

$\mbox{apply}(\mbox{a\_function},a) = b\qquad\mbox{cnf}(\mbox{applying\_the\_function},\mbox{hypothesis})$

$\neg\,\mbox{ordered\_pair}(a,b)\in \mbox{a\_function}\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_in\_function},\mbox{negated\_conjecture})$

\medskip\textbf{SET469-6.p} Sets with at most one member 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{singleton}(\mbox{member\_of}(x)) = x\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_one\_member3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,x),\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_one\_member3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET924+1.p} difference(singleton(A),B) = singleton(A) $\leq$$>$ ~ in(A,B)






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

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

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

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{singleton}(a)\iff \neg\,\mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{t67\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{singleton}(a)\iff \neg\,\mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l34\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET777-1.p} Set theory membership and subsets axioms






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

\medskip\textbf{SET187-6.p} Subclass property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_property5}_1,\mbox{negated\_conjecture})$

$\mbox{intersection}(y',x)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_subclass\_property5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET109-6.p} 1st is the ordered pair, first condition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{ordered\_pair}(\mbox{first}(x),\mbox{second}(x))\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_1st\_and\_2nd\_2}_1,\mbox{negated\_conjecture})$

$\mbox{first}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_1st\_and\_2nd\_2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET199$\wedge$5.p} TPS problem BOOL-PROP-39

Trybulec's 39th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( z@ \mbox{xx} ) \ \Rightarrow \ ( x@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( z@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( z@ \mbox{xx} ) \ \Rightarrow \ ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_39\_pme},\mbox{conjecture})$

\medskip\textbf{SET362-6.p} Monotonicity of sum






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_sum}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(x),\mbox{sum\_class}(y))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_sum}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET938+1.p} subset(union(pset(diff(A,B)),pset(diff(B,A))),pset(symdiff(A,B)))






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

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

$\forall a,b{:}\;\mbox{symmetric\_difference}(a,b) = \mbox{symmetric\_difference}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k5\_xboole}_0,\mbox{axiom})$

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

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

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

$\forall a,b{:}\;\mbox{symmetric\_difference}(a,b) = \mbox{set\_union}_2(\mbox{set\_difference}(a,b),\mbox{set\_difference}(b,a))\qquad\mbox{fof}(\mbox{d6\_xboole}_0,\mbox{axiom})$

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

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

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

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

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

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

$\forall a,b{:}\;\mbox{set\_union}_2(\mbox{powerset}(\mbox{set\_difference}(a,b)),\mbox{powerset}(\mbox{set\_difference}(b,a)))\subseteq \mbox{powerset}(\mbox{symmetric\_difference}(a,b))\qquad\mbox{fof}(\mbox{t86\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET055+1.p} Reflexivity of equality






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;x = x\qquad\mbox{fof}(\mbox{reflexivity},\mbox{conjecture})$

\medskip\textbf{SET054-6.p} Subclass is reflexive






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{prove\_subclass\_is\_reflexive}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET169+4.p} Distribution of intersection over union






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,c{:}\;\mbox{equal\_set}(\mbox{intersection}(a,\mbox{union}(b,c)),\mbox{union}(\mbox{intersection}(a,b),\mbox{intersection}(a,c)))\qquad\mbox{fof}(\mbox{thI}_{10},\mbox{conjecture})$

\medskip\textbf{SET728+4.p} If GoF and FoH are identities, then G = H






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,a)\mbox{ and }\mbox{maps}(h,b,a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(g,f,a,b,a),a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(f,h,b,a,b),b) ) \ \Rightarrow \ \mbox{equal\_maps}(g,h,b,a) )\qquad\mbox{fof}(\mbox{thII}_{19},\mbox{conjecture})$

\medskip\textbf{SET027-4.p} Transitivity of subset






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$a\subseteq b\qquad\mbox{cnf}(\mbox{a\_subset\_b},\mbox{hypothesis})$

$b\subseteq c\qquad\mbox{cnf}(\mbox{b\_subset\_c},\mbox{hypothesis})$

$\neg\,a\subseteq c\qquad\mbox{cnf}(\mbox{prove\_a\_subset\_c},\mbox{negated\_conjecture})$

\medskip\textbf{SET443-6.p} Difference of functions is a function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_symmetric\_difference\_of\_functions}_1,\mbox{negated\_conjecture})$

$\mbox{function}(y)\qquad\mbox{cnf}(\mbox{prove\_symmetric\_difference\_of\_functions}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{intersection}(x',y))\qquad\mbox{cnf}(\mbox{prove\_symmetric\_difference\_of\_functions}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET684$\wedge$3.p} $<$x,z$>$ in P(DtoE) o R(EtoF) iff ?y in E:$<$x,y$>$ in P \& $<$y,z$>$ in R

Let P be a relation from D to E, R a relation from E to F, x an
 element of D, and z in F. Then $<$x,z$>$ is in P composed with R if
 and only if there exists an element y in E such that $<$x,y$>$ is in
 P and $<$y,z$>$ is in R.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall p{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( ( \mbox{rel\_composition}@ p@ r@ x@ z )\iff \exists y{:}\;\mbox{\$i}{:}\;( p@ x@ y\mbox{ and }r@ y@ z ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET050-6.p} Corollary to Unordered pair axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_unordered\_pair}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_unordered\_pair}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET908+1.p} union(singleton(A),B) != empty\_set






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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

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

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

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

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

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

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

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

$\forall a,b{:}\;\mbox{set\_union}_2(\mbox{singleton}(a),b)\ne \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t49\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET961+1.p} cart\_prod(A,B) = cart\_prod(B,A) =$>$ ( A=empty | B=empty | A = B )






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

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

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

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

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b{:}\;( \mbox{cartesian\_product}_2(a,b) = \mbox{cartesian\_product}_2(b,a) \ \Rightarrow \ ( a = \mbox{empty\_set}\mbox{ or }b = \mbox{empty\_set}\mbox{ or }a = b ) )\qquad\mbox{fof}(\mbox{t114\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

\medskip\textbf{SET312-6.p} Range only considers ordered pairs






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{range\_of}(\mbox{intersection}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})))\ne \mbox{range\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_range\_does\_ordered\_pairs}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET244-6.p} Restriction with universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{universal\_class},x,y)\ne \mbox{cross\_product}(x,y)\qquad\mbox{cnf}(\mbox{prove\_restriction\_with\_universal\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET609$\wedge$3.p} X \ (Y \ Z) = (X \ Y) U X $\wedge$ Z

The difference of X and (the difference of Y and Z) is the union
 of (the difference of X and Y) and the intersection of X and Z.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},z{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{setminus}@ x@ ( \mbox{setminus}@ y@ z ) ) = ( \mbox{union}@ ( \mbox{setminus}@ x@ y )@ ( \mbox{intersection}@ x@ z ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET741$\wedge$4.p} Problem on composition of mappings 6

Consider three mappings F from A to B,G from B to C,H from C to A.  If HoGoF is injective and FoHoG and GoFoH surjective, then H is
 one-to-one.




$\mbox{include}(\mbox{'Axioms/SET008\string^1.ax'})$

$\forall f{:}\;\mbox{\$i}\to \mbox{\$i},g{:}\;\mbox{\$i}\to \mbox{\$i},h{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{fun\_injective}@ ( \mbox{fun\_composition}@ ( \mbox{fun\_composition}@ f@ g )@ h )\mbox{ and }\mbox{fun\_surjective}@ ( \mbox{fun\_composition}@ ( \mbox{fun\_composition}@ g@ h )@ f )\mbox{ and }\mbox{fun\_surjective}@ ( \mbox{fun\_composition}@ ( \mbox{fun\_composition}@ h@ f )@ g ) ) \ \Rightarrow \ ( \mbox{fun\_bijective}@ h ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET363-6.p} Power class alternative definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{power\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_alternative\_defn1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(z,x)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_alternative\_defn1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET066-7.p} Unordered pair is commutative






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{special\_classes\_lemma},\mbox{axiom})$

$\neg\,z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{existence\_of\_null\_class},\mbox{axiom})$

$\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{cnf}(\mbox{null\_class\_is\_subclass},\mbox{axiom})$

$\mbox{subclass}(x,\mbox{null\_class}) \ \Rightarrow \ x = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{corollary\_of\_null\_class\_is\_subclass},\mbox{axiom})$

$z = \mbox{null\_class}\mbox{ or }\mbox{not\_subclass\_element}(z,\mbox{null\_class})\in z\qquad\mbox{cnf}(\mbox{null\_class\_is\_unique},\mbox{axiom})$

$\mbox{null\_class}\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{null\_class\_is\_a\_set},\mbox{axiom})$

$\mbox{unordered\_pair}(x,y)\ne \mbox{unordered\_pair}(y,x)\qquad\mbox{cnf}(\mbox{prove\_commutativity\_of\_unordered\_pair}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET162+3.p} The union of X and the empty set is X






$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b{:}\;\mbox{union}(b,\mbox{empty\_set}) = b\qquad\mbox{fof}(\mbox{prove\_union\_empty\_set},\mbox{conjecture})$

\medskip\textbf{SET729+4.p} F is one-to-one and inverse(F)=F iff FoF is the identity






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a{:}\;( \mbox{maps}(f,a,a) \ \Rightarrow \ ( ( \mbox{one\_to\_one}(f,a,a)\mbox{ and }\mbox{inverse\_predicate}(f,f,a,a) )\iff \mbox{identity}(\mbox{compose\_function}(f,f,a,a,a),a) ) )\qquad\mbox{fof}(\mbox{thII}_{20},\mbox{conjecture})$

\medskip\textbf{SET032-6.p} Range of composition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{xr}\circ \mbox{yr},\mbox{cross\_product}(\mbox{domain\_of}(\mbox{yr}),\mbox{range\_of}(\mbox{xr})))\qquad\mbox{cnf}(\mbox{prove\_composition\_domain\_and\_range}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET186-6.p} Subclass property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,y) = y\qquad\mbox{cnf}(\mbox{prove\_subclass\_property4}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_property4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET045$\wedge$5.p} TPS problem TTTP5243

Comprehension Theorem.




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

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

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

$\exists u{:}\;a\to b{:}\;\forall v{:}\;a{:}\;( u@ v ) = \mbox{cA}\qquad\mbox{thf}(\mbox{cTTTP}_{5243},\mbox{conjecture})$

\medskip\textbf{SET062+3.p} The empty set is a subset of X






$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b{:}\;\mbox{empty\_set}\subseteq b\qquad\mbox{fof}(\mbox{prove\_empty\_set\_subset},\mbox{conjecture})$

\medskip\textbf{SET635+3.p} X $\wedge$ (Y \ Z) = X $\wedge$ Y \ X $\wedge$ Z

The intersection of X and the difference of Y and Z is the
 intersection of X and the difference of Y and the intersection
 of X and Z.




$\forall b,c{:}\;\mbox{intersection}(b,c)\subseteq b\qquad\mbox{fof}(\mbox{intersection\_is\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b\setminus c = \mbox{empty\_set}\iff b\subseteq c )\qquad\mbox{fof}(\mbox{difference\_empty\_set},\mbox{axiom})$

$\forall b{:}\;\mbox{union}(b,\mbox{empty\_set}) = b\qquad\mbox{fof}(\mbox{union\_empty\_set},\mbox{axiom})$

$\forall b,c,d{:}\;b\setminus \mbox{intersection}(c,d) = \mbox{union}(b\setminus c,b\setminus d)\qquad\mbox{fof}(\mbox{difference\_and\_intersection\_and\_union},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{intersection}(b,c\setminus d) = \mbox{intersection}(b,c)\setminus d\qquad\mbox{fof}(\mbox{difference\_and\_intersection},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{intersection}(b,c\setminus d) = \mbox{intersection}(b,c)\setminus \mbox{intersection}(b,d)\qquad\mbox{fof}(\mbox{prove\_th}_{117},\mbox{conjecture})$

\medskip\textbf{SET020-3.p} 1st is unique when x is an ordered pair of sets






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{b\_little\_set},\mbox{hypothesis})$

$\mbox{first}(\mbox{ordered\_pair}(a,b))\ne a\qquad\mbox{cnf}(\mbox{prove\_first\_is\_first},\mbox{negated\_conjecture})$

\medskip\textbf{SET536-6.p} Corollary 1 to application property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{xf},\mbox{singleton}(x)),\mbox{power\_class}(\mbox{apply}(\mbox{xf},x)))\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_application\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET901+1.p} Basic properties of sets, theorem 42

subset(A,unordered\_pair(B,C)) $\leq$$>$ ~ ( A != empty\_set \&
 A != singleton(B) \& A != singleton(C) \& A != unordered\_pair(B,C) )




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

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

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

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

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

$\forall a,b,c{:}\;( a\subseteq \mbox{unordered\_pair}(b,c)\iff \neg\,a\ne \mbox{empty\_set}\mbox{ and }a\ne \mbox{singleton}(b)\mbox{ and }a\ne \mbox{singleton}(c)\mbox{ and }a\ne \mbox{unordered\_pair}(b,c) )\qquad\mbox{fof}(\mbox{t42\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( a\subseteq \mbox{unordered\_pair}(b,c)\iff \neg\,a\ne \mbox{empty\_set}\mbox{ and }a\ne \mbox{singleton}(b)\mbox{ and }a\ne \mbox{singleton}(c)\mbox{ and }a\ne \mbox{unordered\_pair}(b,c) )\qquad\mbox{fof}(\mbox{l46\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET904+1.p} subset(set\_union2(singleton(A),B),B) =$>$ in(A,B)






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

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

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

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

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

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

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

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

$\forall a,b{:}\;( \mbox{set\_union}_2(\mbox{singleton}(a),b)\subseteq b \ \Rightarrow \ \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{t45\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{set\_union}_2(\mbox{singleton}(a),b)\subseteq b \ \Rightarrow \ \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l21\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET012-3.p} Complement is an involution






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{as}' = \mbox{bs}\qquad\mbox{cnf}(\mbox{complement\_of\_a\_is\_b},\mbox{hypothesis})$

$\mbox{bs}' = \mbox{cs}\qquad\mbox{cnf}(\mbox{complement\_of\_b\_is\_c},\mbox{hypothesis})$

$\mbox{as}\ne \mbox{cs}\qquad\mbox{cnf}(\mbox{prove\_a\_equals\_c},\mbox{negated\_conjecture})$

\medskip\textbf{SET090+1.p} Uniqueness of member\_of of a singleton set






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,u{:}\;( ( u\in \mbox{universal\_class}\mbox{ and }x = \mbox{singleton}(u) ) \ \Rightarrow \ \mbox{member\_of}(x) = u )\qquad\mbox{fof}(\mbox{member\_of\_singleton},\mbox{conjecture})$

\medskip\textbf{SET574+3.p} Trybulec's 13th Boolean property of sets






$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\in c\mbox{ and }b\in d ) \ \Rightarrow \ \mbox{intersect}(c,d) )\qquad\mbox{fof}(\mbox{prove\_th}_{13},\mbox{conjecture})$

\medskip\textbf{SET079-6.p} Corollary to a set belongs to its singleton






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_set\_in\_its\_singleton}_1,\mbox{negated\_conjecture})$

$\mbox{singleton}(x) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_set\_in\_its\_singleton}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET098+1.p} Corollary 1 to a class contains 0, 1, or at least 2 members






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;( x = \mbox{null\_class}\mbox{ or }\exists y{:}\;\mbox{singleton}(y) = x\mbox{ or }\exists v{:}\;( v\in x\mbox{ and }\exists w{:}\;( w\in \mbox{intersection}(\mbox{singleton}(v)',x)\mbox{ and }w\in x ) ) )\qquad\mbox{fof}(\mbox{corollary\_1\_to\_number\_of\_elements\_in\_class},\mbox{conjecture})$

\medskip\textbf{SET344-6.p} Sum class alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{sum\_class}(y)\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{range}(\mbox{element\_relation},x,y)\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET069-7.p} Proper class in an unordered pair, part 3

If one argument of an unordered pair is a proper class, the
 pair contains only the other.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{special\_classes\_lemma},\mbox{axiom})$

$\neg\,z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{existence\_of\_null\_class},\mbox{axiom})$

$\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{cnf}(\mbox{null\_class\_is\_subclass},\mbox{axiom})$

$\mbox{subclass}(x,\mbox{null\_class}) \ \Rightarrow \ x = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{corollary\_of\_null\_class\_is\_subclass},\mbox{axiom})$

$z = \mbox{null\_class}\mbox{ or }\mbox{not\_subclass\_element}(z,\mbox{null\_class})\in z\qquad\mbox{cnf}(\mbox{null\_class\_is\_unique},\mbox{axiom})$

$\mbox{null\_class}\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{null\_class\_is\_a\_set},\mbox{axiom})$

$\mbox{unordered\_pair}(x,y) = \mbox{unordered\_pair}(y,x)\qquad\mbox{cnf}(\mbox{commutativity\_of\_unordered\_pair},\mbox{axiom})$

$\neg\,y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_equals\_singleton1}_1,\mbox{negated\_conjecture})$

$\mbox{unordered\_pair}(x,y)\ne \mbox{singleton}(x)\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_equals\_singleton1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET180-6.p} Distribution property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x',\mbox{intersection}(x,z))\ne \mbox{union}(x',z)\qquad\mbox{cnf}(\mbox{prove\_distribution\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET159$\wedge$5.p} TPS problem BOOL-PROP-64

Trybulec's 64th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }y@ \mbox{xz}\mbox{ or }z@ \mbox{xz} ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }y@ \mbox{xz}\mbox{ or }z@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_64\_pme},\mbox{conjecture})$

\medskip\textbf{SET355+4.p} If X is in Y, then X is a subset of the sum of Y






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,x{:}\;( x\in a \ \Rightarrow \ x\subseteq \mbox{\mbox{sum}}(a) )\qquad\mbox{fof}(\mbox{thI}_{43},\mbox{conjecture})$

\medskip\textbf{SET411-6.p} Compose condition for singleton membership 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{singleton\_relation}\qquad\mbox{cnf}(\mbox{prove\_compose\_condition\_for\_singleton\_membership1}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_compose\_condition\_for\_singleton\_membership1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET057-7.p} Expanded equality definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y\qquad\mbox{cnf}(\mbox{prove\_equality2}_1,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_equality2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{prove\_equality2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET185+3.p} If X is a subset of  Y, then the union of X and Y is Y






$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c \ \Rightarrow \ \mbox{union}(b,c) = c )\qquad\mbox{fof}(\mbox{prove\_subset\_union},\mbox{conjecture})$

\medskip\textbf{SET797+4.p} If X subset Y, then an upper bound of Y is an upper bound of X






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e{:}\;( \mbox{order}(r,e) \ \Rightarrow \ \forall x,y{:}\;( ( x\subseteq e\mbox{ and }y\subseteq e\mbox{ and }x\subseteq y ) \ \Rightarrow \ \forall m{:}\;( b \ \Rightarrow \ b ) ) )\qquad\mbox{fof}(\mbox{thIV}_9,\mbox{conjecture})$

\medskip\textbf{SET851-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_1,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_2,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_y},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_x},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_U},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_U} = \mbox{v\_m} )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET487-6.p} Replacement property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{power\_class}(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property9}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property9}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET311-6.p} Range property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{intersection}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})),\mbox{cross\_product}(\mbox{universal\_class},\mbox{range\_of}(x)))\qquad\mbox{cnf}(\mbox{prove\_range\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET426-6.p} Single valued class alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{ordered\_pair}(\mbox{single\_valued}_3(x),\mbox{single\_valued}_1(x))\in x\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{single\_valued\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET359-6.p} Sum class property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{sum\_class}(\mbox{intersection}(x,y)),\mbox{intersection}(\mbox{sum\_class}(x),\mbox{sum\_class}(y)))\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET033-3.p} Domain of composition






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(a)\subseteq \mbox{domain\_of}(b)\qquad\mbox{cnf}(\mbox{range\_subset\_of\_domain},\mbox{hypothesis})$

$\mbox{domain\_of}(a)\ne \mbox{domain\_of}(b\circ a)\qquad\mbox{cnf}(\mbox{prove\_domain\_of\_composition},\mbox{negated\_conjecture})$

\medskip\textbf{SET778-1.p} Set theory membership and union axioms






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-1.ax'})$

\medskip\textbf{SET836-2.p} Problem about set theory






$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET576$\wedge$7.p} Trybulec's 17th Boolean property of sets






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

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

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

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mequiv}@ ( \mbox{intersect}@ b@ c )@ ( \mbox{mexists\_ind}@ \lambda d{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \in @ d@ b )@ ( \in @ d@ c ) ) ) ) ) )\qquad\mbox{thf}(\mbox{intersect\_defn},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mequiv}@ ( \mbox{disjoint}@ b@ c )@ ( \mbox{mnot}@ ( \mbox{intersect}@ b@ c ) ) ) ) )\qquad\mbox{thf}(\mbox{disjoint\_defn},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{intersect}@ b@ c )@ ( \mbox{intersect}@ c@ b ) ) ) )\qquad\mbox{thf}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mforall\_ind}@ \lambda d{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \in @ d@ b )@ ( \mbox{mnot}@ ( \in @ d@ c ) ) ) )@ ( \mbox{disjoint}@ b@ c ) ) ) )\qquad\mbox{thf}(\mbox{prove\_th}_{17},\mbox{conjecture})$

\medskip\textbf{SET086-6.p} A singleton set has a member, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_exists1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{member\_of}(\mbox{singleton}(y))\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_exists1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET732+4.p} Property of restriction 3

If F is a mapping from A to B,and G equal to F on A and
 C =image2(F,A) and F is injective, then G is one-to-one.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }c\subseteq b\mbox{ and }\mbox{image}_2(f,a) = c\mbox{ and }\forall x,y{:}\;( ( x\in a\mbox{ and }y\in c ) \ \Rightarrow \ ( \mbox{apply}(g,x,y)\iff \mbox{apply}(f,x,y) ) )\mbox{ and }\mbox{injective}(f,a,b) ) \ \Rightarrow \ \mbox{one\_to\_one}(g,a,c) )\qquad\mbox{fof}(\mbox{thII}_{23},\mbox{conjecture})$

\medskip\textbf{SET585$\wedge$5.p} TPS problem BOOL-PROP-38

Trybulec's 38th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) \ \Rightarrow \ ( x@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_38\_pme},\mbox{conjecture})$

\medskip\textbf{SET253-6.p} Restriction property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{restrict}(x,y,z),\mbox{cross\_product}(y,z))\qquad\mbox{cnf}(\mbox{prove\_restriction\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET618+3.p} The symmetric difference of X and X is the empty set






$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b{:}\;\mbox{union}(b,b) = b\qquad\mbox{fof}(\mbox{idempotency\_of\_union},\mbox{axiom})$

$\forall b{:}\;b\setminus b = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{self\_difference\_is\_empty\_set},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b{:}\;\mbox{symmetric\_difference}(b,b) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{prove\_th}_{93},\mbox{conjecture})$

\medskip\textbf{SET706+4.p} Property of difference

The difference of C in A is equal to the union of the difference
 of C in B and the difference of B in A.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,c{:}\;( ( c\subseteq b\mbox{ and }b\subseteq a ) \ \Rightarrow \ \mbox{equal\_set}(a\setminus c,\mbox{union}(b\setminus c,a\setminus b)) )\qquad\mbox{fof}(\mbox{thI}_{49},\mbox{conjecture})$

\medskip\textbf{SET018+1.p} Second components of equal ordered pairs are equal






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall w,x,y,z{:}\;( ( \mbox{ordered\_pair}(w,x) = \mbox{ordered\_pair}(y,z)\mbox{ and }x\in \mbox{universal\_class} ) \ \Rightarrow \ x = z )\qquad\mbox{fof}(\mbox{ordered\_pair\_determines\_components}_2,\mbox{conjecture})$

\medskip\textbf{SET024+1.p} A set belongs to its singleton






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;( x\in \mbox{universal\_class} \ \Rightarrow \ x\in \mbox{singleton}(x) )\qquad\mbox{fof}(\mbox{set\_in\_its\_singleton},\mbox{conjecture})$

\medskip\textbf{SET096+1.p} There are at most two subsets of a singleton set






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( \mbox{subclass}(x,\mbox{singleton}(y)) \ \Rightarrow \ ( x = \mbox{null\_class}\mbox{ or }\mbox{singleton}(y) = x ) )\qquad\mbox{fof}(\mbox{two\_subsets\_of\_singleton},\mbox{conjecture})$

\medskip\textbf{SET440-6.p} The restriction of function is function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_restriction\_of\_function\_is\_function}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{restrict}(\mbox{xf},x,y))\qquad\mbox{cnf}(\mbox{prove\_restriction\_of\_function\_is\_function}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET351+4.p} Sum of a singleton is the singleton member






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a{:}\;\mbox{equal\_set}(\mbox{\mbox{sum}}(\mbox{singleton}(a)),a)\qquad\mbox{fof}(\mbox{thI}_{39},\mbox{conjecture})$

\medskip\textbf{SET586+3.p} If X (= Y, then X $\wedge$ Z (= Y $\wedge$ Z

If X is a subset of Y, then the intersection of X and Z is a
 subset of the intersection of Y and Z.




$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq c \ \Rightarrow \ \mbox{intersection}(b,d)\subseteq \mbox{intersection}(c,d) )\qquad\mbox{fof}(\mbox{prove\_intersection\_of\_subset},\mbox{conjecture})$

\medskip\textbf{SET899+1.p} subset(A,B) =$>$ ( in(C,A) | subset(A,difference(B,singleton(C))) )






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

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

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

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

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ ( \mbox{in}(c,a)\mbox{ or }a\subseteq \mbox{set\_difference}(b,\mbox{singleton}(c)) ) )\qquad\mbox{fof}(\mbox{t40\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ ( \mbox{in}(c,a)\mbox{ or }a\subseteq \mbox{set\_difference}(b,\mbox{singleton}(c)) ) )\qquad\mbox{fof}(\mbox{l3\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET614$\wedge$5.p} TPS problem BOOL-PROP-88

Trybulec's 88th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_88\_pme},\mbox{conjecture})$

\medskip\textbf{SET029-4.p} Relationship between apply and image, part 2 of 2






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\neg\,\mbox{image}(\mbox{singleton\_set}(\mbox{element}),\mbox{a\_function})\subseteq \mbox{apply}(\mbox{a\_function},\mbox{element})\qquad\mbox{cnf}(\mbox{prove\_image\_and\_apply}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET855-2.p} Problem about Zorn's lemma






$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_n},\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{v\_n}\ne \mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET707+4.p} Components of equal ordered pairs are equal

If {{A},{A,B}} = {{U},{U,V}} then A = U and B = V.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,u,v{:}\;( \mbox{equal\_set}(\mbox{unordered\_pair}(\mbox{singleton}(a),\mbox{unordered\_pair}(a,b)),\mbox{unordered\_pair}(\mbox{singleton}(u),\mbox{unordered\_pair}(u,v))) \ \Rightarrow \ ( a = u\mbox{ and }b = v ) )\qquad\mbox{fof}(\mbox{thI}_{50},\mbox{conjecture})$

\medskip\textbf{SET105-6.p} Special member 3 of an ordered pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{unordered\_pair}(\mbox{null\_class},\mbox{singleton}(\mbox{null\_class}))\ne \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_property\_3\_of\_ordered\_pair}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_property\_3\_of\_ordered\_pair}_2,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_property\_3\_of\_ordered\_pair}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET211-6.p} Corollary 2 to cross product product monotonicity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(y,z),\mbox{cross\_product}(\mbox{union}(x,y),z))\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_X\_product\_monotonicity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET018-1.p} Second components of equal ordered pairs are equal






$x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{singleton}_1,\mbox{axiom})$

$x\in \mbox{singleton\_set}(y) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{singleton}_2,\mbox{axiom})$

$x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{unordered\_pair}_1,\mbox{axiom})$

$y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{unordered\_pair}_2,\mbox{axiom})$

$x\in \mbox{unordered\_pair}(y,z) \ \Rightarrow \ ( x = y\mbox{ or }x = z )\qquad\mbox{cnf}(\mbox{unordered\_pair}_3,\mbox{axiom})$

$\mbox{ordered\_pair}(x,y) = \mbox{unordered\_pair}(\mbox{singleton\_set}(x),\mbox{unordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(m_1,r_1) = \mbox{ordered\_pair}(m_2,r_2)\qquad\mbox{cnf}(\mbox{equal\_ordered\_pairs},\mbox{hypothesis})$

$r_1\ne r_2\qquad\mbox{cnf}(\mbox{prove\_second\_components\_equal},\mbox{negated\_conjecture})$

\medskip\textbf{SET271-6.p} Corollary to domain property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(\mbox{cross\_product}(x,x))\ne x\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_domain\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET239-6.p} Restriction alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{restrict}(\mbox{xr},x,y)\qquad\mbox{cnf}(\mbox{prove\_restriction\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,z\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_restriction\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET002-6.p} Idempotency of union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,x)\ne x\qquad\mbox{cnf}(\mbox{prove\_idempotency\_of\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET790+4.p} The least element, if it existes, is unique






$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e,m{:}\;( ( \mbox{order}(r,e)\mbox{ and }\mbox{least}(m,r,e) ) \ \Rightarrow \ \forall x{:}\;( \mbox{least}(x,r,e) \ \Rightarrow \ m = x ) )\qquad\mbox{fof}(\mbox{thIV}_2,\mbox{conjecture})$

\medskip\textbf{SET478-6.p} Axiom of subsets 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_axiom\_of\_subsets2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{intersection}(x,y)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_axiom\_of\_subsets2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET225-6.p} Inverse of cross product squared






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(x,x)'\ne \mbox{cross\_product}(x,x)\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_square}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET151-6.p} Complement of null class is universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{null\_class}'\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_complement\_of\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET438-6.p} In a function, the image of each domain element is a singleton






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_images\_of\_functions\_are\_singletons}_1,\mbox{negated\_conjecture})$

$z\in \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_images\_of\_functions\_are\_singletons}_2,\mbox{negated\_conjecture})$

$\mbox{singleton}(\mbox{member\_of}(\mbox{image}(x,\mbox{singleton}(z))))\ne \mbox{image}(x,\mbox{singleton}(z))\qquad\mbox{cnf}(\mbox{prove\_images\_of\_functions\_are\_singletons}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET018-4.p} Second components of equal ordered pairs are equal






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{b\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(c)\qquad\mbox{cnf}(\mbox{c\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(d)\qquad\mbox{cnf}(\mbox{d\_little\_set},\mbox{hypothesis})$

$\mbox{ordered\_pair}(a,b) = \mbox{ordered\_pair}(c,d)\qquad\mbox{cnf}(\mbox{equal\_ordered\_pair},\mbox{hypothesis})$

$b\ne d\qquad\mbox{cnf}(\mbox{prove\_second\_components\_equal},\mbox{negated\_conjecture})$

\medskip\textbf{SET053-6.p} Corollary to Cartesian product axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_cartesian\_product}_1,\mbox{negated\_conjecture})$

$\neg\,v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_cartesian\_product}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET318-6.p} Corollary 2 to range property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(\mbox{range\_of}(\mbox{restrict}(x,y,z)),u),\mbox{cross\_product}(\mbox{intersection}(z,\mbox{range\_of}(x)),u))\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_range\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET366-6.p} The empty set is in every power set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{null\_class}\in \mbox{power\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_null\_class\_in\_power\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET475-6.p} Restricted domain






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{cross\_product}(y,y),\mbox{union}(x,\mbox{identity\_relation}))\qquad\mbox{cnf}(\mbox{prove\_restricted\_domain}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{restrict}(x,y,y))\ne y\qquad\mbox{cnf}(\mbox{prove\_restricted\_domain}_2,\mbox{negated\_conjecture})$

$\mbox{singleton}(\mbox{member\_of}(y))\ne y\qquad\mbox{cnf}(\mbox{prove\_restricted\_domain}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET750+4.p} Property of isomorphism

If F is one-to-one, then F is an isomorphism if and only if
 F and its inverse are increasing.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,r,s{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{one\_to\_one}(f,a,b) ) \ \Rightarrow \ ( \mbox{isomorphism}(f,a,r,b,s)\iff ( \mbox{increasing}(f,a,r,b,s)\mbox{ and }\mbox{increasing}(\mbox{inverse\_function}(f,a,b),b,s,a,r) ) ) )\qquad\mbox{fof}(\mbox{thII}_{41},\mbox{conjecture})$

\medskip\textbf{SET358-6.p} Sum class distributes over union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{sum\_class}(\mbox{union}(x,y))\ne \mbox{union}(\mbox{sum\_class}(x),\mbox{sum\_class}(y))\qquad\mbox{cnf}(\mbox{prove\_sum\_over\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET621+3.p} (X sym\ Y) \ Z = (X \ (Y U Z)) U (Y \ (X U Z))

The difference of (the symmetric difference of X and Y) and Z
 is the union of (the difference of X and (the union of Y and Z))
 and (the difference of Y and (the union of X and Z)).




$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( b\setminus c )\setminus d = b\setminus \mbox{union}(c,d)\qquad\mbox{fof}(\mbox{difference\_difference\_union},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{union}(b,c)\setminus d = \mbox{union}(b\setminus d,c\setminus d)\qquad\mbox{fof}(\mbox{difference\_distributes\_over\_union},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{symmetric\_difference}(b,c)\setminus d = \mbox{union}(b\setminus \mbox{union}(c,d),c\setminus \mbox{union}(b,d))\qquad\mbox{fof}(\mbox{prove\_th}_{97},\mbox{conjecture})$

\medskip\textbf{SET531-6.p} Application property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_application\_property1}_1,\mbox{negated\_conjecture})$

$x\in \mbox{domain\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_application\_property1}_2,\mbox{negated\_conjecture})$

$\mbox{member\_of}(\mbox{image}(z,\mbox{singleton}(x)))\ne \mbox{apply}(z,x)\qquad\mbox{cnf}(\mbox{prove\_application\_property1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET057-6.p} Expanded equality definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{not\_subclass\_element}(x,y)\in y\qquad\mbox{cnf}(\mbox{prove\_equality2}_1,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_equality2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{prove\_equality2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET136-6.p} Corollary 4 to membership in unordered triple






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_4\_to\_member\_of\_triple}_1,\mbox{negated\_conjecture})$

$\mbox{set\_builder}(x,\mbox{set\_builder}(y,\mbox{set\_builder}(u,\mbox{null\_class}))) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_4\_to\_member\_of\_triple}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET590$\wedge$5.p} TPS problem BOOL-PROP-49

Trybulec's 49th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) \ \Rightarrow \ ( x@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_49\_pme},\mbox{conjecture})$

\medskip\textbf{SET949+1.p} ~ ( in(A,cartesian\_product(B,C)) \& ordered\_pair(D,E) != A )






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

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

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

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

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

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

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,\mbox{cartesian\_product}_2(b,c))\mbox{ and }\forall d,e{:}\;\mbox{ordered\_pair}(d,e)\ne a\qquad\mbox{fof}(\mbox{t102\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET625$\wedge$5.p} TPS problem BOOL-PROP-101

Trybulec's 101th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( y@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) ) ) \ \Rightarrow \ \exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_101\_pme},\mbox{conjecture})$

\medskip\textbf{SET398-6.p} Associativity of composition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$( \mbox{xr}\circ \mbox{yr} )\circ \mbox{zr}\ne \mbox{xr}\circ ( \mbox{yr}\circ \mbox{zr} )\qquad\mbox{cnf}(\mbox{prove\_associativity\_of\_composition}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET824-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_U},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{SET044$\wedge$5.p} TPS problem PELL40

If there were an anti-Russell set (a set that contains exactly
 those sets that are members of themselves), then not every set
 has a complement.




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

$\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xx}@ \mbox{xy} )\iff ( \mbox{cF}@ \mbox{xx}@ \mbox{xx} ) ) \ \Rightarrow \ \neg\,\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xz}{:}\;\mbox{\$i}{:}\;( ( \mbox{cF}@ \mbox{xz}@ \mbox{xy} )\iff \neg\,\mbox{cF}@ \mbox{xz}@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cPELL}_{40},\mbox{conjecture})$

\medskip\textbf{SET331-6.p} Range is image of the domain






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(\mbox{xr},\mbox{domain\_of}(\mbox{xr}))\ne \mbox{range\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_image\_of\_domain}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET781-1.p} Set theory axioms






$\mbox{include}(\mbox{'Axioms/SET002-0.ax'})$

\medskip\textbf{SET035-6.p} Maps for composition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{maps}(\mbox{xf},u,v)\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_mappings}_1,\mbox{negated\_conjecture})$

$\mbox{maps}(\mbox{xg},v,w)\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_mappings}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{maps}(\mbox{xg}\circ \mbox{xf},u,w)\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_mappings}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET690+4.p} Property of union and intersection






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,c{:}\;( \mbox{equal\_set}(\mbox{union}(\mbox{intersection}(a,b),c),\mbox{intersection}(a,\mbox{union}(b,c)))\iff c\subseteq a )\qquad\mbox{fof}(\mbox{thI}_{12},\mbox{conjecture})$

\medskip\textbf{SET159+3.p} Associativity of union

The union of (the union of X and Y) and Z is the union of X and
 (the union of Y and Z).




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{union}(\mbox{union}(b,c),d) = \mbox{union}(b,\mbox{union}(c,d))\qquad\mbox{fof}(\mbox{prove\_associativity\_of\_union},\mbox{conjecture})$

\medskip\textbf{SET550-6.p} Application property 18






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{function}(\mbox{image}(\mbox{application\_function},\mbox{singleton}(x)))\qquad\mbox{cnf}(\mbox{prove\_application\_property15}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET835-2.p} Problem about set theory






$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET027+3.p} Transitivity of subset

If X is a subset of Y and Y is a subset of Z, then X is a
 subset of Z.




$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }c\subseteq d ) \ \Rightarrow \ b\subseteq d )\qquad\mbox{fof}(\mbox{prove\_transitivity\_of\_subset},\mbox{conjecture})$

\medskip\textbf{SET484-6.p} Replacement property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_replacement\_property6}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{xf})\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property6}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{xf}\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property6}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET076+1.p} If both members of a pair belong to a set, the pair is a subset






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y,z{:}\;( ( x\in z\mbox{ and }y\in z ) \ \Rightarrow \ \mbox{subclass}(\mbox{unordered\_pair}(x,y),z) )\qquad\mbox{fof}(\mbox{unordered\_pair\_is\_subset},\mbox{conjecture})$

\medskip\textbf{SET220-6.p} Cross product distributes over intersection 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{cross\_product}(x,z),\mbox{cross\_product}(y,z))\ne \mbox{cross\_product}(\mbox{intersection}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_over\_intersection1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET134-6.p} Corollary 2 to membership in unordered triple






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_member\_of\_triple}_1,\mbox{negated\_conjecture})$

$\mbox{set\_builder}(u,\mbox{set\_builder}(y,\mbox{set\_builder}(z,\mbox{null\_class}))) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_member\_of\_triple}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET772+4.p} Belonging of the same member of a partition is an equivalence






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall a,e,r{:}\;( \mbox{partition}(a,e) \ \Rightarrow \ ( \forall x,y{:}\;( ( x\in e\mbox{ and }y\in e ) \ \Rightarrow \ ( \mbox{apply}(r,x,y)\iff \exists z{:}\;( z\in a\mbox{ and }x\in z\mbox{ and }y\in z ) ) ) \ \Rightarrow \ \mbox{equivalence}(r,e) ) )\qquad\mbox{fof}(\mbox{thIII}_{08},\mbox{conjecture})$

\medskip\textbf{SET020-6.p} 1st is unique when x is an ordered pair of sets






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_unique\_1st\_and\_2nd\_in\_pair\_of\_sets1}_1,\mbox{negated\_conjecture})$

$\mbox{first}(\mbox{ordered\_pair}(u,v))\ne u\qquad\mbox{cnf}(\mbox{prove\_unique\_1st\_and\_2nd\_in\_pair\_of\_sets1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET584$\wedge$5.p} TPS problem BOOL-PROP-33

Trybulec's 33rd Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_33\_pme},\mbox{conjecture})$

\medskip\textbf{SET633+3.p} If X \ Y (= Z and Y \ X (= Z, then X sym\ Y (= Z

If the difference of X and Y is a subset of Z and the
 difference of Y and X is a subset of Z, then the symmetric
 difference of X and Y is a subset of Z.




$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }d\subseteq c ) \ \Rightarrow \ \mbox{union}(b,d)\subseteq c )\qquad\mbox{fof}(\mbox{union\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( ( ( b\setminus c )\subseteq d\mbox{ and }( c\setminus b )\subseteq d ) \ \Rightarrow \ \mbox{symmetric\_difference}(b,c)\subseteq d )\qquad\mbox{fof}(\mbox{prove\_th}_{115},\mbox{conjecture})$

\medskip\textbf{SET710+4.p} Associativity of composition






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c,d{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,c,d) ) \ \Rightarrow \ \mbox{equal\_maps}(\mbox{compose\_function}(h,\mbox{compose\_function}(g,f,a,b,c),a,c,d),\mbox{compose\_function}(\mbox{compose\_function}(h,g,b,c,d),f,a,b,d),a,d) )\qquad\mbox{fof}(\mbox{thII}_{02},\mbox{conjecture})$

\medskip\textbf{SET462-6.p} Range of restricted identity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{range\_of}(\mbox{restrict}(\mbox{identity\_relation},x,y))\ne \mbox{intersection}(x,y)\qquad\mbox{cnf}(\mbox{prove\_range\_of\_restricted\_identity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET927+1.p} diff(uno\_pair(A,B),C) = sgtn(A) $\leq$$>$ (~in(A,C) \& (in(B,C) | A = B))






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

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

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

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

$\forall a,b,c{:}\;( \mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c) = \mbox{singleton}(a)\iff ( \neg\,\mbox{in}(a,c)\mbox{ and }( \mbox{in}(b,c)\mbox{ or }a = b ) ) )\qquad\mbox{fof}(\mbox{t70\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( \mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c) = \mbox{singleton}(a)\iff ( \neg\,\mbox{in}(a,c)\mbox{ and }( \mbox{in}(b,c)\mbox{ or }a = b ) ) )\qquad\mbox{fof}(\mbox{l39\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET963+1.p} Basic properties of sets, theorem 116






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

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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

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

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

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

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

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

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

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ a\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{l50\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,\mbox{cartesian\_product}_2(b,c))\mbox{ and }\forall d,e{:}\;\mbox{ordered\_pair}(d,e)\ne a\qquad\mbox{fof}(\mbox{t102\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;( a\subseteq \mbox{cartesian\_product}_2(a,a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t116\_zfmisc}_1,\mbox{conjecture})$

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

\medskip\textbf{SET444-6.p} Function property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_function\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{function}(y)\qquad\mbox{cnf}(\mbox{prove\_function\_property1}_2,\mbox{negated\_conjecture})$

$\mbox{intersection}(\mbox{domain\_of}(x),\mbox{domain\_of}(y)) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_function\_property1}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{union}(x,y))\qquad\mbox{cnf}(\mbox{prove\_function\_property1}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET400-6.p} Right compose with null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\circ \mbox{null\_class}\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_right\_compose\_with\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET061-7.p} Existence of the null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{special\_classes\_lemma},\mbox{axiom})$

$z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET873+1.p} union(singleton(A),singleton(B)) = singleton(A) =$>$ A = B






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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

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

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

$\forall a,b{:}\;( \mbox{set\_union}_2(\mbox{singleton}(a),b)\subseteq b \ \Rightarrow \ \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l21\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b{:}\;( \mbox{set\_union}_2(\mbox{singleton}(a),\mbox{singleton}(b)) = \mbox{singleton}(a) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t13\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET024-4.p} A set belongs to its singleton






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\neg\,a\in \mbox{singleton\_set}(a)\qquad\mbox{cnf}(\mbox{prove\_membership\_of\_singleton\_set},\mbox{negated\_conjecture})$

\medskip\textbf{SET494-6.p} Diagonalization alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(\mbox{intersection}(x',\mbox{identity\_relation}))\ne \mbox{diagonalise}(x)\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET129-6.p} Membership in a built unordered triple






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$u\in \mbox{set\_builder}(x,\mbox{set\_builder}(y,\mbox{set\_builder}(z,\mbox{null\_class})))\qquad\mbox{cnf}(\mbox{prove\_members\_of\_built\_triple}_1,\mbox{negated\_conjecture})$

$u\ne x\qquad\mbox{cnf}(\mbox{prove\_members\_of\_built\_triple}_2,\mbox{negated\_conjecture})$

$u\ne y\qquad\mbox{cnf}(\mbox{prove\_members\_of\_built\_triple}_3,\mbox{negated\_conjecture})$

$u\ne z\qquad\mbox{cnf}(\mbox{prove\_members\_of\_built\_triple}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET780-1.p} Set theory membership and difference axioms






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-3.ax'})$

\medskip\textbf{SET291-6.p} Proof of Goedel's axiom B6, part 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(v,u)\in x\qquad\mbox{cnf}(\mbox{prove\_inverse\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_inverse\_property3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,v)\in x'\qquad\mbox{cnf}(\mbox{prove\_inverse\_property3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET527-6.p} Ordered pair components are sets 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(\mbox{first}(x),\mbox{second}(x)) = x\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_components\_are\_sets2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{second}(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_components\_are\_sets2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET260-6.p} Domain alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_domain\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_domain\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{domain\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_domain\_alternate\_defn3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET223-6.p} Cross product property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(\mbox{intersection}(w,y),\mbox{intersection}(x,z)),\mbox{cross\_product}(w,z))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property5}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET124-6.p} Alternative definition of set builder, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_set\_builder\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{set\_builder}(x,z)\qquad\mbox{cnf}(\mbox{prove\_set\_builder\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET237-6.p} Restriction alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{restrict}(\mbox{xr},x,y),\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_restriction\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET973+1.p} Basic properties of sets, theorem 126






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

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

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

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

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

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

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

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

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

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

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

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

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

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

$\forall a,b,c,d{:}\;\mbox{cartesian\_product}_2(\mbox{set\_intersection}_2(a,b),\mbox{set\_intersection}_2(c,d)) = \mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,d))\qquad\mbox{fof}(\mbox{t123\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{cartesian\_product}_2(\mbox{set\_difference}(a,b),c) = \mbox{set\_difference}(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,c))\mbox{ and }\mbox{cartesian\_product}_2(c,\mbox{set\_difference}(a,b)) = \mbox{set\_difference}(\mbox{cartesian\_product}_2(c,a),\mbox{cartesian\_product}_2(c,b)) )\qquad\mbox{fof}(\mbox{t125\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;\mbox{set\_difference}(\mbox{cartesian\_product}_2(a,b),\mbox{cartesian\_product}_2(c,d)) = \mbox{set\_union}_2(\mbox{cartesian\_product}_2(\mbox{set\_difference}(a,c),b),\mbox{cartesian\_product}_2(a,\mbox{set\_difference}(b,d)))\qquad\mbox{fof}(\mbox{t126\_zfmisc}_1,\mbox{conjecture})$

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

$\forall a,b,c{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_difference}(c,b)\subseteq \mbox{set\_difference}(c,a) )\qquad\mbox{fof}(\mbox{t34\_xboole}_1,\mbox{axiom})$

$\forall a,b,c{:}\;\mbox{set\_difference}(a,\mbox{set\_intersection}_2(b,c)) = \mbox{set\_union}_2(\mbox{set\_difference}(a,b),\mbox{set\_difference}(a,c))\qquad\mbox{fof}(\mbox{t54\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SET533-6.p} The range of Z is the class of applications of Z to Z's domain 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_class\_of\_applications\_to\_domain1}_1,\mbox{negated\_conjecture})$

$x\in \mbox{domain\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_class\_of\_applications\_to\_domain1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{apply}(z,x)\in \mbox{range\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_class\_of\_applications\_to\_domain1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET277-6.p} Corollary 3 to domain property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(u,\mbox{domain\_of}(\mbox{restrict}(x,y,z))),\mbox{cross\_product}(u,\mbox{intersection}(\mbox{domain\_of}(x),y)))\qquad\mbox{cnf}(\mbox{prove\_corollary\_3\_to\_domain\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET604$\wedge$5.p} TPS problem BOOL-PROP-75

Trybulec's 75th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( \mbox{\$false}\mbox{ and }\neg\,x@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_75\_pme},\mbox{conjecture})$

\medskip\textbf{SET934+1.p} subset(union(powerset(A),powerset(B)),powerset(union(A,B)))






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

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

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

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

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

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

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

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

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

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

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

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

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

$\forall a,b{:}\;\mbox{set\_union}_2(\mbox{powerset}(a),\mbox{powerset}(b))\subseteq \mbox{powerset}(\mbox{set\_union}_2(a,b))\qquad\mbox{fof}(\mbox{t81\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET458-6.p} Corollary to identity is a function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{function}(\mbox{restrict}(\mbox{identity\_relation},x,y))\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_identity\_is\_a\_function}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET612$\wedge$5.p} TPS problem BOOL-PROP-85

Trybulec's 85th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx}\mbox{ and }x@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_85\_pme},\mbox{conjecture})$

\medskip\textbf{SET128-6.p} Building a triple






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$\mbox{union}(\mbox{singleton}(x),\mbox{union}(\mbox{singleton}(y),\mbox{singleton}(z)))\ne \mbox{set\_builder}(x,\mbox{set\_builder}(y,\mbox{set\_builder}(z,\mbox{null\_class})))\qquad\mbox{cnf}(\mbox{prove\_building\_a\_triple}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET119+1.p} Corollary 1 to components of equal ordered pairs being equal






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( x\in \mbox{universal\_class}\mbox{ or }\mbox{first}(\mbox{ordered\_pair}(x,y)) = \mbox{ordered\_pair}(x,y) )\qquad\mbox{fof}(\mbox{corollary\_1\_to\_OP\_determines\_components}_1,\mbox{conjecture})$

\medskip\textbf{SET403-6.p} Domain of composition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{domain\_of}(\mbox{yr}),\mbox{range\_of}(\mbox{xr}))\qquad\mbox{cnf}(\mbox{prove\_dual\_of\_boyer\_lemma\_18}_1,\mbox{negated\_conjecture})$

$\mbox{range\_of}(\mbox{yr}\circ \mbox{xr})\ne \mbox{range\_of}(\mbox{yr})\qquad\mbox{cnf}(\mbox{prove\_dual\_of\_boyer\_lemma\_18}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET068-6.p} Proper class in an unordered pair, part 2

If one argument of an unordered pair is a proper class, the
 pair contains only the other.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{singleton}(y),\mbox{unordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{prove\_singleton\_in\_unordered\_pair2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET629$\wedge$5.p} TPS problem BOOL-PROP-111

Trybulec's 111th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;\neg\,\exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx}\mbox{ and }x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_111\_pme},\mbox{conjecture})$

\medskip\textbf{SET034-6.p} The composition of functions is a function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_functions}_1,\mbox{negated\_conjecture})$

$\mbox{function}(\mbox{yf})\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_functions}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{xf}\circ \mbox{yf})\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_functions}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET062+1.p} The empty set is a subset of X






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{fof}(\mbox{null\_class\_is\_subclass},\mbox{conjecture})$

\medskip\textbf{SET407-6.p} Corollary 2 monotonicity of composition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{xr},\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_monotonicity\_of\_composition}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{zr}\circ ( \mbox{xr}\circ \mbox{ur} ),\mbox{zr}\circ \mbox{ur})\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_monotonicity\_of\_composition}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET921+1.p} in(A,difference(B,singleton(C))) $\leq$$>$ ( in(A,B) \& A != C )






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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

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

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

$\forall a,b,c{:}\;( \mbox{in}(a,\mbox{set\_difference}(b,\mbox{singleton}(c)))\iff ( \mbox{in}(a,b)\mbox{ and }a\ne c ) )\qquad\mbox{fof}(\mbox{t64\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET156-6.p} De Morgans law 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x,y)'\ne \mbox{union}(x',y')\qquad\mbox{cnf}(\mbox{prove\_demorgans\_law2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET781-3.p} Set theory axioms based on NBG set theory






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

\medskip\textbf{SET856-2.p} Problem about Zorn's lemma






$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_n},\mbox{t\_a}),\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_n},\mbox{t\_a}),\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET554-6.p} Cantor class alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{cantor}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_cantor\_class\_property1}_1,\mbox{negated\_conjecture})$

$z\in \mbox{apply}(\mbox{xr},z)\qquad\mbox{cnf}(\mbox{prove\_cantor\_class\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET601$\wedge$3.p} X $\wedge$ Y U Y $\wedge$ Z U Z $\wedge$ X = (X U Y) $\wedge$ (Y U Z) $\wedge$ (Z U X)

The intersection of X and the union of Y and the intersection
 of Y and the union of Z and the intersection of Z and X is the
 intersection of (the union of X and Y) and the intersection of
 (the union of Y and Z) and (the union of Z and X).




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},z{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{union}@ ( \mbox{intersection}@ x@ y )@ ( \mbox{union}@ ( \mbox{intersection}@ y@ z )@ ( \mbox{intersection}@ z@ x ) ) ) = ( \mbox{intersection}@ ( \mbox{union}@ x@ y )@ ( \mbox{intersection}@ ( \mbox{union}@ y@ z )@ ( \mbox{union}@ z@ x ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET417-6.p} Composition property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{composition\_function},\mbox{singleton}(x)),\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_composition\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET016-3.p} First components of equal ordered pairs are equal






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{little\_set\_a},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{little\_set\_b},\mbox{hypothesis})$

$\mbox{ordered\_pair}(a,c) = \mbox{ordered\_pair}(b,d)\qquad\mbox{cnf}(\mbox{equal\_ordered\_pairs},\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{prove\_first\_components\_equal},\mbox{negated\_conjecture})$

\medskip\textbf{SET467-6.p} Sets with at most one member 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{cross\_product}(x,x),\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_one\_member1}_1,\mbox{negated\_conjecture})$

$\mbox{singleton}(\mbox{not\_subclass\_element}(x,\mbox{null\_class}))\ne x\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_one\_member1}_2,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_one\_member1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET353-6.p} Corollary to sum of pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sum\_of\_pair}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{union}(x,y)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sum\_of\_pair}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET815+4.p} An ordinal number is equal to the sum of its successor






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+4.ax'})$

$\forall a,x{:}\;( x\in a \ \Rightarrow \ \mbox{singleton}(x)\subseteq a )\qquad\mbox{fof}(\mbox{thI}_{44},\mbox{axiom})$

$\forall a{:}\;( a\in \mbox{on} \ \Rightarrow \ \mbox{equal\_set}(\mbox{\mbox{sum}}(\mbox{suc}(a)),a) )\qquad\mbox{fof}(\mbox{thV}_{15},\mbox{conjecture})$

\medskip\textbf{SET116-6.p} 2nd is unique if x is not an ordered pair of sets, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(\mbox{first}_1(x),\mbox{second}_1(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_unique\_1st\_and\_2nd\_in\_pair\_of\_non\_sets4}_1,\mbox{negated\_conjecture})$

$\mbox{second}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_unique\_2nd\_in\_pair\_of\_non\_sets},\mbox{negated\_conjecture})$

\medskip\textbf{SET756+4.p} Inverse image of union equals the union of inverse images






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,x,y{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }x\subseteq b\mbox{ and }y\subseteq b ) \ \Rightarrow \ \mbox{equal\_set}(\mbox{inverse\_image}_2(f,\mbox{union}(x,y)),\mbox{union}(\mbox{inverse\_image}_2(f,x),\mbox{inverse\_image}_2(f,y))) )\qquad\mbox{fof}(\mbox{thIIa}_{06},\mbox{conjecture})$

\medskip\textbf{SET121+1.p} Corollary 3 to components of equal ordered pairs being equal






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( y\in \mbox{universal\_class}\mbox{ or }\mbox{first}(\mbox{ordered\_pair}(x,y)) = \mbox{ordered\_pair}(x,y) )\qquad\mbox{fof}(\mbox{corollary\_1\_to\_OP\_determines\_components}_2,\mbox{conjecture})$

\medskip\textbf{SET752$\wedge$4.p} The image of union is equal to the union of images






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^1.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \mbox{fun\_image}@ f@ ( \mbox{union}@ x@ y ) ) = ( \mbox{union}@ ( \mbox{fun\_image}@ f@ x )@ ( \mbox{fun\_image}@ f@ y ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET361-6.p} Range is sum squared






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{range\_of}(x),\mbox{sum\_class}(\mbox{sum\_class}(\mbox{sum\_class}(x))))\qquad\mbox{cnf}(\mbox{prove\_range\_is\_sum\_squared}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET988+1.p} Functions and their basic properties, theorem 2






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

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

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

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

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

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

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

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

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{fc2\_subset}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc3\_subset}_1,\mbox{axiom})$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\forall a{:}\;( ( \forall b{:}\;\neg\,\mbox{in}(b,a)\mbox{ and }\forall c,d{:}\;\mbox{ordered\_pair}(c,d)\ne b\mbox{ and }\forall b,c,d{:}\;( ( \mbox{in}(\mbox{ordered\_pair}(b,c),a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(b,d),a) ) \ \Rightarrow \ c = d ) ) \ \Rightarrow \ ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) ) )\qquad\mbox{fof}(\mbox{t2\_funct}_1,\mbox{conjecture})$

$\forall a{:}\;( \mbox{function}(a)\iff \forall b,c,d{:}\;( ( \mbox{in}(\mbox{ordered\_pair}(b,c),a)\mbox{ and }\mbox{in}(\mbox{ordered\_pair}(b,d),a) ) \ \Rightarrow \ c = d ) )\qquad\mbox{fof}(\mbox{d1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{relation}(a)\iff \forall b{:}\;\neg\,\mbox{in}(b,a)\mbox{ and }\forall c,d{:}\;b\ne \mbox{ordered\_pair}(c,d) )\qquad\mbox{fof}(\mbox{d1\_relat}_1,\mbox{axiom})$

\medskip\textbf{SET061-6.p} Existence of the null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET019-4.p} Two sets that contain one another are equal






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$b\subseteq a\qquad\mbox{cnf}(\mbox{a\_contains\_b},\mbox{hypothesis})$

$a\subseteq b\qquad\mbox{cnf}(\mbox{b\_contains\_a},\mbox{hypothesis})$

$a\ne b\qquad\mbox{cnf}(\mbox{prove\_\_a\_equals\_b},\mbox{negated\_conjecture})$

\medskip\textbf{SET325-6.p} Image alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{image}(\mbox{xr},x)\qquad\mbox{cnf}(\mbox{prove\_image\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(1_{\text{dom}(\mbox{xr})},y)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_image\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET123-6.p} Alternative definition of set builder, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$x\in \mbox{set\_builder}(y,z)\qquad\mbox{cnf}(\mbox{prove\_set\_builder\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_set\_builder\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

$\neg\,x\in z\qquad\mbox{cnf}(\mbox{prove\_set\_builder\_alternate\_defn1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET716+4.p} The composition of injective mappings is injective






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{injective}(f,a,b)\mbox{ and }\mbox{injective}(g,b,c) ) \ \Rightarrow \ \mbox{injective}(\mbox{compose\_function}(g,f,a,b,c),a,c) )\qquad\mbox{fof}(\mbox{thII}_{07},\mbox{conjecture})$

\medskip\textbf{SET176-6.p} Corollary to absorbtion for union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,\mbox{union}(y,\mbox{intersection}(x,z)))\ne \mbox{union}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_absorbtion\_for\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET466-6.p} Inverse of identity is identity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{identity\_relation}'\ne \mbox{identity\_relation}\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_identity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET171-6.p} Union distributes over intersection

The union of X and (the intersection of Y and Z) is the
 intersection of (the union of X and Y) and (the union of X and Z).




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{union}(x,y),\mbox{union}(x,z))\ne \mbox{union}(x,\mbox{intersection}(y,z))\qquad\mbox{cnf}(\mbox{prove\_union\_over\_intersection1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET015-3.p} The union of sets is commutative






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{union}(\mbox{as},\mbox{bs}) = \mbox{cs}\qquad\mbox{cnf}(\mbox{a\_union\_b\_is\_c},\mbox{hypothesis})$

$\mbox{union}(\mbox{bs},\mbox{as}) = \mbox{ds}\qquad\mbox{cnf}(\mbox{b\_union\_a\_is\_d},\mbox{hypothesis})$

$\mbox{cs}\ne \mbox{ds}\qquad\mbox{cnf}(\mbox{prove\_c\_equals\_d},\mbox{negated\_conjecture})$

\medskip\textbf{SET044-5.p} Anti-Russell Sets

If there were an anti-Russell set (a set that contains
 exactly those sets that are members of themselves), then not
 every set has a complement.




$\mbox{element}(x,a) \ \Rightarrow \ \mbox{element}(x,x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\mbox{element}(x,x) \ \Rightarrow \ \mbox{element}(x,a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\mbox{element}(y,f(x)) \ \Rightarrow \ \neg\,\mbox{element}(y,x)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

$\mbox{element}(y,x)\mbox{ or }\mbox{element}(y,f(x))\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET317-6.p} Corollary 1 to range property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{range\_of}(\mbox{restrict}(x,y,z)),\mbox{intersection}(z,\mbox{range\_of}(x)))\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_range\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET683$\wedge$3.p} !y in E : y in range of R (X to Y) ?x in D : x in domain of R

For every element y in E such that y is in the range in a relation  R from X to Y there exists an element x in D such that x is in the  domain of R.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{is\_rel\_on}@ r@ x@ y ) \ \Rightarrow \ \forall v{:}\;\mbox{\$i}{:}\;( ( y@ v ) \ \Rightarrow \ ( ( \mbox{rel\_codomain}@ r@ v ) \ \Rightarrow \ \exists u{:}\;\mbox{\$i}{:}\;( x@ u\mbox{ and }\mbox{rel\_domain}@ r@ u ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET894+1.p} cart\_prod(singleton(A),singleton(B)) = singleton(o\_pair(A,B))






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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{cartesian\_product}_2(\mbox{singleton}(a),\mbox{singleton}(b)) = \mbox{singleton}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{t35\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET622$\wedge$5.p} TPS problem BOOL-PROP-98

Trybulec's 98th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,( y@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} )\mbox{ or }( z@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz}\mbox{ or }z@ \mbox{xz} )\mbox{ or }( x@ \mbox{xz}\mbox{ and }y@ \mbox{xz}\mbox{ and }z@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_98\_pme},\mbox{conjecture})$

\medskip\textbf{SET514-6.p} Class of ordered pairs is not a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\in x\qquad\mbox{cnf}(\mbox{prove\_class\_of\_ordered\_pairs\_not\_set}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET840-2.p} Problem about set theory






$\neg\,\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_insert}(\mbox{v\_U},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_b})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_b})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_b})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_S},\mbox{t\_b}),\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_b}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_A},\mbox{v\_X},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_X},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_A},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnionI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_a},\mbox{c\_insert}(\mbox{v\_b},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{c\_insert}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{v\_a} = \mbox{v\_b} )\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

\medskip\textbf{SET861-2.p} Problem about Zorn's lemma






$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_z},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OHausdorff\_}_1(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OHausdorff}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_subset\_\_chain}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_xa}(\mbox{v\_U}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_xa}(\mbox{v\_U}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_U} = \mbox{v\_xa}(\mbox{v\_U}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_V},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_V},\mbox{v\_x}(\mbox{v\_U}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET409-6.p} Composition of element relation 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{element\_relation}\circ \mbox{element\_relation}\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_element\_relation1}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{sum\_class}(y)\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_element\_relation1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET577+3.p} Trybulec's 18th Boolean property of sets






$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( \forall e{:}\;( e\in b\iff ( e\in c\mbox{ or }e\in d ) ) \ \Rightarrow \ b = \mbox{union}(c,d) )\qquad\mbox{fof}(\mbox{prove\_th}_{18},\mbox{conjecture})$

\medskip\textbf{SET573+3.p} Trybulec's 12th Boolean property of sets






$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{disjoint}(b,c)\iff \neg\,\mbox{intersect}(b,c) )\qquad\mbox{fof}(\mbox{disjoint\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\in c\mbox{ and }\mbox{disjoint}(c,d) ) \ \Rightarrow \ \neg\,b\in d )\qquad\mbox{fof}(\mbox{prove\_th}_{12},\mbox{conjecture})$

\medskip\textbf{SET300-6.p} Inverse of inverse






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$( x' )'\ne \mbox{restrict}(x,\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_inverse}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET146-6.p} The intersection of X and the empty set is the empty set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{null\_class},x)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_intersection\_with\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET580+3.p} x is in X sym Y iff x is in X iff x is not in Y

x is in the symmetric difference of X and Y iff it is not the
 case x is in X iff x is in Y.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( b\in \mbox{symmetric\_difference}(c,d)\iff \neg\,b\in c\iff b\in d )\qquad\mbox{fof}(\mbox{prove\_th}_{23},\mbox{conjecture})$

\medskip\textbf{SET094+1.p} Property 1 of singletons






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( ( \mbox{singleton}(\mbox{member\_of}(x)) = x\mbox{ and }y\in x ) \ \Rightarrow \ \mbox{member\_of}(x) = y )\qquad\mbox{fof}(\mbox{property\_of\_singletons1}_1,\mbox{conjecture})$

\medskip\textbf{SET833-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_V},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_V},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_V},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET033-6.p} Domain of composition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{range\_of}(\mbox{xr}),\mbox{domain\_of}(\mbox{yr}))\qquad\mbox{cnf}(\mbox{prove\_boyer\_lemma\_18}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{yr}\circ \mbox{xr})\ne \mbox{domain\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_boyer\_lemma\_18}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET047+1.p} Set equality is symmetric

Define set equality as having exactly the same members. Prove
 set equality is symmetric.




$\forall x,y{:}\;( \mbox{set\_equal}(x,y)\iff \forall z{:}\;( \mbox{element}(z,x)\iff \mbox{element}(z,y) ) )\qquad\mbox{fof}(\mbox{pel43}_1,\mbox{axiom})$

$\forall x,y{:}\;( \mbox{set\_equal}(x,y)\iff \mbox{set\_equal}(y,x) )\qquad\mbox{fof}(\mbox{pel}_{43},\mbox{conjecture})$

\medskip\textbf{SET597+3.p} X = Y U Z iff Y (= X, Z (= X, !V: Y (= V \& Z (= V, X (= V

X is the union of Y and Z if and only if the following
 conditions are satisfied: 1. Y is a subset of X, 2. Z is a
 subset of X, and 3. for every V such that Y is a subset of V
 and Z is a subset of V : X is a subset of V.




$\forall b,c{:}\;b\subseteq \mbox{union}(b,c)\qquad\mbox{fof}(\mbox{subset\_of\_union},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }d\subseteq c ) \ \Rightarrow \ \mbox{union}(b,d)\subseteq c )\qquad\mbox{fof}(\mbox{union\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( b = \mbox{union}(c,d)\iff ( c\subseteq b\mbox{ and }d\subseteq b\mbox{ and }\forall e{:}\;( ( c\subseteq e\mbox{ and }d\subseteq e ) \ \Rightarrow \ b\subseteq e ) ) )\qquad\mbox{fof}(\mbox{prove\_th}_{56},\mbox{conjecture})$

\medskip\textbf{SET006-1.p} A = A $\wedge$ B if A (= B

If the intersection of two sets is the first of the two sets,
 then the first is a subset of the second.




$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-2.ax'})$

$\mbox{intersection}(d,a,d)\qquad\mbox{cnf}(\mbox{d\_intersection\_a\_is\_d},\mbox{hypothesis})$

$\neg\,d\subseteq a\qquad\mbox{cnf}(\mbox{prove\_d\_is\_a\_subset\_of\_a},\mbox{negated\_conjecture})$

\medskip\textbf{SET069-6.p} Proper class in an unordered pair, part 3

If one argument of an unordered pair is a proper class, the
 pair contains only the other.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_equals\_singleton1}_1,\mbox{negated\_conjecture})$

$\mbox{unordered\_pair}(x,y)\ne \mbox{singleton}(x)\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_equals\_singleton1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET862-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OHausdorff\_}_1(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OHausdorff}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_z},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_1,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_subset\_\_chain}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_y},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_x}(\mbox{v\_U}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{v\_U} = \mbox{v\_x}(\mbox{v\_U}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SET076$\wedge$1.p} If both members of a pair belong to a set, the pair is a subset






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( z@ x\mbox{ and }z@ y ) \ \Rightarrow \ ( \subseteq @ ( \mbox{unord\_pair}@ x@ y )@ z ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET031-3.p} The composition of two sets is a relation






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\neg\,\mbox{relation}(a\circ b)\qquad\mbox{cnf}(\mbox{prove\_composition\_is\_a\_relation},\mbox{negated\_conjecture})$

\medskip\textbf{SET173$\wedge$5.p} TPS problem BOOL-PROP-68

Trybulec's 68th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} ) ) ) = x\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_68\_pme},\mbox{conjecture})$

\medskip\textbf{SET036-6.p} Properties of apply for functions, part 1 of 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_application\_property9}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{xf}\qquad\mbox{cnf}(\mbox{prove\_application\_property9}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_application\_property9}_3,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{xf},x)\ne y\qquad\mbox{cnf}(\mbox{prove\_application\_property9}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET918+1.p} ~ ( intersection(uno\_pair(A,B),C) = sgtn(A) \& in(B,C) \& A != B )






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

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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

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

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

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

$\forall a,b,c{:}\;\neg\,\mbox{set\_intersection}_2(\mbox{unordered\_pair}(a,b),c) = \mbox{singleton}(a)\mbox{ and }\mbox{in}(b,c)\mbox{ and }a\ne b\qquad\mbox{fof}(\mbox{t59\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET030-6.p} Function values are little sets






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_application\_property8}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{apply}(\mbox{xf},x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_application\_property8}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET857-2.p} Problem about Zorn's lemma






$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) = \mbox{v\_m} ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_2,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_3,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_n},\mbox{t\_a}),\mbox{v\_x},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_V},\mbox{v\_U},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_V},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_V},\mbox{t\_a}),\mbox{v\_U},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_V} = \mbox{v\_U} )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SET035-3.p} Maps for composition






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(x)\subseteq \mbox{domain\_of}(y) \ \Rightarrow \ \mbox{domain\_of}(x) = \mbox{domain\_of}(y\circ x)\qquad\mbox{cnf}(\mbox{domain\_of\_composition},\mbox{axiom})$

$( \mbox{function}(x)\mbox{ and }\mbox{function}(y) ) \ \Rightarrow \ \mbox{function}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_function},\mbox{axiom})$

$\mbox{maps}(\mbox{function}_1,a,b)\qquad\mbox{cnf}(\mbox{one\_mapping},\mbox{hypothesis})$

$\mbox{maps}(\mbox{function}_2,c,d)\qquad\mbox{cnf}(\mbox{another\_mapping},\mbox{hypothesis})$

$\neg\,\mbox{maps}(\mbox{function}_2\circ \mbox{function}_1,a,d)\qquad\mbox{cnf}(\mbox{prove\_maps\_for\_composition},\mbox{negated\_conjecture})$

\medskip\textbf{SET031-4.p} The composition of two sets is a relation






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\neg\,\mbox{relation}(a\circ b)\qquad\mbox{cnf}(\mbox{prove\_composition\_is\_a\_relation},\mbox{negated\_conjecture})$

\medskip\textbf{SET206-6.p} Cross product with null class 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(\mbox{null\_class},y)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_with\_null\_class2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET119-6.p} Corollary 1 to components of equal ordered pairs being equal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_OP\_determines\_components1}_1,\mbox{negated\_conjecture})$

$\mbox{first}(\mbox{ordered\_pair}(x,y))\ne \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_OP\_determines\_components1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET313-6.p} Range property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{range\_of}(\mbox{cross\_product}(x,y))\ne y\qquad\mbox{cnf}(\mbox{prove\_range\_property2}_1,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_range\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET722+4.p} If the composition of F and G is surjective, then F is surjective






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{surjective}(\mbox{compose\_function}(g,f,a,b,c),a,c) ) \ \Rightarrow \ \mbox{surjective}(g,b,c) )\qquad\mbox{fof}(\mbox{thII}_{13},\mbox{conjecture})$

\medskip\textbf{SET306-6.p} Range of universal class is the universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{range\_of}(\mbox{universal\_class})\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_range\_of\_universal\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET983+1.p} Basic properties of sets, theorem 137






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

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

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

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

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

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

$\forall a,b,c,d{:}\;\mbox{cartesian\_product}_2(\mbox{set\_intersection}_2(a,b),\mbox{set\_intersection}_2(c,d)) = \mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,d))\qquad\mbox{fof}(\mbox{t123\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d,e{:}\;( ( \mbox{in}(a,\mbox{cartesian\_product}_2(b,c))\mbox{ and }\mbox{in}(a,\mbox{cartesian\_product}_2(d,e)) ) \ \Rightarrow \ \mbox{in}(a,\mbox{cartesian\_product}_2(\mbox{set\_intersection}_2(b,d),\mbox{set\_intersection}_2(c,e))) )\qquad\mbox{fof}(\mbox{t137\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET704+4.p} If X is a member of A, then product(A) is a subset of X






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,x{:}\;( x\in a \ \Rightarrow \ \mbox{\mbox{product}}(a)\subseteq x )\qquad\mbox{fof}(\mbox{thI}_{42},\mbox{conjecture})$

\medskip\textbf{SET086+1.p} A singleton set has a member






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;\exists u{:}\;( ( u\in \mbox{universal\_class}\mbox{ and }x = \mbox{singleton}(u) )\mbox{ or }( \neg\,\exists y{:}\;( y\in \mbox{universal\_class}\mbox{ and }x = \mbox{singleton}(y) )\mbox{ and }u = x ) )\qquad\mbox{fof}(\mbox{member\_of\_substitution},\mbox{conjecture})$

\medskip\textbf{SET914+1.p} ~ ( disjoint(unordered\_pair(A,B),C) \& in(A,C) )






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

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

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

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

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

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

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

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

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

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

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

$\forall a,b,c{:}\;\neg\,\mbox{disjoint}(\mbox{unordered\_pair}(a,b),c)\mbox{ and }\mbox{in}(a,c)\qquad\mbox{fof}(\mbox{t55\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET791+4.p} The greatest element, if it exists, is maximal






$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e,m{:}\;( ( \mbox{order}(r,e)\mbox{ and }\mbox{greatest}(m,r,e) ) \ \Rightarrow \ \mbox{max}(m,r,e) )\qquad\mbox{fof}(\mbox{thIV}_3,\mbox{conjecture})$

\medskip\textbf{SET065+1.p} Null class is a set (follows from axiom of infinity)






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\mbox{null\_class}\in \mbox{universal\_class}\qquad\mbox{fof}(\mbox{null\_class\_is\_a\_set},\mbox{conjecture})$

\medskip\textbf{SET301-6.p} Inverse commutes with restriction






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{xr},y,x)'\ne \mbox{restrict}(\mbox{xr}',x,y)\qquad\mbox{cnf}(\mbox{prove\_inverse\_commutes\_restriction}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET715+4.p} The composition of F and its inverse is the identity






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{one\_to\_one}(f,a,b) ) \ \Rightarrow \ \mbox{identity}(\mbox{compose\_function}(f,\mbox{inverse\_function}(f,a,b),b,a,b),b) )\qquad\mbox{fof}(\mbox{thII}_{06},\mbox{conjecture})$

\medskip\textbf{SET219-6.p} Cross product distributes over union 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{cross\_product}(x,y),\mbox{cross\_product}(x,z))\ne \mbox{cross\_product}(x,\mbox{union}(y,z))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_over\_union2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET021-4.p} 2nd is unique when x is an ordered pair of sets






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{b\_little\_set},\mbox{hypothesis})$

$\mbox{second}(\mbox{ordered\_pair}(a,b))\ne b\qquad\mbox{cnf}(\mbox{prove\_second\_is\_second},\mbox{negated\_conjecture})$

\medskip\textbf{SET575+3.p} Trybulec's 15th Boolean property of sets






$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{prove\_th}_{15},\mbox{conjecture})$

\medskip\textbf{SET561-6.p} Compatible function property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{operation}(\mbox{xf}_1)\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{compatible}(\mbox{xh},\mbox{xf}_1,\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property1}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{domain\_of}(\mbox{xh}),\mbox{domain\_of}(\mbox{xh}))\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property1}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{apply}(\mbox{xf}_1,\mbox{ordered\_pair}(x,y))\in \mbox{domain\_of}(\mbox{xh})\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property1}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET441-6.p} The intersection of functions is a function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_intersection\_of\_functions}_1,\mbox{negated\_conjecture})$

$\mbox{function}(y)\qquad\mbox{cnf}(\mbox{prove\_intersection\_of\_functions}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{intersection}(x,y))\qquad\mbox{cnf}(\mbox{prove\_intersection\_of\_functions}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET612$\wedge$3.p} X \ (Y U Z) = (X \ Y) $\wedge$ (X \ Z)

The difference of X and (the union of Y and Z) is the
 intersection of (the difference of X and Y) and (the difference
 of X and Z).




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},z{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{setminus}@ x@ ( \mbox{union}@ y@ z ) ) = ( \mbox{intersection}@ ( \mbox{setminus}@ x@ y )@ ( \mbox{setminus}@ x@ z ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET480-6.p} Replacement property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{range\_of}(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET201$\wedge$5.p} TPS problem BOOL-PROP-41

Trybulec's 41st Boolean property of sets




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

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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( z@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cV}@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx}\mbox{ and }\mbox{cV}@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_41\_pme},\mbox{conjecture})$

\medskip\textbf{SET185$\wedge$5.p} TPS problem BOOL-PROP-35

Trybulec's 35th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_35\_pme},\mbox{conjecture})$

\medskip\textbf{SET072+1.p} Right cancellation for unordered pairs






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y,z{:}\;( ( x\in \mbox{universal\_class}\mbox{ and }y\in \mbox{universal\_class}\mbox{ and }\mbox{unordered\_pair}(x,z) = \mbox{unordered\_pair}(y,z) ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{right\_cancellation},\mbox{conjecture})$

\medskip\textbf{SET045$\wedge$7.p} No Universal Set






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda z{:}\;\mbox{mu}{:}\;( \mbox{mexists\_ind}@ \lambda y{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{element}@ x@ y ) )@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{element}@ x@ z ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ ( \mbox{element}@ x@ x ) ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{element}@ x@ z ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mbox\_s}_4@ ( \mbox{element}@ x@ x ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{element}@ x@ y ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{pel41}_1,\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mnot}@ ( \mbox{mexists\_ind}@ \lambda z{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda x{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{element}@ x@ z ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{pel}_{41},\mbox{conjecture})$

\medskip\textbf{SET018-3.p} Second components of equal ordered pairs are equal






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{b\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(c)\qquad\mbox{cnf}(\mbox{c\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(d)\qquad\mbox{cnf}(\mbox{d\_little\_set},\mbox{hypothesis})$

$\mbox{ordered\_pair}(a,b) = \mbox{ordered\_pair}(c,d)\qquad\mbox{cnf}(\mbox{equal\_ordered\_pair},\mbox{hypothesis})$

$b\ne d\qquad\mbox{cnf}(\mbox{prove\_second\_components\_equal},\mbox{negated\_conjecture})$

\medskip\textbf{SET014$\wedge$5.p} TPS problem BOOL-PROP-32

Trybulec's 32nd Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( y@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_32\_pme},\mbox{conjecture})$

\medskip\textbf{SET064-6.p} The null class is unique






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$z\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_unique}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(z,\mbox{null\_class})\in z\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_unique}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET746+4.p} If F and G and increasing, then GoF is increasing






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c,r,s,t{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{increasing}(f,a,r,b,s)\mbox{ and }\mbox{increasing}(g,b,s,c,t) ) \ \Rightarrow \ \mbox{increasing}(\mbox{compose\_function}(g,f,a,b,c),a,r,c,t) )\qquad\mbox{fof}(\mbox{thII}_{37},\mbox{conjecture})$

\medskip\textbf{SET864-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OHausdorff\_}_1(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OHausdorff}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_u},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_c},\mbox{t\_a}),\mbox{v\_u},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_Union}(\mbox{v\_c},\mbox{t\_a}) = \mbox{v\_u}\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_Zorn}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_subset\_\_chain}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_c},\mbox{t\_a}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_x}(\mbox{v\_U}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{v\_U} = \mbox{v\_x}(\mbox{v\_U}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET745+4.p} Problem on composition of mappings 10

Consider three mappings F1 from A1 to B,F2 from A2 to B,
 F which is equal to F1 on A1 and to F2 on A2, then F is a
 mapping from union(A1,A2) to B if and only if F1 and F2 are
 equal on the intersection of A1 and A2.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f_1,f_2,f,a_1,a_2,b{:}\;( ( \mbox{maps}(f_1,a_1,b)\mbox{ and }\mbox{maps}(f_2,a_2,b)\mbox{ and }\forall x,y{:}\;( ( x\in \mbox{union}(a_1,a_2)\mbox{ and }y\in b ) \ \Rightarrow \ ( \mbox{apply}(f,x,y)\iff ( ( x\in a_1\mbox{ and }\mbox{apply}(f_1,x,y) )\mbox{ or }( x\in a_2\mbox{ and }\mbox{apply}(f_2,x,y) ) ) ) ) ) \ \Rightarrow \ ( \mbox{maps}(f,\mbox{union}(a_1,a_2),b)\iff \forall x,y_1,y_2{:}\;( ( x\in a_1\mbox{ and }x\in a_2\mbox{ and }y_1\in b\mbox{ and }y_2\in b\mbox{ and }\mbox{apply}(f_1,x,y_1)\mbox{ and }\mbox{apply}(f_2,x,y_2) ) \ \Rightarrow \ y_1 = y_2 ) ) )\qquad\mbox{fof}(\mbox{thII}_{36},\mbox{conjecture})$

\medskip\textbf{SET596+3.p} If X (= Y and Y $\wedge$ Z = the empty set, then X $\wedge$ Z = the empty set

If X is a subset of Y and the intersection of Y and Z is the
 empty set, then the intersection of X and Z is the empty set.




$\forall b{:}\;( b\subseteq \mbox{empty\_set} \ \Rightarrow \ b = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{subset\_of\_empty\_set\_is\_empty\_set},\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq c \ \Rightarrow \ \mbox{intersection}(b,d)\subseteq \mbox{intersection}(c,d) )\qquad\mbox{fof}(\mbox{intersection\_of\_subset},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }\mbox{intersection}(c,d) = \mbox{empty\_set} ) \ \Rightarrow \ \mbox{intersection}(b,d) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{prove\_th}_{55},\mbox{conjecture})$

\medskip\textbf{SET559-6.p} Compatible functions alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{operation}(\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_compatible\_functions\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\mbox{compatible}(\mbox{xh},\mbox{xf}_1,\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_compatible\_functions\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(\mbox{range\_of}(\mbox{xh}),\mbox{range\_of}(\mbox{xh})),\mbox{domain\_of}(\mbox{xf}_2))\qquad\mbox{cnf}(\mbox{prove\_compatible\_functions\_alternate\_defn2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET183$\wedge$5.p} TPS problem BOOL-PROP-42

Trybulec's 42nd Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) \ \Rightarrow \ ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) ) = x )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_42\_pme},\mbox{conjecture})$

\medskip\textbf{SET546-6.p} Application special case 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{apply}(\mbox{null\_class},x)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_application\_special\_case2}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_application\_special\_case2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET218-6.p} Cross product distributes over union 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{cross\_product}(x,z),\mbox{cross\_product}(y,z))\ne \mbox{cross\_product}(\mbox{union}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_over\_union1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET592+3.p} If X (= Y and X (= Z and Y $\wedge$ Z = empty set, then X = empty set

If X is a subset of Y and X is a subset of Z and the
 intersection of Y and Z is the empty set, then X is the empty
 set.




$\forall b{:}\;( b\subseteq \mbox{empty\_set} \ \Rightarrow \ b = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{subset\_of\_empty\_set\_is\_empty\_set},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }b\subseteq d ) \ \Rightarrow \ b\subseteq \mbox{intersection}(c,d) )\qquad\mbox{fof}(\mbox{intersection\_of\_subsets},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }b\subseteq d\mbox{ and }\mbox{intersection}(c,d) = \mbox{empty\_set} ) \ \Rightarrow \ b = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{prove\_th}_{51},\mbox{conjecture})$

\medskip\textbf{SET067-7.p} Proper class in an unordered pair, part 1

If one argument of an unordered pair is a proper class, the
 pair contains only the other.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{special\_classes\_lemma},\mbox{axiom})$

$\neg\,z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{existence\_of\_null\_class},\mbox{axiom})$

$\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{cnf}(\mbox{null\_class\_is\_subclass},\mbox{axiom})$

$\mbox{subclass}(x,\mbox{null\_class}) \ \Rightarrow \ x = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{corollary\_of\_null\_class\_is\_subclass},\mbox{axiom})$

$z = \mbox{null\_class}\mbox{ or }\mbox{not\_subclass\_element}(z,\mbox{null\_class})\in z\qquad\mbox{cnf}(\mbox{null\_class\_is\_unique},\mbox{axiom})$

$\mbox{null\_class}\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{null\_class\_is\_a\_set},\mbox{axiom})$

$\mbox{unordered\_pair}(x,y) = \mbox{unordered\_pair}(y,x)\qquad\mbox{cnf}(\mbox{commutativity\_of\_unordered\_pair},\mbox{axiom})$

$\neg\,\mbox{subclass}(\mbox{singleton}(x),\mbox{unordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{prove\_singleton\_in\_unordered\_pair1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET208-6.p} Cross product is monotonic 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_is\_monotonic1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,z),\mbox{cross\_product}(y,z))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_is\_monotonic1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET305-6.p} Range of null class is the null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{range\_of}(\mbox{null\_class})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_range\_of\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET302-6.p} Range alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{range\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_range\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(1_{\text{dom}(\mbox{xr})},y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_range\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET310-6.p} Monotonicity of range 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{range\_of}(\mbox{intersection}(x,y)),\mbox{range\_of}(x))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_range3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET639+3.p} Trybulec's 121th Boolean property of sets






$\forall b,c{:}\;( b\subseteq c \ \Rightarrow \ \mbox{intersection}(b,c) = b )\qquad\mbox{fof}(\mbox{subset\_intersection},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( ( b\subseteq c\mbox{ and }\mbox{intersection}(c,b) = \mbox{empty\_set} ) \ \Rightarrow \ b = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{prove\_th}_{121},\mbox{conjecture})$

\medskip\textbf{SET038-6.p} Properties of apply for functions, part 3 of 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{maps}(\mbox{xf},\mbox{xd},\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_mapping\_property1}_1,\mbox{negated\_conjecture})$

$x\in \mbox{xd}\qquad\mbox{cnf}(\mbox{prove\_mapping\_property1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{apply}(\mbox{xf},x)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_mapping\_property1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET386-6.p} Relation property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(x,\mbox{universal\_class},\mbox{universal\_class}) = x\qquad\mbox{cnf}(\mbox{prove\_relation\_property4}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_relation\_property4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET427-6.p} Single valued class alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{ordered\_pair}(\mbox{single\_valued}_3(x),\mbox{single\_valued}_2(x))\in x\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{single\_valued\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET156+4.p} De Morgans law 2






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,e{:}\;( ( a\subseteq e\mbox{ and }b\subseteq e ) \ \Rightarrow \ \mbox{equal\_set}(e\setminus \mbox{intersection}(a,b),\mbox{union}(e\setminus a,e\setminus b)) )\qquad\mbox{fof}(\mbox{thI}_{25},\mbox{conjecture})$

\medskip\textbf{SET755+4.p} If X is a subset of Y, then f-1(X) is a subset of f-1(Y)






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,x,y{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }x\subseteq b\mbox{ and }y\subseteq b\mbox{ and }x\subseteq y ) \ \Rightarrow \ \mbox{inverse\_image}_2(f,x)\subseteq \mbox{inverse\_image}_2(f,y) )\qquad\mbox{fof}(\mbox{thIIa}_{05},\mbox{conjecture})$

\medskip\textbf{SET671$\wedge$3.p} X a subset of X1 =$>$ R (X to Y) restricted to X1 is R

If X is a subset of X1 then a relation R from X to Y restricted
 to X1 is R.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall z{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{is\_rel\_on}@ r@ x@ y\mbox{ and }\subseteq @ x@ z ) \ \Rightarrow \ ( \mbox{restrict\_rel\_domain}@ r@ z ) = r )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET343-6.p} Image of intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(\mbox{xr},\mbox{intersection}(y,z))\ne \mbox{intersection}(\mbox{image}(\mbox{xr},y),\mbox{image}(\mbox{xr},z))\qquad\mbox{cnf}(\mbox{prove\_image\_of\_intersection}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET618$\wedge$5.p} TPS problem BOOL-PROP-93

Trybulec's 93th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,x@ \mbox{xz} )\mbox{ or }( x@ \mbox{xz}\mbox{ and }\neg\,x@ \mbox{xz} ) ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_93\_pme},\mbox{conjecture})$

\medskip\textbf{SET030-4.p} Function values are little sets






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{function}(a)\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$\neg\,\mbox{little\_set}(\mbox{apply}(a,b))\qquad\mbox{cnf}(\mbox{prove\_function\_values\_are\_small},\mbox{negated\_conjecture})$

\medskip\textbf{SET805+4.p} Order relation on E is an order relation on a subset of E






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e{:}\;( \mbox{order}(r,e) \ \Rightarrow \ \forall x{:}\;( x\subseteq e \ \Rightarrow \ \mbox{order}(r,x) ) )\qquad\mbox{fof}(\mbox{thIV}_{17},\mbox{conjecture})$

\medskip\textbf{SET893+1.p} in(o\_pair(A,B),cart\_prod(sgtn(C),sgtn(D))) $\leq$$>$ ( A = C \& B = D )






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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(\mbox{singleton}(c),\mbox{singleton}(d)))\iff ( a = c\mbox{ and }b = d ) )\qquad\mbox{fof}(\mbox{t34\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET877+1.p} intersection(singleton(A),singleton(B)) = singleton(A) =$>$ A = B






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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

$\forall a,b{:}\;( \mbox{set\_intersection}_2(a,\mbox{singleton}(b)) = \mbox{singleton}(b) \ \Rightarrow \ \mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{l30\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b{:}\;( \mbox{set\_intersection}_2(\mbox{singleton}(a),\mbox{singleton}(b)) = \mbox{singleton}(a) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t18\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET082-6.p} The singleton of a non-set is the null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_singleton\_is\_null\_class}_1,\mbox{negated\_conjecture})$

$\mbox{singleton}(x)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_singleton\_is\_null\_class}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET560-6.p} Compatible functions alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(\mbox{xh})\qquad\mbox{cnf}(\mbox{prove\_compatible\_functions\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\mbox{cross\_product}(\mbox{domain\_of}(\mbox{xh}),\mbox{domain\_of}(\mbox{xh})) = \mbox{domain\_of}(\mbox{xf}_1)\qquad\mbox{cnf}(\mbox{prove\_compatible\_functions\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{cross\_product}(\mbox{range\_of}(\mbox{xh}),\mbox{range\_of}(\mbox{xh})),\mbox{domain\_of}(\mbox{xf}_2))\qquad\mbox{cnf}(\mbox{prove\_compatible\_functions\_alternate\_defn3}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{compatible}(\mbox{xh},\mbox{xf}_1,\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_compatible\_functions\_alternate\_defn3}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET016+1.p} First components of equal ordered pairs are equal






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall w,x,y,z{:}\;( ( \mbox{ordered\_pair}(w,x) = \mbox{ordered\_pair}(y,z)\mbox{ and }w\in \mbox{universal\_class} ) \ \Rightarrow \ w = y )\qquad\mbox{fof}(\mbox{ordered\_pair\_determines\_components}_1,\mbox{conjecture})$

\medskip\textbf{SET043-5.p} Russell's Paradox

Russell's paradox : there is no Russell set (a set which
 contains exactly those sets which are not members
 of themselves).




$\mbox{element}(x,a) \ \Rightarrow \ \neg\,\mbox{element}(x,x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\mbox{element}(x,x)\mbox{ or }\mbox{element}(x,a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET499-6.p} The Russell class not a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{diagonalise}(\mbox{element\_relation})\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_russell\_class\_not\_a\_set}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET463-6.p} Corollary to domain and range of identity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(\mbox{identity\_relation},x)\ne x\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_domain\_and\_range\_of\_identity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET278-6.p} Corollary 4 to domain property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(\mbox{domain\_of}(\mbox{restrict}(x_1,y_1,z_1)),\mbox{domain\_of}(\mbox{restrict}(x_2,y_2,z_2))),\mbox{cross\_product}(y_1,y_2))\qquad\mbox{cnf}(\mbox{prove\_corollary\_4\_to\_domain\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET016-6.p} First components of equal ordered pairs are equal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(w,x) = \mbox{ordered\_pair}(y,z)\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_determines\_components1}_1,\mbox{negated\_conjecture})$

$w\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_determines\_components1}_2,\mbox{negated\_conjecture})$

$w\ne y\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_determines\_components1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET248-6.p} Restriction preserves intersections






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{restrict}(\mbox{xr}_1,x_1,y_1),\mbox{restrict}(\mbox{xr}_2,x_2,y_2))\ne \mbox{restrict}(\mbox{intersection}(\mbox{xr}_1,\mbox{xr}_2),\mbox{intersection}(x_1,x_2),\mbox{intersection}(y_1,y_2))\qquad\mbox{cnf}(\mbox{prove\_restriction\_preserves\_intersections}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET819-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_D},\mbox{v\_F},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_V},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_V},\mbox{v\_F},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET091+1.p} Uniqueness of member\_of when X is not a singleton of a set






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,u{:}\;( ( \neg\,\exists y{:}\;( y\in \mbox{universal\_class}\mbox{ and }x = \mbox{singleton}(y) )\mbox{ and }x = u ) \ \Rightarrow \ \mbox{member\_of}(x) = u )\qquad\mbox{fof}(\mbox{member\_when\_not\_a\_singleton},\mbox{conjecture})$

\medskip\textbf{SET526-6.p} Ordered pair components are sets 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(\mbox{first}(x),\mbox{second}(x)) = x\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_components\_are\_sets1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{first}(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_components\_are\_sets1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET060-7.p} Nothing in the intersection of a set and its complement






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{prove\_special\_classes\_lemma}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET847-2.p} Problem about Zorn's lemma






$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_minus}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}) )\qquad\mbox{cnf}(\mbox{cls\_Set\_ODiffI}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{v\_a},\mbox{c\_emptyset},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OemptyE}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_emptyset},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Oempty\_\_subsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_refl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_chain\_\_lemma4}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oequal\_\_succ\_\_Union}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_minus}(\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_c},\mbox{t\_a})\ne \mbox{v\_c}\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_not\_\_equals}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{SET015-4.p} The union of sets is commutative






$x\in y \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(a_2,\mbox{axiom})$

$\mbox{little\_set}(f_1(x,y))\mbox{ or }x = y\qquad\mbox{cnf}(\mbox{extensionality}_1,\mbox{axiom})$

$f_1(x,y)\in x\mbox{ or }f_1(x,y)\in y\mbox{ or }x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$( f_1(x,y)\in x\mbox{ and }f_1(x,y)\in y ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_3,\mbox{axiom})$

$z\in \mbox{intersection}(x,y) \ \Rightarrow \ z\in x\qquad\mbox{cnf}(\mbox{intersection}_1,\mbox{axiom})$

$z\in \mbox{intersection}(x,y) \ \Rightarrow \ z\in y\qquad\mbox{cnf}(\mbox{intersection}_2,\mbox{axiom})$

$( z\in x\mbox{ and }z\in y ) \ \Rightarrow \ z\in \mbox{intersection}(x,y)\qquad\mbox{cnf}(\mbox{intersection}_3,\mbox{axiom})$

$z\in x' \ \Rightarrow \ \neg\,z\in x\qquad\mbox{cnf}(\mbox{complement}_1,\mbox{axiom})$

$\mbox{little\_set}(z) \ \Rightarrow \ ( z\in x'\mbox{ or }z\in x )\qquad\mbox{cnf}(\mbox{complement}_2,\mbox{axiom})$

$\mbox{union}(x,y) = \mbox{intersection}(x',y')'\qquad\mbox{cnf}(\mbox{union},\mbox{axiom})$

$\neg\,z\in \mbox{empty\_set}\qquad\mbox{cnf}(\mbox{empty\_set},\mbox{axiom})$

$\mbox{little\_set}(z) \ \Rightarrow \ z\in \mbox{universal\_set}\qquad\mbox{cnf}(\mbox{universal\_set},\mbox{axiom})$

$\mbox{union}(\mbox{as},\mbox{bs}) = \mbox{cs}\qquad\mbox{cnf}(\mbox{a\_union\_b\_is\_c},\mbox{hypothesis})$

$\mbox{union}(\mbox{bs},\mbox{as}) = \mbox{ds}\qquad\mbox{cnf}(\mbox{b\_union\_a\_is\_d},\mbox{hypothesis})$

$\mbox{cs}\ne \mbox{ds}\qquad\mbox{cnf}(\mbox{prove\_c\_equals\_d},\mbox{negated\_conjecture})$

\medskip\textbf{SET324-6.p} Image alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{image}(\mbox{xr},x)\qquad\mbox{cnf}(\mbox{prove\_image\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(1_{\text{dom}(\mbox{xr})},y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_image\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET977+1.p} Basic properties of sets, theorem 130






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

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

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

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

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

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

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(\mbox{singleton}(c),d))\iff ( a = c\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{t128\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,\mbox{singleton}(d)))\iff ( \mbox{in}(a,c)\mbox{ and }b = d ) )\qquad\mbox{fof}(\mbox{t129\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\ne \mbox{empty\_set} \ \Rightarrow \ ( \mbox{cartesian\_product}_2(\mbox{singleton}(b),a)\ne \mbox{empty\_set}\mbox{ and }\mbox{cartesian\_product}_2(a,\mbox{singleton}(b))\ne \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{t130\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET474-6.p} Lemma 2 to restricted domain






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{cross\_product}(y,y),\mbox{union}(x,\mbox{identity\_relation}))\qquad\mbox{cnf}(\mbox{prove\_lemma\_2\_to\_restricted\_domain}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{restrict}(x,y,y))\ne y\qquad\mbox{cnf}(\mbox{prove\_lemma\_2\_to\_restricted\_domain}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(y,y),\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_lemma\_2\_to\_restricted\_domain}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET842-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$( \mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_n},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_n} = \mbox{v\_m} )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_subsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET195-6.p} Lattice upper bound 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(y,\mbox{union}(x,y))\qquad\mbox{cnf}(\mbox{prove\_lattice\_upper\_bound2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET741+4.p} Problem on composition of mappings 6

Consider three mappings F from A to B,G from B to C,H from C to A.
 If HoGoF is injective and FoHoG and GoFoH surjective, then H is
 one-to-one.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,c,a)\mbox{ and }\mbox{injective}(\mbox{compose\_function}(h,\mbox{compose\_function}(g,f,a,b,c),a,c,a),a,a)\mbox{ and }\mbox{surjective}(\mbox{compose\_function}(f,\mbox{compose\_function}(h,g,b,c,a),b,a,b),b,b)\mbox{ and }\mbox{surjective}(\mbox{compose\_function}(g,\mbox{compose\_function}(f,h,c,a,b),c,b,c),c,c) ) \ \Rightarrow \ \mbox{one\_to\_one}(h,c,a) )\qquad\mbox{fof}(\mbox{thII}_{32},\mbox{conjecture})$

\medskip\textbf{SET734+4.p} If GoF is the identity, then G is surjective






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b{:}\;( ( \mbox{maps}(g,a,b)\mbox{ and }\mbox{maps}(f,b,a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(g,f,b,a,b),b) ) \ \Rightarrow \ \mbox{surjective}(g,a,b) )\qquad\mbox{fof}(\mbox{thII}_{25},\mbox{conjecture})$

\medskip\textbf{SET397-6.p} Range of composition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{range\_of}(\mbox{xr}\circ \mbox{yr}),\mbox{range\_of}(\mbox{xr}))\qquad\mbox{cnf}(\mbox{prove\_range\_of\_composition}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET864-2.p} Problem about Zorn's lemma






$( \mbox{c\_in}(\mbox{v\_u},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_c},\mbox{t\_a}),\mbox{v\_u},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_Union}(\mbox{v\_c},\mbox{t\_a}) = \mbox{v\_u}\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_Zorn}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_c},\mbox{t\_a}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_x}(\mbox{v\_U}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{v\_U} = \mbox{v\_x}(\mbox{v\_U}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET452-6.p} Subset relation alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{subset\_relation}\qquad\mbox{cnf}(\mbox{prove\_subset\_relation\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subset\_relation\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET268-6.p} Domain property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{intersection}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})),\mbox{cross\_product}(\mbox{domain\_of}(x),\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_domain\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET062+4.p} The empty set is a subset of all sets






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a{:}\;\mbox{empty\_set}\subseteq a\qquad\mbox{fof}(\mbox{thI}_{15},\mbox{conjecture})$

\medskip\textbf{SET583+3.p} Extensionality






$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( ( b\subseteq c\mbox{ and }c\subseteq b ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{prove\_extensionality},\mbox{conjecture})$

\medskip\textbf{SET488-6.p} Replacement property 10






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,y)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property10}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property10}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET011-1.p} X \ (X \ Y) = X $\wedge$ Y

The difference of a first set and the set which is the
 difference of the first set and a second set, is the
 intersection of the two sets.




$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-3.ax'})$

$a\setminus b{=}\mbox{aDb}\qquad\mbox{cnf}(\mbox{a\_minus\_b},\mbox{hypothesis})$

$a\setminus \mbox{aDb}{=}\mbox{aD\_aDb}\qquad\mbox{cnf}(\mbox{a\_minus\_aDb},\mbox{hypothesis})$

$\neg\,\mbox{intersection}(a,b,\mbox{aD\_aDb})\qquad\mbox{cnf}(\mbox{prove\_a\_intersection\_b\_is\_aD\_aDb},\mbox{negated\_conjecture})$

\medskip\textbf{SET168-6.p} Members of union 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in z\qquad\mbox{cnf}(\mbox{prove\_members\_of\_union3}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{union}(y,z)\qquad\mbox{cnf}(\mbox{prove\_members\_of\_union3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET274-6.p} Domain property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(\mbox{restrict}(x,y,\mbox{universal\_class}))\ne \mbox{intersection}(\mbox{domain\_of}(x),y)\qquad\mbox{cnf}(\mbox{prove\_domain\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET013-1.p} The intersection of sets is commutative






$\mbox{include}(\mbox{'Axioms/SET002-0.ax'})$

$\neg\,\mbox{equal\_sets}(\mbox{intersection}(\mbox{as},\mbox{bs}),\mbox{intersection}(\mbox{bs},\mbox{as}))\qquad\mbox{cnf}(\mbox{prove\_commutativity},\mbox{negated\_conjecture})$

\medskip\textbf{SET161-6.p} Commutativity outside union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,\mbox{union}(y,z))\ne \mbox{union}(y,\mbox{union}(x,z))\qquad\mbox{cnf}(\mbox{prove\_commutativity\_outside\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET599+3.p} X \ Y (= X sym\ Y

The difference of X and Y is a subset of the symmetric
 difference of X and Y.




$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;b\subseteq \mbox{union}(b,c)\qquad\mbox{fof}(\mbox{subset\_of\_union},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b\setminus c )\subseteq \mbox{symmetric\_difference}(b,c)\qquad\mbox{fof}(\mbox{prove\_th}_{58},\mbox{conjecture})$

\medskip\textbf{SET365-6.p} Monotonicity of power






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_power}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{power\_class}(x),\mbox{power\_class}(y))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_power}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET087-6.p} A singleton set has a member, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_exists2}_1,\mbox{negated\_conjecture})$

$\mbox{singleton}(\mbox{member\_of}(\mbox{singleton}(y)))\ne \mbox{singleton}(y)\qquad\mbox{cnf}(\mbox{prove\_member\_exists2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET775+4.p} Pre-order and equivalence

If P is a pre-order relation,and R defined by R(A,B) if and
 only if P(A,B) and P(B,A), then R is an equivalence relation.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall e,p,r{:}\;( ( \mbox{pre\_order}(p,e)\mbox{ and }\forall a,b{:}\;( ( a\in e\mbox{ and }b\in e ) \ \Rightarrow \ ( \mbox{apply}(r,a,b)\iff ( \mbox{apply}(p,a,b)\mbox{ and }\mbox{apply}(p,b,a) ) ) ) ) \ \Rightarrow \ \mbox{equivalence}(r,e) )\qquad\mbox{fof}(\mbox{thIII}_{11},\mbox{conjecture})$

\medskip\textbf{SET033-4.p} Domain of composition






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{range\_of}(a)\subseteq \mbox{domain\_of}(b)\qquad\mbox{cnf}(\mbox{range\_subset\_of\_domain},\mbox{hypothesis})$

$\mbox{domain\_of}(a)\ne \mbox{domain\_of}(b\circ a)\qquad\mbox{cnf}(\mbox{prove\_domain\_of\_composition},\mbox{negated\_conjecture})$

\medskip\textbf{SET468-6.p} Sets with at most one member 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_one\_member2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,x),\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_one\_member2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET991+1.p} Functions and their basic properties, theorem 12






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

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

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

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

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

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

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

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

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_rng}(a)) )\qquad\mbox{fof}(\mbox{fc6\_relat}_1,\mbox{axiom})$

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

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_rng}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_rng}(a)) ) )\qquad\mbox{fof}(\mbox{fc8\_relat}_1,\mbox{axiom})$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\forall a,b{:}\;( ( \mbox{relation}(b)\mbox{ and }\mbox{function}(b) ) \ \Rightarrow \ ( \mbox{in}(a,\mbox{relation\_dom}(b)) \ \Rightarrow \ \mbox{in}(\mbox{apply}(b,a),\mbox{relation\_rng}(b)) ) )\qquad\mbox{fof}(\mbox{t12\_funct}_1,\mbox{conjecture})$

$\forall a{:}\;( ( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) ) \ \Rightarrow \ \forall b{:}\;( b = \mbox{relation\_rng}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(d,\mbox{relation\_dom}(a))\mbox{ and }c = \mbox{apply}(a,d) ) ) ) )\qquad\mbox{fof}(\mbox{d5\_funct}_1,\mbox{axiom})$

\medskip\textbf{SET556-6.p} Cantor class property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{domain\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_cantor\_class\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{xr},z) = \mbox{cantor}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_cantor\_class\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET368-6.p} Power class of universal class is universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{power\_class}(\mbox{universal\_class})\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_power\_class\_of\_universal\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET130-6.p} Membership in unordered triple, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple1}_1,\mbox{negated\_conjecture})$

$\neg\,u\in \mbox{set\_builder}(u,\mbox{set\_builder}(y,\mbox{set\_builder}(z,\mbox{null\_class})))\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET034-4.p} The composition of functions is a function






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{function}(a)\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$\mbox{function}(b)\qquad\mbox{cnf}(\mbox{b\_function},\mbox{hypothesis})$

$\neg\,\mbox{function}(b\circ a)\qquad\mbox{cnf}(\mbox{prove\_their\_composition\_is\_a\_function},\mbox{negated\_conjecture})$

\medskip\textbf{SET045-5.p} No Universal Set

The restricted comprehension axiom says : given a set
 z, there is a set all of whose members are drawn from z and
 which satisfy some property. If there were a universal set,
 then the Russell set could be formed, using this axiom.
 So given the appropriate instance of this axiom, there
 is no universal set.




$\mbox{element}(x,f(y)) \ \Rightarrow \ \mbox{element}(x,y)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{axiom})$

$\mbox{element}(x,f(y)) \ \Rightarrow \ \neg\,\mbox{element}(x,x)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{axiom})$

$\mbox{element}(x,y) \ \Rightarrow \ ( \mbox{element}(x,x)\mbox{ or }\mbox{element}(x,f(y)) )\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$\mbox{element}(x,a)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET610+3.p} (X U Y) \ Y = X \ Y

The difference of (the union of X and Y) and Y is the
 difference of X and Y.




$\forall b,c{:}\;( \forall d{:}\;( d\in b\iff d\in c ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{member\_equal},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c)\setminus c = b\setminus c\qquad\mbox{fof}(\mbox{prove\_th}_{83},\mbox{conjecture})$

\medskip\textbf{SET097+1.p} A class contains 0, 1 or at least 2 members.






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;( x = \mbox{null\_class}\mbox{ or }\exists y{:}\;\mbox{singleton}(y) = x\mbox{ or }\exists v{:}\;( v\in x\mbox{ and }\exists w{:}\;w\in \mbox{intersection}(\mbox{singleton}(v)',x) ) )\qquad\mbox{fof}(\mbox{number\_of\_elements\_in\_class},\mbox{conjecture})$

\medskip\textbf{SET604+3.p} The difference of the empty set and X is the empty set






$\forall b{:}\;\mbox{empty\_set}\subseteq b\qquad\mbox{fof}(\mbox{empty\_set\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b\setminus c = \mbox{empty\_set}\iff b\subseteq c )\qquad\mbox{fof}(\mbox{difference\_empty\_set},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b{:}\;\mbox{empty\_set}\setminus b = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{prove\_no\_difference\_with\_empty\_set},\mbox{conjecture})$

\medskip\textbf{SET782-1.p} Set theory (Boolean algebra) axioms based on NBG set theory






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

\medskip\textbf{SET879+1.p} difference(singleton(A),singleton(B)) = singleton(A) $\leq$$>$ A != B






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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{singleton}(a)\iff \neg\,\mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l34\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),\mbox{singleton}(b)) = \mbox{singleton}(a)\iff a\ne b )\qquad\mbox{fof}(\mbox{t20\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET875+1.p} ~ ( disjoint(singleton(A),singleton(B)) \& A = B )






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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{disjoint}(\mbox{singleton}(a),b)\mbox{ and }\mbox{in}(a,b)\qquad\mbox{fof}(\mbox{l25\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b{:}\;\neg\,\mbox{disjoint}(\mbox{singleton}(a),\mbox{singleton}(b))\mbox{ and }a = b\qquad\mbox{fof}(\mbox{t16\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET011$\wedge$5.p} TPS problem BOOL-PROP-82

Trybulec's 82nd Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_82\_pme},\mbox{conjecture})$

\medskip\textbf{SET903+1.p} ~ ( sgtn(A) = union(B,C) \& B != C \& B != empty \& C != empty )






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

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

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

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

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

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

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

$\forall a,b,c{:}\;\neg\,\mbox{singleton}(a) = \mbox{set\_union}_2(b,c)\mbox{ and }\neg\,b = \mbox{singleton}(a)\mbox{ and }c = \mbox{singleton}(a)\mbox{ and }\neg\,b = \mbox{empty\_set}\mbox{ and }c = \mbox{singleton}(a)\mbox{ and }\neg\,b = \mbox{singleton}(a)\mbox{ and }c = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t43\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{singleton}(a) = \mbox{set\_union}_2(b,c)\mbox{ and }b\ne c\mbox{ and }b\ne \mbox{empty\_set}\mbox{ and }c\ne \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t44\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET863-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OHausdorff\_}_1(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OHausdorff}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_z},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_1,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_subset\_\_chain}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_y},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_x},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_xa},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_xa},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SET944+1.p} subset(union(intersection(A,B)),intersection(union(A),union(B)))






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

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

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

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

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

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

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

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

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

$\forall a,b{:}\;\mbox{union}(\mbox{set\_intersection}_2(a,b))\subseteq \mbox{set\_intersection}_2(\mbox{union}(a),\mbox{union}(b))\qquad\mbox{fof}(\mbox{t97\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET079+1.p} A set belongs to its singleton






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;( x\in \mbox{universal\_class} \ \Rightarrow \ \mbox{singleton}(x)\ne \mbox{null\_class} )\qquad\mbox{fof}(\mbox{corollary\_to\_set\_in\_its\_singleton},\mbox{conjecture})$

\medskip\textbf{SET819-2.p} Problem about set theory






$\neg\,\mbox{c\_in}(\mbox{v\_a},\mbox{c\_emptyset},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OemptyE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET818-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_U},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_x}(\mbox{v\_U}),c_0,\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET196-6.p} The intersection of X and Y is a subset of X






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{intersection}(x,y),x)\qquad\mbox{cnf}(\mbox{prove\_lattice\_lower\_bound1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET046+1.p} No set of non-circular sets

A set is circular if it is a member of another set which
 in turn is a member of the orginal. Intuitively all sets are
 non-circular. Prove there is no set of non-circular sets.




$\neg\,\exists y{:}\;\forall x{:}\;( \mbox{element}(x,y)\iff \neg\,\exists z{:}\;( \mbox{element}(x,z)\mbox{ and }\mbox{element}(z,x) ) )\qquad\mbox{fof}(\mbox{pel}_{42},\mbox{conjecture})$

\medskip\textbf{SET357-6.p} Sum class alternate definition 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(\mbox{element\_relation}',x)\ne \mbox{sum\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_alternate\_defn4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET551-6.p} Application property 19






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_application\_property16}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_application\_property16}_2,\mbox{negated\_conjecture})$

$\mbox{image}(\mbox{application\_function},\mbox{singleton}(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_application\_property16}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET158-6.p} Corollary to complement axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in x\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_complement\_axiom}_1,\mbox{negated\_conjecture})$

$z\in x'\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_complement\_axiom}_2,\mbox{negated\_conjecture})$

$y = z\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_complement\_axiom}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET153-6.p} Intersection with complement is null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x',x)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_intersection\_with\_complement}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET852-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_1,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_2,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_y},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_U},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_U},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_U} = \mbox{v\_m} )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET825-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_Q}(\mbox{v\_n})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{v\_Q}(\mbox{v\_m})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(c_0,c_0,\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\mbox{ or }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_xa}(\mbox{v\_U}),\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Suc}(\mbox{v\_x}(\mbox{v\_U})),\mbox{c\_Suc}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(c_0,c_0,\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET165-6.p} Corollary to idempotency of union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,\mbox{union}(x,y))\ne \mbox{union}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_idempotency\_of\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET200$\wedge$5.p} TPS problem BOOL-PROP-34

Trybulec's 34th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o},v{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( z@ \mbox{xx} ) \ \Rightarrow \ ( v@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx}\mbox{ or }v@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_34\_pme},\mbox{conjecture})$

\medskip\textbf{SET786+1.p} Peter Andrews Problem THM25






$\neg\,\exists y{:}\;\forall x{:}\;( \mbox{element}(x,y)\iff \neg\,\exists z{:}\;( \mbox{element}(x,z)\mbox{ and }\mbox{element}(z,x) ) )\qquad\mbox{fof}(\mbox{thm}_{25},\mbox{conjecture})$

\medskip\textbf{SET162$\wedge$5.p} TPS problem BOOL-PROP-60

Trybulec's 60th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }\mbox{\$false} ) ) = x\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_60\_pme},\mbox{conjecture})$

\medskip\textbf{SET610$\wedge$5.p} TPS problem BOOL-PROP-83

Trybulec's 83th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} )\mbox{ and }\neg\,y@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_83\_pme},\mbox{conjecture})$

\medskip\textbf{SET092-6.p} Member\_of(X) is unique if X is not a singleton, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{singleton}(\mbox{member\_of}_1(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_member\_of\_non\_singleton\_unique2}_1,\mbox{negated\_conjecture})$

$\mbox{member\_of}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_member\_of\_non\_singleton\_unique2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET231-6.p} Cross product property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property6}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(\mbox{first}(\mbox{not\_subclass\_element}(\mbox{cross\_product}(x,y),z)),\mbox{second}(\mbox{not\_subclass\_element}(\mbox{cross\_product}(x,y),z)))\ne \mbox{not\_subclass\_element}(\mbox{cross\_product}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property6}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET051-6.p} Corollary to Unordered pair axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_unordered\_pair}_1,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_unordered\_pair}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET084+1.p} A singleton set is determined by its element






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( ( \mbox{singleton}(x) = \mbox{singleton}(y)\mbox{ and }y\in \mbox{universal\_class} ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{singleton\_identified\_by\_element}_2,\mbox{conjecture})$

\medskip\textbf{SET102+1.p} Ordered pair member of ordered pair






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;\mbox{unordered\_pair}(x,\mbox{singleton}(y))\in \mbox{ordered\_pair}(x,y)\qquad\mbox{fof}(\mbox{unordered\_pair\_member\_of\_ordered\_pair},\mbox{conjecture})$

\medskip\textbf{SET384-6.p} Corollary 1 to relation property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$( ( x' )' )'\ne x'\qquad\mbox{cnf}(\mbox{prove\_relation\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET189-6.p} Corollary to subclass property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(y',x) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_subclass\_property6}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(y,x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_subclass\_property6}_2,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_subclass\_property6}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET956+1.p} Basic properties of sets, theorem 109

( subset(A,cartesian\_product2(B,C)) \&
 ( in(ordered\_pair(E,F),A) =$>$ in(ordered\_pair(E,F),D) ) )
 =$>$ subset(A,D)




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

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

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

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

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

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

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

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

$\forall a,b,c,d{:}\;\neg\,a\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }\mbox{in}(d,a)\mbox{ and }\forall e,f{:}\;\neg\,\mbox{in}(e,b)\mbox{ and }\mbox{in}(f,c)\mbox{ and }d = \mbox{ordered\_pair}(e,f)\qquad\mbox{fof}(\mbox{t103\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( a\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }\forall e,f{:}\;( \mbox{in}(\mbox{ordered\_pair}(e,f),a) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(e,f),d) ) ) \ \Rightarrow \ a\subseteq d )\qquad\mbox{fof}(\mbox{t109\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET808+4.p} The members of an ordinal number are ordinal numbers






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+4.ax'})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{thI}_3,\mbox{axiom})$

$\forall a{:}\;( a\in \mbox{on} \ \Rightarrow \ a\subseteq \mbox{on} )\qquad\mbox{fof}(\mbox{thV}_1,\mbox{conjecture})$

\medskip\textbf{SET371-6.p} Power class property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(x,\mbox{sum\_class}(\mbox{power\_class}(x)))\qquad\mbox{cnf}(\mbox{prove\_power\_class\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET623$\wedge$3.p} (X sym\ Y) sym\ Z = X sym\ (Y sym\ Z)

The symmetric difference of (the symmetric difference of X and Y)
 and Z is the symmetric difference of X and (the symmetric
 difference of Y and Z).




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},z{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{excl\_union}@ ( \mbox{excl\_union}@ x@ y )@ z ) = ( \mbox{excl\_union}@ x@ ( \mbox{excl\_union}@ y@ z ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET032-4.p} Range of composition






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\neg\,\mbox{range\_of}(a\circ b)\subseteq \mbox{range\_of}(a)\qquad\mbox{cnf}(\mbox{prove\_range\_of\_composition},\mbox{negated\_conjecture})$

\medskip\textbf{SET909+1.p} union(unordered\_pair(A,B),C) != empty\_set






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

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

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

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

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

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

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

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

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

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

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

$\forall a,b,c{:}\;\mbox{set\_union}_2(\mbox{unordered\_pair}(a,b),c)\ne \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t50\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET390-6.p} Composition alternate definition 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{xf}\circ \mbox{yf}\qquad\mbox{cnf}(\mbox{prove\_composition\_alternate\_defn4}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(1_{\text{dom}(\mbox{xf})},v)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_composition\_alternate\_defn4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET077+1.p} Every singleton is a set






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;\mbox{singleton}(x)\in \mbox{universal\_class}\qquad\mbox{fof}(\mbox{singletons\_are\_sets},\mbox{conjecture})$

\medskip\textbf{SET900+1.p} ~ ( A != singleton(B) \& A != empty\_set \& ~ ( in(C,A) \& C != B ) )






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

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

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

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

$\forall a,b{:}\;\neg\,a\ne \mbox{singleton}(b)\mbox{ and }a\ne \mbox{empty\_set}\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }c\ne b\qquad\mbox{fof}(\mbox{t41\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,a\ne \mbox{singleton}(b)\mbox{ and }a\ne \mbox{empty\_set}\mbox{ and }\forall c{:}\;\neg\,\mbox{in}(c,a)\mbox{ and }c\ne b\qquad\mbox{fof}(\mbox{l45\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET858-2.p} Problem about Zorn's lemma






$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_m},\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) = \mbox{v\_m} ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_2,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_3,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_U},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_U} = \mbox{v\_m} )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma2}_0,\mbox{axiom})$

\medskip\textbf{SET613$\wedge$5.p} TPS problem BOOL-PROP-87

Trybulec's 87th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} )\mbox{ and }\neg\,x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} )\mbox{ or }( y@ \mbox{xz}\mbox{ and }\neg\,x@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_87\_pme},\mbox{conjecture})$

\medskip\textbf{SET548-6.p} Application property 16






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_application\_property13}_1,\mbox{negated\_conjecture})$

$u\in \mbox{domain\_of}(\mbox{intersection}(y,x))\qquad\mbox{cnf}(\mbox{prove\_application\_property13}_2,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{intersection}(y,x),u)\ne \mbox{apply}(x,u)\qquad\mbox{cnf}(\mbox{prove\_application\_property13}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET624$\wedge$3.p} X intersects Y U Z iff X intersects Y or X intersects Z

X intersects the union of Y and Z iff X intersects Y or X
 intersects Z.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},z{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{meets}@ x@ ( \mbox{union}@ y@ z ) )\iff ( \mbox{meets}@ x@ y\mbox{ or }\mbox{meets}@ x@ z ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET290-6.p} Proof of Goedel's axiom B6, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(u,v)\in x'\qquad\mbox{cnf}(\mbox{prove\_inverse\_property2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(v,u)\in x\qquad\mbox{cnf}(\mbox{prove\_inverse\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET383-6.p} Corollary 2 to relation property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{restrict}(x,\mbox{universal\_class},\mbox{universal\_class}),\mbox{cross\_product}(\mbox{domain\_of}(x),\mbox{range\_of}(x)))\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_relation\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET733+4.p} If GoF is the identity, then F is injective






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(g,f,a,b,a),a) ) \ \Rightarrow \ \mbox{injective}(f,a,b) )\qquad\mbox{fof}(\mbox{thII}_{24},\mbox{conjecture})$

\medskip\textbf{SET118-6.p} Corollary 2 to every ordered pair being a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$x\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_ordered\_pairs\_are\_sets}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_ordered\_pairs\_are\_sets}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET828-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET185-6.p} If X is a subset of  Y, then the union of X and Y is Y






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{union}(x,y)\ne y\qquad\mbox{cnf}(\mbox{prove\_subclass\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET056-6.p} Expanded equality definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_equality1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(x,y)\in x\qquad\mbox{cnf}(\mbox{prove\_equality1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{prove\_equality1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET850-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_y},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET567-6.p} Compatible function special case






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{operation}(\mbox{null\_class})\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_special\_case}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET336-6.p} Corollary 1 image property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(z,\mbox{null\_class})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_image\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET779-1.p} Set theory membership and intersection axioms






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-2.ax'})$

\medskip\textbf{SET095+4.p} If X is in Y, then the singleton containing X is a subset of Y






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,x{:}\;( x\in a \ \Rightarrow \ \mbox{singleton}(x)\subseteq a )\qquad\mbox{fof}(\mbox{thI}_{44},\mbox{conjecture})$

\medskip\textbf{SET886+1.p} subset(uno\_pair(A,B),singleton(C)) =$>$ uno\_pair(A,B) = singleton(C)






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

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

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

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

$\forall a,b,c{:}\;( \mbox{unordered\_pair}(a,b)\subseteq \mbox{singleton}(c) \ \Rightarrow \ a = c )\qquad\mbox{fof}(\mbox{t26\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{unordered\_pair}(a,b)\subseteq \mbox{singleton}(c) \ \Rightarrow \ \mbox{unordered\_pair}(a,b) = \mbox{singleton}(c) )\qquad\mbox{fof}(\mbox{t27\_zfmisc}_1,\mbox{conjecture})$

$\forall a{:}\;\mbox{unordered\_pair}(a,a) = \mbox{singleton}(a)\qquad\mbox{fof}(\mbox{t69\_enumset}_1,\mbox{axiom})$

\medskip\textbf{SET807+4.p} Inclusion of sets defines a pre-order relation






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall x,y{:}\;( \mbox{apply}(\mbox{subset\_predicate},x,y)\iff x\subseteq y )\qquad\mbox{fof}(\mbox{rel\_subset},\mbox{hypothesis})$

$\forall e{:}\;\mbox{pre\_order}(\mbox{subset\_predicate},\mbox{power\_set}(e))\qquad\mbox{fof}(\mbox{thIV18a},\mbox{conjecture})$

\medskip\textbf{SET144$\wedge$5.p} TPS problem BOOL-PROP-44

Trybulec's 44th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) ) \ \Rightarrow \ ( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }( y@ \mbox{xz}\mbox{ and }z@ \mbox{xz} ) ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} )\mbox{ and }z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_44\_pme},\mbox{conjecture})$

\medskip\textbf{SET945+1.p} ( in(C,A) =$>$ disjoint(C,B) ) =$>$ disjoint(union(A),B)






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

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

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

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

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

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

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

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

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

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{disjoint}(c,b) ) \ \Rightarrow \ \mbox{disjoint}(\mbox{union}(a),b) )\qquad\mbox{fof}(\mbox{t98\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET019+4.p} Two sets that contain one another are equal






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;( ( a\subseteq b\mbox{ and }b\subseteq a ) \ \Rightarrow \ \mbox{equal\_set}(a,b) )\qquad\mbox{fof}(\mbox{thI}_{02},\mbox{conjecture})$

\medskip\textbf{SET840-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_insert}(\mbox{v\_U},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_b})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_b})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_b})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_S},\mbox{t\_b}),\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_b}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET126-6.p} Relation to singleton






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$\mbox{set\_builder}(x,\mbox{null\_class})\ne \mbox{singleton}(x)\qquad\mbox{cnf}(\mbox{prove\_set\_builder\_and\_singleton}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET648$\wedge$3.p} Range of R (X to Y) a subset of Y =$>$ R is (domain of R to Y)

If the range of a relation R from X to Y is a subset of Y, R is
 a relation from the domain of a relation R from X to Y and Y.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \subseteq @ ( \mbox{rel\_codomain}@ r )@ y ) \ \Rightarrow \ ( \mbox{sub\_rel}@ r@ ( \mbox{cartesian\_product}@ ( \mbox{rel\_domain}@ r )@ y ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET258-6.p} Domain alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{domain\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_domain\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(x,\mbox{range}(\mbox{xr},x,\mbox{universal\_class}))\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_domain\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET266-6.p} Domain is monotonic 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{domain\_of}(\mbox{intersection}(x,y)),\mbox{domain\_of}(y))\qquad\mbox{cnf}(\mbox{prove\_domain\_is\_monotonic3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET017$\wedge$1.p} Left cancellation for unordered pairs






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i},z{:}\;\mbox{\$i}{:}\;( ( \mbox{unord\_pair}@ x@ y ) = ( \mbox{unord\_pair}@ x@ z ) \ \Rightarrow \ y = z )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET620$\wedge$5.p} TPS problem BOOL-PROP-96

Trybulec's 96th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} )\mbox{ or }( y@ \mbox{xz}\mbox{ and }\neg\,x@ \mbox{xz} ) ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} )\mbox{ and }\neg\,x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_96\_pme},\mbox{conjecture})$

\medskip\textbf{SET063+1.p} If X is a subset of the empty set, then X is the empty set






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;( \mbox{subclass}(x,\mbox{null\_class}) \ \Rightarrow \ x = \mbox{null\_class} )\qquad\mbox{fof}(\mbox{corollary\_of\_null\_class\_is\_subclass},\mbox{conjecture})$

\medskip\textbf{SET101-6.p} Singleton of the first is a member of an ordered pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{singleton}(x)\in \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_singleton\_member\_of\_ordered\_pair}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET329-6.p} Image alternate definition 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_image\_alternate\_defn4}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_image\_alternate\_defn4}_2,\mbox{negated\_conjecture})$

$x\in z\qquad\mbox{cnf}(\mbox{prove\_image\_alternate\_defn4}_3,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{image}(\mbox{xr},z)\qquad\mbox{cnf}(\mbox{prove\_image\_alternate\_defn4}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET101+1.p} Singleton of the first is a member of an ordered pair






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;\mbox{singleton}(x)\in \mbox{ordered\_pair}(x,y)\qquad\mbox{fof}(\mbox{singleton\_member\_of\_ordered\_pair},\mbox{conjecture})$

\medskip\textbf{SET849-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_OsuccI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_chain\_\_lemma4}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{v\_m} = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oequal\_\_succ\_\_Union}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oequal\_\_succ\_\_Union}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_minus}(\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_c},\mbox{t\_a})\ne \mbox{v\_c}\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_not\_\_equals}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\ne \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET020-4.p} 1st is unique when x is an ordered pair of sets






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{b\_little\_set},\mbox{hypothesis})$

$\mbox{first}(\mbox{ordered\_pair}(a,b))\ne a\qquad\mbox{cnf}(\mbox{prove\_first\_is\_first},\mbox{negated\_conjecture})$

\medskip\textbf{SET326-6.p} Corollary to image alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{image}(\mbox{xr},x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_image\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,1_{\text{dom}(\mbox{xr})}\in \mbox{domain\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_image\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET709+4.p} The composition of mappings is a mapping






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c) ) \ \Rightarrow \ \mbox{maps}(\mbox{compose\_function}(g,f,a,b,c),a,c) )\qquad\mbox{fof}(\mbox{thII}_{01},\mbox{conjecture})$

\medskip\textbf{SET859-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) = \mbox{v\_m} ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_2,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_3,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_U},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_U} = \mbox{v\_m} )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma2}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_n},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_m},\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET481-6.p} Replacement property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_replacement\_property3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{cross\_product}(x,y)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET191-6.p} Subclass property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x',y) = \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_subclass\_property8}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_property8}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET625+3.p} If X intersects Y and Y is a subset of Z, then X intersects Z






$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( ( \mbox{intersect}(b,c)\mbox{ and }c\subseteq d ) \ \Rightarrow \ \mbox{intersect}(b,d) )\qquad\mbox{fof}(\mbox{prove\_th}_{101},\mbox{conjecture})$

\medskip\textbf{SET204-6.p} Cross product property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(v,u)\in \mbox{cross\_product}(y,x)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET590+3.p} The difference of X and Y is a subset of X






$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b\setminus c )\subseteq b\qquad\mbox{fof}(\mbox{prove\_th}_{49},\mbox{conjecture})$

\medskip\textbf{SET688+4.p} Property of proper subset

If A is a proper subset of B and B a proper subset of C, then
 A is not equal to C.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }\neg\,\mbox{equal\_set}(a,b)\mbox{ and }b\subseteq c\mbox{ and }\neg\,\mbox{equal\_set}(b,c) ) \ \Rightarrow \ \neg\,\mbox{equal\_set}(a,c) )\qquad\mbox{fof}(\mbox{thI}_{04},\mbox{conjecture})$

\medskip\textbf{SET025-9.p} Ordered pairs are little sets






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{ordered\_pair\_predicate}(a)\qquad\mbox{cnf}(\mbox{an\_ordered\_pair\_predicate},\mbox{hypothesis})$

$\neg\,\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{prove\_predicate\_is\_small},\mbox{negated\_conjecture})$

\medskip\textbf{SET839-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_insert}(\mbox{v\_U},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_V},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_V},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET056+1.p} Expanded equality definition






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( x = y\mbox{ or }\exists u{:}\;( u\in x\mbox{ and }\neg\,u\in y )\mbox{ or }\exists w{:}\;( w\in y\mbox{ and }\neg\,w\in x ) )\qquad\mbox{fof}(\mbox{equality}_1,\mbox{conjecture})$

\medskip\textbf{SET632+3.p} If X (= Y \& X (= Z \& Y disjoint from Z, then X = empty set

If X is a subset of Y and X is a subset of Z and Y is disjoint
 from Z, then X is the empty set.




$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{disjoint}(b,c)\iff \neg\,\mbox{intersect}(b,c) )\qquad\mbox{fof}(\mbox{disjoint\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }b\subseteq d\mbox{ and }\mbox{disjoint}(c,d) ) \ \Rightarrow \ b = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{prove\_th}_{114},\mbox{conjecture})$

\medskip\textbf{SET473-6.p} Lemma 1 to restricted domain






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_lemma\_1\_to\_restricted\_domain}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{restrict}(x,y,y))\ne y\qquad\mbox{cnf}(\mbox{prove\_lemma\_1\_to\_restricted\_domain}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET247-6.p} Restriction with null class 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{xr},x,\mbox{null\_class})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_restriction\_with\_null\_class3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET541-6.p} Application property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{apply}(z,x)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_application\_property7}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{domain\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_application\_property7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET524-6.p} Equivalent condition 1 for x not to be an ordered pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{first}(x) = x\qquad\mbox{cnf}(\mbox{prove\_condition\_for\_x\_not\_to\_be\_an\_ordered\_pair1}_1,\mbox{negated\_conjecture})$

$\mbox{second}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_condition\_for\_x\_not\_to\_be\_an\_ordered\_pair1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET696+4.p} If A (= E, then (E / A) $\wedge$ A = empty set






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,e{:}\;( a\subseteq e \ \Rightarrow \ \mbox{equal\_set}(\mbox{intersection}(e\setminus a,a),\mbox{empty\_set}) )\qquad\mbox{fof}(\mbox{thI}_{28},\mbox{conjecture})$

\medskip\textbf{SET635$\wedge$5.p} TPS problem BOOL-PROP-117

Trybulec's 117th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_117\_pme},\mbox{conjecture})$

\medskip\textbf{SET071-6.p} Null unordered pair

If both arguments of an unordered pair are proper classes,
 the pair is null.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{unordered\_pair}(x,y)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_null\_unordered\_pair}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_null\_unordered\_pair}_2,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_null\_unordered\_pair}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET060+1.p} Nothing in the intersection of a set and its complement






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{fof}(\mbox{special\_classes\_lemma},\mbox{conjecture})$

\medskip\textbf{SET490-6.p} Replacement property 12






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(x,y)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property12}_1,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property12}_2,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property12}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET611+3.p} X $\wedge$ Y = the empty set iff X \ Y = X

The intersection of X and Y is the empty set iff the difference
 of X and Y is X.




$\forall b,c{:}\;( \forall d{:}\;( d\in b\iff d\in c ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{member\_equal},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersection}(b,c) = \mbox{empty\_set}\iff b\setminus c = b )\qquad\mbox{fof}(\mbox{prove\_th}_{84},\mbox{conjecture})$

\medskip\textbf{SET972+1.p} Basic properties of sets, theorem 125






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

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

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d)) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(b,a),\mbox{cartesian\_product}_2(d,c)) )\qquad\mbox{fof}(\mbox{t107\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d,e,f{:}\;( ( a\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }d\subseteq \mbox{cartesian\_product}_2(e,f)\mbox{ and }\forall g,h{:}\;( \mbox{in}(\mbox{ordered\_pair}(g,h),a)\iff \mbox{in}(\mbox{ordered\_pair}(g,h),d) ) ) \ \Rightarrow \ a = d )\qquad\mbox{fof}(\mbox{t110\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{cartesian\_product}_2(\mbox{set\_difference}(a,b),c) = \mbox{set\_difference}(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,c))\mbox{ and }\mbox{cartesian\_product}_2(c,\mbox{set\_difference}(a,b)) = \mbox{set\_difference}(\mbox{cartesian\_product}_2(c,a),\mbox{cartesian\_product}_2(c,b)) )\qquad\mbox{fof}(\mbox{t125\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;\mbox{set\_difference}(a,b)\subseteq a\qquad\mbox{fof}(\mbox{t36\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SET335-6.p} Image property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x,\mbox{domain\_of}(z)) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_image\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{image}(z,x)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_image\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET613+3.p} (X U Y) \ X $\wedge$ Y = (X \ Y) U (Y \ X)

The difference of (the union of X and Y) and the intersection
 of X and Y is the union of (the difference of X and Y) and (the
 difference of Y and X).




$\forall b,c{:}\;( \forall d{:}\;( d\in b\iff d\in c ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{member\_equal},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c)\setminus \mbox{intersection}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{prove\_difference\_union\_intersection},\mbox{conjecture})$

\medskip\textbf{SET103-6.p} Special member 1 of an ordered pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{unordered\_pair}(\mbox{singleton}(x),\mbox{unordered\_pair}(x,\mbox{null\_class}))\ne \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_property\_1\_of\_ordered\_pair}_1,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_property\_1\_of\_ordered\_pair}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET226-6.p} Cross product left cancellation 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(u,v) = \mbox{cross\_product}(w,x)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_left\_cancellation1}_1,\mbox{negated\_conjecture})$

$u\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_left\_cancellation1}_2,\mbox{negated\_conjecture})$

$v\ne x\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_left\_cancellation1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET096$\wedge$1.p} There are at most two subsets of a singleton set






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}{:}\;( ( \subseteq @ x@ ( \mbox{singleton}@ y ) ) \ \Rightarrow \ ( x = \mbox{emptyset}\mbox{ or }x = ( \mbox{singleton}@ y ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET628+3.p} X intersects X iff X is not the empty set






$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{not\_equal}(b,c)\iff b\ne c )\qquad\mbox{fof}(\mbox{not\_equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b{:}\;( \mbox{intersect}(b,b)\iff \mbox{not\_equal}(b,\mbox{empty\_set}) )\qquad\mbox{fof}(\mbox{prove\_th}_{110},\mbox{conjecture})$

\medskip\textbf{SET016+4.p} First components of equal ordered pairs are equal

If {{A},{A,B}} = {{U},{U,V}} then A = U.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,u,v{:}\;( \mbox{equal\_set}(\mbox{unordered\_pair}(\mbox{singleton}(a),\mbox{unordered\_pair}(a,b)),\mbox{unordered\_pair}(\mbox{singleton}(u),\mbox{unordered\_pair}(u,v))) \ \Rightarrow \ a = u )\qquad\mbox{fof}(\mbox{thI50a},\mbox{conjecture})$

\medskip\textbf{SET009-1.p} If X is a subset of Y, then Z \ Y is a subset of Z \ X






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-3.ax'})$

$d\subseteq a\qquad\mbox{cnf}(\mbox{d\_is\_a\_subset\_of\_a},\mbox{hypothesis})$

$b\setminus a{=}\mbox{bDa}\qquad\mbox{cnf}(\mbox{b\_minus\_a},\mbox{hypothesis})$

$b\setminus d{=}\mbox{bDd}\qquad\mbox{cnf}(\mbox{b\_minus\_d},\mbox{hypothesis})$

$\neg\,\mbox{bDa}\subseteq \mbox{bDd}\qquad\mbox{cnf}(\mbox{prove\_bDa\_is\_a\_subset\_of\_bDd},\mbox{negated\_conjecture})$

\medskip\textbf{SET626$\wedge$5.p} TPS problem BOOL-PROP-102

Trybulec's 102th Boolean property of sets




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

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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx}\mbox{ and }\mbox{cZ}@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_102\_pme},\mbox{conjecture})$

\medskip\textbf{SET630$\wedge$5.p} TPS problem BOOL-PROP-112

Trybulec's 112th Boolean property of sets.




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;\neg\,\exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx}\mbox{ and }( ( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} )\mbox{ or }( y@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_112\_pme},\mbox{conjecture})$

\medskip\textbf{SET553-6.p} Cantor class alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cantor}(x),\mbox{domain\_of}(x))\qquad\mbox{cnf}(\mbox{prove\_cantor\_class\_alternate\_defn}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET603+3.p} The difference of X and the empty set is X






$\forall b,c{:}\;( \forall d{:}\;( d\in b\iff d\in c ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{member\_equal},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b{:}\;b\setminus \mbox{empty\_set} = b\qquad\mbox{fof}(\mbox{prove\_th}_{74},\mbox{conjecture})$

\medskip\textbf{SET099-6.p} Corollary 2 to a class contains 0, 1, or at least 2 members






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{not\_subclass\_element}(\mbox{intersection}(\mbox{singleton}(\mbox{not\_subclass\_element}(x,\mbox{null\_class}))',x),\mbox{null\_class}) = \mbox{not\_subclass\_element}(x,\mbox{null\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_number\_of\_elements\_in\_class}_1,\mbox{negated\_conjecture})$

$\mbox{singleton}(\mbox{not\_subclass\_element}(x,\mbox{null\_class}))\ne x\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_number\_of\_elements\_in\_class}_2,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_number\_of\_elements\_in\_class}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET764+4.p} The inverse image of empty set is empty






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b{:}\;( \mbox{maps}(f,a,b) \ \Rightarrow \ \mbox{equal\_set}(\mbox{inverse\_image}_2(f,\mbox{empty\_set}),\mbox{empty\_set}) )\qquad\mbox{fof}(\mbox{thIIa}_{14},\mbox{conjecture})$

\medskip\textbf{SET895+1.p} Basic properties of sets, theorem 36

cartesian\_product2(singleton(A),unordered\_pair(B,C)) =
 unordered\_pair(ordered\_pair(A,B),ordered\_pair(A,C)) \&
 cartesian\_product2(unordered\_pair(A,B),singleton(C)) =
 unordered\_pair(ordered\_pair(A,C),ordered\_pair(B,C))




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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{cartesian\_product}_2(\mbox{singleton}(a),\mbox{unordered\_pair}(b,c)) = \mbox{unordered\_pair}(\mbox{ordered\_pair}(a,b),\mbox{ordered\_pair}(a,c))\mbox{ and }\mbox{cartesian\_product}_2(\mbox{unordered\_pair}(a,b),\mbox{singleton}(c)) = \mbox{unordered\_pair}(\mbox{ordered\_pair}(a,c),\mbox{ordered\_pair}(b,c)) )\qquad\mbox{fof}(\mbox{t36\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET221-6.p} Cross product distributes over intersection 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{cross\_product}(x,y),\mbox{cross\_product}(x,z))\ne \mbox{cross\_product}(x,\mbox{intersection}(y,z))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_over\_intersection2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET845-2.p} Problem about Zorn's lemma






$\mbox{c\_in}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_upper}_0,\mbox{axiom})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_OsuccI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_m}\ne \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET334-6.p} Monotonicity of image 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{xr},\mbox{yr})\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_image2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{xr},z),\mbox{image}(\mbox{yr},z))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_image2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET978+1.p} Basic properties of sets, theorem 131






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

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

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

$\forall a,b,c,d{:}\;( ( \mbox{disjoint}(a,b)\mbox{ or }\mbox{disjoint}(c,d) ) \ \Rightarrow \ \mbox{disjoint}(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,d)) )\qquad\mbox{fof}(\mbox{t127\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( a\ne b \ \Rightarrow \ ( \mbox{disjoint}(\mbox{cartesian\_product}_2(\mbox{singleton}(a),c),\mbox{cartesian\_product}_2(\mbox{singleton}(b),d))\mbox{ and }\mbox{disjoint}(\mbox{cartesian\_product}_2(c,\mbox{singleton}(a)),\mbox{cartesian\_product}_2(d,\mbox{singleton}(b))) ) )\qquad\mbox{fof}(\mbox{t131\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( a\ne b \ \Rightarrow \ \mbox{disjoint}(\mbox{singleton}(a),\mbox{singleton}(b)) )\qquad\mbox{fof}(\mbox{t17\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET214-6.p} Corollary 5 to cross product product monotonicity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(\mbox{intersection}(x,y),z),\mbox{cross\_product}(x,z))\qquad\mbox{cnf}(\mbox{prove\_corollary\_5\_to\_X\_product\_monotonicity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET865-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OHausdorff\_}_1(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OHausdorff}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_u},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_c},\mbox{t\_a}),\mbox{v\_u},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_Union}(\mbox{v\_c},\mbox{t\_a}) = \mbox{v\_u}\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_Zorn}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_subset\_\_chain}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_U},\mbox{t\_a}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_x}(\mbox{v\_U}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_U} = \mbox{v\_x}(\mbox{v\_U}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET847-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_OsuccI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_chain\_\_lemma4}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{v\_m} = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oequal\_\_succ\_\_Union}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oequal\_\_succ\_\_Union}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_minus}(\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_c},\mbox{t\_a})\ne \mbox{v\_c}\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_not\_\_equals}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

\medskip\textbf{SET564-6.p} Corollary 1 to compatible function property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{compatible}(\mbox{xh}_1,\mbox{xf}_1,\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_compatible\_function\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{compatible}(\mbox{xh}_2,\mbox{xf}_2,\mbox{xf}_3)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_compatible\_function\_property3}_2,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{xh}_2\circ \mbox{xh}_1)\ne \mbox{domain\_of}(\mbox{xh}_1)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_compatible\_function\_property3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET061+1.p} Existence of a null class






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\exists x{:}\;\forall z{:}\;\neg\,z\in x\qquad\mbox{fof}(\mbox{existence\_of\_null\_class},\mbox{conjecture})$

\medskip\textbf{SET491-6.p} Diagonalization lemma 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_lemma1}_0,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,x)\in y\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_lemma1}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{domain\_of}(\mbox{intersection}(y,\mbox{identity\_relation}))\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_lemma1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET839-2.p} Problem about set theory






$\neg\,\mbox{c\_lessequals}(\mbox{v\_S},\mbox{c\_insert}(\mbox{v\_U},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_V},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_V},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

\medskip\textbf{SET173-6.p} Absorbtion for intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x,\mbox{union}(x,y))\ne x\qquad\mbox{cnf}(\mbox{prove\_absorbtion\_for\_intersection}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET850-2.p} Problem about Zorn's lemma






$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_y},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET631$\wedge$5.p} TPS problem BOOL-PROP-113

Trybulec's 113th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) \ \Rightarrow \ \exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_113\_pme},\mbox{conjecture})$

\medskip\textbf{SET792+4.p} The least element, if it existes, is minimal






$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e,m{:}\;( ( \mbox{order}(r,e)\mbox{ and }\mbox{least}(m,r,e) ) \ \Rightarrow \ \mbox{min}(m,r,e) )\qquad\mbox{fof}(\mbox{thIV}_4,\mbox{conjecture})$

\medskip\textbf{SET314-6.p} Range property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{xr},x,\mbox{intersection}(y,\mbox{range\_of}(\mbox{xr})))\ne \mbox{restrict}(\mbox{xr},x,y)\qquad\mbox{cnf}(\mbox{prove\_range\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET418-6.p} Composition property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{function}(\mbox{image}(\mbox{composition\_function},\mbox{singleton}(x)))\qquad\mbox{cnf}(\mbox{prove\_composition\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET012-2.p} Complement is an involution






$\mbox{include}(\mbox{'Axioms/SET002-0.ax'})$

$\mbox{equal\_sets}(a',b)\qquad\mbox{cnf}(\mbox{complement\_of\_a\_is\_b},\mbox{hypothesis})$

$\mbox{equal\_sets}(b',c)\qquad\mbox{cnf}(\mbox{complement\_of\_b\_is\_c},\mbox{hypothesis})$

$\neg\,\mbox{equal\_sets}(a,c)\qquad\mbox{cnf}(\mbox{prove\_a\_equals\_c},\mbox{negated\_conjecture})$

\medskip\textbf{SET580$\wedge$3.p} x is in X sym Y iff x is in X iff x is not in Y

x is in the symmetric difference of X and Y iff it is not the
 case x is in X iff x is in Y.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},u{:}\;\mbox{\$i}{:}\;( ( \mbox{excl\_union}@ x@ y@ u )\iff ( ( x@ u )\iff \neg\,y@ u ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET080-6.p} Corollary to a set belongs to its singleton






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{null\_class}\in \mbox{singleton}(\mbox{null\_class})\qquad\mbox{cnf}(\mbox{prove\_null\_class\_in\_its\_singleton}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET525-6.p} Equivalent condition 2 for x not to be an ordered pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{second}(x) = x\qquad\mbox{cnf}(\mbox{prove\_condition\_for\_x\_not\_to\_be\_an\_ordered\_pair2}_1,\mbox{negated\_conjecture})$

$\mbox{first}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_condition\_for\_x\_not\_to\_be\_an\_ordered\_pair2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET023-4.p} The second component of an ordered pair is a little set






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{ordered\_pair\_predicate}(a)\qquad\mbox{cnf}(\mbox{an\_ordered\_pair\_predicate},\mbox{hypothesis})$

$\neg\,\mbox{little\_set}(\mbox{second}(a))\qquad\mbox{cnf}(\mbox{prove\_second\_component\_is\_small},\mbox{negated\_conjecture})$

\medskip\textbf{SET074+1.p} Corollary to unordered pair axiom






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( y\in \mbox{universal\_class} \ \Rightarrow \ \mbox{unordered\_pair}(x,y)\ne \mbox{null\_class} )\qquad\mbox{fof}(\mbox{corollary1}_2,\mbox{conjecture})$

\medskip\textbf{SET241-6.p} Restriction alternate definition 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{restrict}(\mbox{xr},x,y)\qquad\mbox{cnf}(\mbox{prove\_restriction\_alternate\_defn4}_1,\mbox{negated\_conjecture})$

$\neg\,v\in y\qquad\mbox{cnf}(\mbox{prove\_restriction\_alternate\_defn4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET557$\wedge$1.p} Cantor's theorem






$\neg\,\exists g{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall f{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\exists x{:}\;\mbox{\$i}{:}\;( g@ x ) = f\qquad\mbox{thf}(\mbox{surjectiveCantorThm},\mbox{conjecture})$

\medskip\textbf{SET352+4.p} The sum of an unordered pair is the union of the pair






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;\mbox{equal\_set}(\mbox{\mbox{sum}}(\mbox{unordered\_pair}(a,b)),\mbox{union}(a,b))\qquad\mbox{fof}(\mbox{thI}_{40},\mbox{conjecture})$

\medskip\textbf{SET369-6.p} Power class of set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_power\_class\_of\_set}_1,\mbox{negated\_conjecture})$

$\mbox{image}(\mbox{subset\_relation}',\mbox{singleton}(x))\ne \mbox{power\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_of\_set}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET018+4.p} Second components of equal ordered pairs are equal

If {{A},{A,B}} = {{U},{U,V}} then B = V.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,u,v{:}\;( \mbox{equal\_set}(\mbox{unordered\_pair}(\mbox{singleton}(a),\mbox{unordered\_pair}(a,b)),\mbox{unordered\_pair}(\mbox{singleton}(u),\mbox{unordered\_pair}(u,v))) \ \Rightarrow \ b = v )\qquad\mbox{fof}(\mbox{thI50b},\mbox{conjecture})$

\medskip\textbf{SET580$\wedge$5.p} TPS problem BOOL-PROP-23

Trybulec's 23rd Boolean property of sets




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

$\forall \mbox{xx}{:}\;a,x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( ( ( ( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} )\mbox{ or }( y@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx} ) )\iff ( x@ \mbox{xx} ) )\iff \neg\,y@ \mbox{xx} )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_23\_pme},\mbox{conjecture})$

\medskip\textbf{SET865-2.p} Problem about Zorn's lemma






$\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_U},\mbox{t\_a}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_x}(\mbox{v\_U}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_U} = \mbox{v\_x}(\mbox{v\_U}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OHausdorff\_}_1(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OHausdorff}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_u},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_c},\mbox{t\_a}),\mbox{v\_u},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_Union}(\mbox{v\_c},\mbox{t\_a}) = \mbox{v\_u}\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_Zorn}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_subset\_\_chain}_0,\mbox{axiom})$

\medskip\textbf{SET539-6.p} Application property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{apply}(\mbox{xf},z)\qquad\mbox{cnf}(\mbox{prove\_application\_property5}_1,\mbox{negated\_conjecture})$

$z\in \mbox{diagonalise}(\mbox{element\_relation}'\circ \mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_application\_property5}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET776+4.p} Property of pre-order

If P is a pre-order relation,and R defined by R(A,B) iff P(A,B)
 and P(B,A), then R(X1,Y1) and R(X2,Y2) and P(X1,X2) implies
 P(Y1,Y2).




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall e,p,r{:}\;( ( \mbox{pre\_order}(p,e)\mbox{ and }\forall a,b{:}\;( ( a\in e\mbox{ and }b\in e ) \ \Rightarrow \ ( \mbox{apply}(r,a,b)\iff ( \mbox{apply}(p,a,b)\mbox{ and }\mbox{apply}(p,b,a) ) ) ) ) \ \Rightarrow \ \forall x_1,x_2,y_1,y_2{:}\;( ( x_1\in e\mbox{ and }x_2\in e\mbox{ and }y_1\in e\mbox{ and }y_2\in e ) \ \Rightarrow \ ( ( \mbox{apply}(r,x_1,y_1)\mbox{ and }\mbox{apply}(r,x_2,y_2)\mbox{ and }\mbox{apply}(p,x_1,x_2) ) \ \Rightarrow \ \mbox{apply}(p,y_1,y_2) ) ) )\qquad\mbox{fof}(\mbox{thIII}_{12},\mbox{conjecture})$

\medskip\textbf{SET163-6.p} Union with universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{universal\_class},x)\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_union\_with\_universal\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET148-6.p} Idempotency of intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x,x)\ne x\qquad\mbox{cnf}(\mbox{prove\_idempotency\_of\_intersection}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET602+3.p} The difference of X and X is the empty set






$\forall b,c{:}\;( b\setminus c = \mbox{empty\_set}\iff b\subseteq c )\qquad\mbox{fof}(\mbox{difference\_empty\_set},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b{:}\;b\setminus b = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{prove\_self\_difference\_is\_empty\_set},\mbox{conjecture})$

\medskip\textbf{SET364-6.p} Power class alternative definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(z,x)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_alternative\_defn2}_1,\mbox{negated\_conjecture})$

$z\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_power\_class\_alternative\_defn2}_2,\mbox{negated\_conjecture})$

$\neg\,z\in \mbox{power\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_alternative\_defn2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET016-1.p} First components of equal ordered pairs are equal






$x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{singleton}_1,\mbox{axiom})$

$x\in \mbox{singleton\_set}(y) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{singleton}_2,\mbox{axiom})$

$x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{unordered\_pair}_1,\mbox{axiom})$

$y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{unordered\_pair}_2,\mbox{axiom})$

$x\in \mbox{unordered\_pair}(y,z) \ \Rightarrow \ ( x = y\mbox{ or }x = z )\qquad\mbox{cnf}(\mbox{unordered\_pair}_3,\mbox{axiom})$

$\mbox{ordered\_pair}(x,y) = \mbox{unordered\_pair}(\mbox{singleton\_set}(x),\mbox{unordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(m_1,r_1) = \mbox{ordered\_pair}(m_2,r_2)\qquad\mbox{cnf}(\mbox{equal\_ordered\_pairs},\mbox{hypothesis})$

$m_1\ne m_2\qquad\mbox{cnf}(\mbox{prove\_first\_components\_equal},\mbox{negated\_conjecture})$

\medskip\textbf{SET028-3.p} Relationship between apply and image, part 1 of 2






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\neg\,\mbox{apply}(\mbox{a\_function},\mbox{element})\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(\mbox{element}),\mbox{a\_function}))\qquad\mbox{cnf}(\mbox{prove\_property\_of\_image\_and\_apply}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET299-6.p} Inverse of product






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(x,y)'\ne \mbox{cross\_product}(y,x)\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_product}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET073-6.p} Corollary to unordered pair axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_unordered\_pair\_axiom1}_1,\mbox{negated\_conjecture})$

$\mbox{unordered\_pair}(x,y) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_unordered\_pair\_axiom1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET820-2.p} Problem about set theory






$\neg\,\mbox{c\_in}(\mbox{v\_a},\mbox{c\_emptyset},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OemptyE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{c\_insert}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{v\_a} = \mbox{v\_b} )\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertE}_0,\mbox{axiom})$

$\mbox{v\_P}(\mbox{v\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_V},\mbox{v\_U},\mbox{tc\_IntDef\_Oint}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_U},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_W},\mbox{v\_U},\mbox{tc\_IntDef\_Oint}) \ \Rightarrow \ \neg\,\mbox{v\_P}(\mbox{v\_x}(\mbox{v\_U}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET143$\wedge$5.p} TPS problem BOOL-PROP-67

Trybulec's 67th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_67\_pme},\mbox{conjecture})$

\medskip\textbf{SET703+4.p} Union of singletons

The union of singleton(A) and singleton(B) is equal to the
 unordered\_pair(A,B)




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;\mbox{equal\_set}(\mbox{union}(\mbox{singleton}(a),\mbox{singleton}(b)),\mbox{unordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{thI}_{41},\mbox{conjecture})$

\medskip\textbf{SET333-6.p} Monotonicity of image 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(y,z)\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_image1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{xr},y),\mbox{image}(\mbox{xr},z))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_image1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET155+4.p} De Morgans law 1






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,e{:}\;( ( a\subseteq e\mbox{ and }b\subseteq e ) \ \Rightarrow \ \mbox{equal\_set}(e\setminus \mbox{union}(a,b),\mbox{intersection}(e\setminus a,e\setminus b)) )\qquad\mbox{fof}(\mbox{thI}_{26},\mbox{conjecture})$

\medskip\textbf{SET557-6.p} Cantor's theorem






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_cantors\_theorem}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{xf})\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_cantors\_theorem}_2,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{power\_class}(\mbox{domain\_of}(\mbox{xf})),\mbox{range\_of}(\mbox{xf}))\qquad\mbox{cnf}(\mbox{prove\_cantors\_theorem}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET853-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) = \mbox{v\_m} ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_2,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_3,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_1,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_2,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_y},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_xa},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_xa},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_xa}\ne \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_xa},\mbox{t\_a}),\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_x},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_U},\mbox{t\_a}),\mbox{v\_x},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_U} = \mbox{v\_x} )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET054+1.p} Reflexivity of subclass






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;\mbox{subclass}(x,x)\qquad\mbox{fof}(\mbox{reflexivity\_of\_subclass},\mbox{conjecture})$

\medskip\textbf{SET616$\wedge$5.p} TPS problem BOOL-PROP-90

Trybulec's 90th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( y@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx} ) ) \ \Rightarrow \ x = y )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_90\_pme},\mbox{conjecture})$

\medskip\textbf{SET022-4.p} The first component of an ordered pair is a little set






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{ordered\_pair\_predicate}(a)\qquad\mbox{cnf}(\mbox{an\_ordered\_pair\_predicate},\mbox{hypothesis})$

$\neg\,\mbox{little\_set}(\mbox{first}(a))\qquad\mbox{cnf}(\mbox{prove\_first\_component\_is\_small},\mbox{negated\_conjecture})$

\medskip\textbf{SET104-6.p} Special member 2 of an ordered pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{unordered\_pair}(\mbox{null\_class},\mbox{singleton}(\mbox{singleton}(y)))\ne \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_property\_2\_of\_ordered\_pair}_2,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_property\_2\_of\_ordered\_pair}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET633$\wedge$5.p} TPS problem BOOL-PROP-115

Trybulec's 115th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( y@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( ( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} )\mbox{ or }( y@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx} ) ) \ \Rightarrow \ ( z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_115\_pme},\mbox{conjecture})$

\medskip\textbf{SET846-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Main\_OUnion\_\_least\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_least}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Main\_OUnion\_\_least\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_least}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_OsuccI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oeq\_\_succ\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}),\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET066+1.p} Unordered pair is commutative






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;\mbox{unordered\_pair}(x,y) = \mbox{unordered\_pair}(y,x)\qquad\mbox{fof}(\mbox{commutativity\_of\_unordered\_pair},\mbox{conjecture})$

\medskip\textbf{SET373-6.p} Power class property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{power\_class}(x),\mbox{power\_class}(y))\ne \mbox{power\_class}(\mbox{intersection}(x,y))\qquad\mbox{cnf}(\mbox{prove\_power\_class\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET876+1.p} A != B =$>$ disjoint(singleton(A),singleton(B))






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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{in}(a,b) \ \Rightarrow \ \mbox{disjoint}(\mbox{singleton}(a),b) )\qquad\mbox{fof}(\mbox{l28\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b{:}\;( a\ne b \ \Rightarrow \ \mbox{disjoint}(\mbox{singleton}(a),\mbox{singleton}(b)) )\qquad\mbox{fof}(\mbox{t17\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET787-1.p} un\_eq\_Union\_2\_c2






$u\in \mbox{union}(v) \ \Rightarrow \ u\in \mbox{unionE\_sk}_1(u,v)\qquad\mbox{cnf}(\mbox{clause}_{119},\mbox{axiom})$

$u\in \mbox{union}(v) \ \Rightarrow \ \mbox{unionE\_sk}_1(u,v)\in v\qquad\mbox{cnf}(\mbox{clause}_{120},\mbox{axiom})$

$\mbox{subsetI\_sk}_1(a,b)\in a\mbox{ or }a\subseteq b\qquad\mbox{cnf}(\mbox{clause}_{131},\mbox{axiom})$

$\mbox{subsetI\_sk}_1(a,b)\in b \ \Rightarrow \ a\subseteq b\qquad\mbox{cnf}(\mbox{clause}_{132},\mbox{axiom})$

$u\in \mbox{cons}(v,w) \ \Rightarrow \ ( u = v\mbox{ or }u\in w )\qquad\mbox{cnf}(\mbox{clause}_{112},\mbox{axiom})$

$c\in a \ \Rightarrow \ c\in \mbox{un}(a,b)\qquad\mbox{cnf}(\mbox{unI}_1,\mbox{axiom})$

$c\in b \ \Rightarrow \ c\in \mbox{un}(a,b)\qquad\mbox{cnf}(\mbox{unI}_2,\mbox{axiom})$

$\neg\,x\in \mbox{eptset}\qquad\mbox{cnf}(\mbox{clause}_{158},\mbox{axiom})$

$\mbox{pair}(u,v)\in w \ \Rightarrow \ \mbox{pair}(v,u)\in w^\smallsmile\qquad\mbox{cnf}(\mbox{clause}_{10},\mbox{axiom})$

$\mbox{pair}(a,b)\in r^\smallsmile \ \Rightarrow \ \mbox{pair}(b,a)\in r\qquad\mbox{cnf}(\mbox{converseD},\mbox{axiom})$

$\mbox{yX}\in r^\smallsmile \ \Rightarrow \ \mbox{yX} = \mbox{pair}(\mbox{converseE\_sk}_2(\mbox{yX}),\mbox{converseE\_sk}_1(\mbox{yX}))\qquad\mbox{cnf}(\mbox{converseE}_1,\mbox{axiom})$

$\mbox{yX}\in r^\smallsmile \ \Rightarrow \ \mbox{pair}(\mbox{converse\_sk}_1(\mbox{yX}),\mbox{converse\_sk}_2(\mbox{yX}))\in r\qquad\mbox{cnf}(\mbox{converseE}_2,\mbox{axiom})$

$\mbox{sk}_2\in \mbox{union}(\mbox{cons}(a,\mbox{cons}(b,\mbox{eptset})))\qquad\mbox{cnf}(\mbox{un\_eq\_Union\_2\_c}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{sk}_2\in \mbox{un}(a,b)\qquad\mbox{cnf}(\mbox{un\_eq\_Union\_2\_c}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET298-6.p} Inverse of complement






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x''\ne \mbox{intersection}(( x' )',\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_complement}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET179-6.p} Corollary 2 to distribution property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{intersection}(x,z),\mbox{intersection}(x',\mbox{intersection}(y,z)))\ne \mbox{union}(\mbox{intersection}(x,z),\mbox{intersection}(y,z))\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_distribution\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET424-6.p} Compose class and composition function are related






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in v\qquad\mbox{cnf}(\mbox{prove\_compose\_class\_and\_composition\_function}_1,\mbox{negated\_conjecture})$

$\mbox{image}(\mbox{composition\_function},\mbox{singleton}(x))\ne \mbox{compose\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_compose\_class\_and\_composition\_function}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET249-6.p} Restriction property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{restrict}(\mbox{xr}_1,x,y),\mbox{restrict}(\mbox{xr}_2,x,y))\ne \mbox{restrict}(\mbox{union}(\mbox{xr}_1,\mbox{xr}_2),x,y)\qquad\mbox{cnf}(\mbox{prove\_restriction\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET830-2.p} Problem about set theory






$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OIntE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OIntE}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OIntI}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET602+4.p} The difference of X and X is the empty set






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall e{:}\;\mbox{equal\_set}(e\setminus e,\mbox{empty\_set})\qquad\mbox{fof}(\mbox{thI}_{29},\mbox{conjecture})$

\medskip\textbf{SET413-6.p} Compose condition for singleton membership 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{singleton}(x) = y\qquad\mbox{cnf}(\mbox{prove\_compose\_condition\_for\_singleton\_membership3}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_compose\_condition\_for\_singleton\_membership3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(x,y)\in \mbox{singleton\_relation}\qquad\mbox{cnf}(\mbox{prove\_compose\_condition\_for\_singleton\_membership3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET432-6.p} Function alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(z)\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,v)\in z\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(u,w)\in z\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn1}_3,\mbox{negated\_conjecture})$

$v\ne w\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn1}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET609+3.p} X \ (Y \ Z) = (X \ Y) U X $\wedge$ Z

The difference of X and (the difference of Y and Z) is the union
 of (the difference of X and Y) and the intersection of X and Z.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;b\setminus ( c\setminus d ) = \mbox{union}(b\setminus c,\mbox{intersection}(b,d))\qquad\mbox{fof}(\mbox{prove\_th}_{81},\mbox{conjecture})$

\medskip\textbf{SET751+4.p} If X is a subset of Y, then the image f(X) is a subset of f(Y)






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,x,y{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }x\subseteq a\mbox{ and }y\subseteq a\mbox{ and }x\subseteq y ) \ \Rightarrow \ \mbox{image}_2(f,x)\subseteq \mbox{image}_2(f,y) )\qquad\mbox{fof}(\mbox{thIIa}_{01},\mbox{conjecture})$

\medskip\textbf{SET143+3.p} Associativity of intersection

The intersection of (the intersection of X and Y) and Z is the
 intersection of X and (the intersection of Y and Z).




$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{intersection}(\mbox{intersection}(b,c),d) = \mbox{intersection}(b,\mbox{intersection}(c,d))\qquad\mbox{fof}(\mbox{prove\_associativity\_of\_intersection},\mbox{conjecture})$

\medskip\textbf{SET389-6.p} Composition alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{xf}\circ \mbox{yf}\qquad\mbox{cnf}(\mbox{prove\_composition\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(1_{\text{dom}(\mbox{xf})},v)\in \mbox{xf}\qquad\mbox{cnf}(\mbox{prove\_composition\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET337-6.p} Corollary 2 image property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(\mbox{null\_class},x)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_image\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET470-6.p} Corollary to sets with one member






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{cross\_product}(x,x),\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sets\_with\_one\_member}_1,\mbox{negated\_conjecture})$

$\mbox{singleton}(\mbox{member\_of}(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sets\_with\_one\_member}_2,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_sets\_with\_one\_member}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET448-6.p} Function property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_function\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(y,x)\qquad\mbox{cnf}(\mbox{prove\_function\_property4}_2,\mbox{negated\_conjecture})$

$\mbox{subclass}(z,\mbox{domain\_of}(y))\qquad\mbox{cnf}(\mbox{prove\_function\_property4}_3,\mbox{negated\_conjecture})$

$\mbox{restrict}(x,z,\mbox{universal\_class})\ne \mbox{restrict}(y,z,\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_function\_property4}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET013-2.p} The intersection of sets is commutative






$\mbox{include}(\mbox{'Axioms/SET002-0.ax'})$

$\mbox{equal\_sets}(\mbox{intersection}(\mbox{as},\mbox{bs}),\mbox{cs})\qquad\mbox{cnf}(\mbox{intersection\_of\_a\_and\_b\_is\_c},\mbox{hypothesis})$

$\mbox{equal\_sets}(\mbox{intersection}(\mbox{bs},\mbox{as}),\mbox{ds})\qquad\mbox{cnf}(\mbox{intersection\_of\_b\_and\_a\_is\_d},\mbox{hypothesis})$

$\neg\,\mbox{equal\_sets}(\mbox{cs},\mbox{ds})\qquad\mbox{cnf}(\mbox{prove\_c\_equals\_d},\mbox{negated\_conjecture})$

\medskip\textbf{SET038-4.p} Properties of apply for functions, part 3 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{maps}(\mbox{a\_function},\mbox{a\_domain},\mbox{a\_range})\qquad\mbox{cnf}(\mbox{a\_mapping},\mbox{hypothesis})$

$a\in \mbox{a\_domain}\qquad\mbox{cnf}(\mbox{member\_of\_domain},\mbox{hypothesis})$

$\neg\,\mbox{apply}(\mbox{a\_function},a)\in \mbox{a\_range}\qquad\mbox{cnf}(\mbox{prove\_mapping\_in\_range},\mbox{negated\_conjecture})$

\medskip\textbf{SET442-6.p} Restriction of function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_restrict\_a\_function}_1,\mbox{negated\_conjecture})$

$\mbox{restrict}(\mbox{xf},\mbox{universal\_class},\mbox{universal\_class})\ne \mbox{xf}\qquad\mbox{cnf}(\mbox{prove\_restrict\_a\_function}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET860-2.p} Problem about Zorn's lemma






$\neg\,\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_c},\mbox{c\_uminus}(\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OComplD\_\_dest}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_uminus}(\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OComplI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_uminus}(\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_uminus}(\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OCompl\_\_subset\_\_Compl\_\_iff\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_uminus}(\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_uminus}(\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OCompl\_\_subset\_\_Compl\_\_iff\_\_iff2}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OIntE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OIntE}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OIntI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_UNIV},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUNIV\_\_I}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnCI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) )\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_A},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Main\_OUnionE\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnionE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_A},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_A},\mbox{c\_Main\_OUnionE\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnionE}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_A},\mbox{v\_X},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_X},\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_A},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnionI}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{v\_a},\mbox{c\_emptyset},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OemptyE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_a},\mbox{c\_insert}(\mbox{v\_b},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{c\_insert}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{v\_a} = \mbox{v\_b} )\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertE}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

\medskip\textbf{SET142-6.p} Lexical ordering in unordered triples is irrelevant






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$\mbox{set\_builder}(x,\mbox{set\_builder}(y,\mbox{set\_builder}(z,\mbox{null\_class})))\ne \mbox{set\_builder}(y,\mbox{set\_builder}(x,\mbox{set\_builder}(z,\mbox{null\_class})))\qquad\mbox{cnf}(\mbox{prove\_ordering\_in\_triple}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET638+3.p} If X (= Y U Z and X $\wedge$ Z = the empty set , then X (= Y

If X is a subset of the union of Y and Z and the intersection
 of X and Z is the empty set, then X is a subset of Y.




$\forall b,c{:}\;\mbox{intersection}(b,c)\subseteq b\qquad\mbox{fof}(\mbox{intersection\_is\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c \ \Rightarrow \ \mbox{intersection}(b,c) = b )\qquad\mbox{fof}(\mbox{subset\_intersection},\mbox{axiom})$

$\forall b{:}\;\mbox{union}(b,\mbox{empty\_set}) = b\qquad\mbox{fof}(\mbox{union\_empty\_set},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{intersection}(b,\mbox{union}(c,d)) = \mbox{union}(\mbox{intersection}(b,c),\mbox{intersection}(b,d))\qquad\mbox{fof}(\mbox{intersection\_distributes\_over\_union},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq \mbox{union}(c,d)\mbox{ and }\mbox{intersection}(b,d) = \mbox{empty\_set} ) \ \Rightarrow \ b\subseteq c )\qquad\mbox{fof}(\mbox{prove\_th}_{120},\mbox{conjecture})$

\medskip\textbf{SET726+4.p} If GoF and FoH are identities, then inverse(F) = G






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,a)\mbox{ and }\mbox{maps}(h,b,a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(g,f,a,b,a),a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(f,h,b,a,b),b) ) \ \Rightarrow \ \mbox{equal\_maps}(\mbox{inverse\_function}(f,a,b),g,b,a) )\qquad\mbox{fof}(\mbox{thII}_{17},\mbox{conjecture})$

\medskip\textbf{SET964+1.p} Basic properties of sets, theorem 117






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

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

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

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

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

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b,c{:}\;\neg\,a\ne \mbox{empty\_set}\mbox{ and }( \mbox{cartesian\_product}_2(b,a)\subseteq \mbox{cartesian\_product}_2(c,a)\mbox{ or }\mbox{cartesian\_product}_2(a,b)\subseteq \mbox{cartesian\_product}_2(a,c) )\mbox{ and }\neg\,b\subseteq c\qquad\mbox{fof}(\mbox{t117\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET911+1.p} in(A,B) =$>$ set\_intersection2(B,singleton(A)) = singleton(A)






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

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

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

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

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

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_intersection}_2(b,\mbox{singleton}(a)) = \mbox{singleton}(a) )\qquad\mbox{fof}(\mbox{t52\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_intersection}_2(b,\mbox{singleton}(a)) = \mbox{singleton}(a) )\qquad\mbox{fof}(\mbox{l32\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET587$\wedge$5.p} TPS problem BOOL-PROP-45

Trybulec's 45th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} )\iff \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_45\_pme},\mbox{conjecture})$

\medskip\textbf{SET906+1.p} subset(set\_union2(unordered\_pair(A,B),C),C) =$>$ in(A,C)






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

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

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

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

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

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

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

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

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

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

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

$\forall a,b,c{:}\;( \mbox{set\_union}_2(\mbox{unordered\_pair}(a,b),c)\subseteq c \ \Rightarrow \ \mbox{in}(a,c) )\qquad\mbox{fof}(\mbox{t47\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET027$\wedge$7.p} Transitivity of subset






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

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

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

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mequiv}@ ( \subseteq @ b@ c )@ ( \mbox{mforall\_ind}@ \lambda d{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \in @ d@ b )@ ( \in @ d@ c ) ) ) ) ) )\qquad\mbox{thf}(\mbox{subset\_defn},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \subseteq @ b@ b ) )\qquad\mbox{thf}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mforall\_ind}@ \lambda d{:}\;\mbox{mu}{:}\;( \mbox{mimplies}@ ( \mbox{mand}@ ( \subseteq @ b@ c )@ ( \subseteq @ c@ d ) )@ ( \subseteq @ b@ d ) ) ) ) )\qquad\mbox{thf}(\mbox{prove\_transitivity\_of\_subset},\mbox{conjecture})$

\medskip\textbf{SET943+1.p} union(union(A,B)) = union(union(A),union(B))






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

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

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

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

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

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

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

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

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

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

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

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

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

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

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{union}(a)\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{t95\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{union}(\mbox{set\_union}_2(a,b)) = \mbox{set\_union}_2(\mbox{union}(a),\mbox{union}(b))\qquad\mbox{fof}(\mbox{t96\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET283-6.p} Apply domain relation






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_apply\_domain\_relation}_1,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{domain\_relation},x)\ne \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_apply\_domain\_relation}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET537-6.p} Corollary 2 to application property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{xf}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_application\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_application\_property3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(y,\mbox{apply}(\mbox{xf},x))\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_application\_property3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET039-3.p} Properties of apply for composition of functions, 1 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(x)\subseteq \mbox{domain\_of}(y) \ \Rightarrow \ \mbox{domain\_of}(x) = \mbox{domain\_of}(y\circ x)\qquad\mbox{cnf}(\mbox{domain\_of\_composition},\mbox{axiom})$

$( \mbox{function}(x)\mbox{ and }\mbox{function}(y) ) \ \Rightarrow \ \mbox{function}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_function},\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},u,v)\mbox{ and }\mbox{maps}(\mbox{xg},v,w) ) \ \Rightarrow \ \mbox{maps}(\mbox{xg}\circ \mbox{xf},u,w)\qquad\mbox{cnf}(\mbox{maps\_for\_composition},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{function}(\mbox{xf})\mbox{ and }\mbox{ordered\_pair}(x,y)\in \mbox{xf} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x) = y\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_1,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf})\mbox{ and }\mbox{apply}(\mbox{xf},x) = y ) \ \Rightarrow \ \mbox{ordered\_pair}(x,y)\in \mbox{xf}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_2,\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},\mbox{xd},\mbox{xr})\mbox{ and }x\in \mbox{xd} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_3,\mbox{axiom})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$a\in \mbox{domain\_of}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{member\_of\_domain},\mbox{hypothesis})$

$\neg\,\mbox{apply}(\mbox{another\_function},\mbox{apply}(\mbox{a\_function},a))\subseteq \mbox{apply}(\mbox{another\_function}\circ \mbox{a\_function},a)\qquad\mbox{cnf}(\mbox{prove\_apply\_for\_composition}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET259-6.p} Domain alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{domain\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_domain\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(x,\mbox{range}(\mbox{xr},x,\mbox{universal\_class}))\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_domain\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET351-6.p} Sum class of a singleton is the singleton member






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_sum\_of\_singleton}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(\mbox{singleton}(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_sum\_of\_singleton}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET980+1.p} Basic properties of sets, theorem 134






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

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

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

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

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b{:}\;( \mbox{cartesian\_product}_2(a,b) = \mbox{empty\_set}\iff ( a = \mbox{empty\_set}\mbox{ or }b = \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{t113\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{cartesian\_product}_2(a,b) = \mbox{cartesian\_product}_2(c,d) \ \Rightarrow \ ( a = \mbox{empty\_set}\mbox{ or }b = \mbox{empty\_set}\mbox{ or }( a = c\mbox{ and }b = d ) ) )\qquad\mbox{fof}(\mbox{t134\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

\medskip\textbf{SET093-6.p} Corollary to every singleton is a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{singleton}(\mbox{member\_of}(x)) = x\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_singletons\_are\_sets}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_singletons\_are\_sets}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET439-6.p} Null class is a function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{function}(\mbox{null\_class})\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_a\_function}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET631+3.p} If X intersects the difference of Y and Z, then X intersects Y






$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b,c,d{:}\;( \mbox{intersect}(b,c\setminus d) \ \Rightarrow \ \mbox{intersect}(b,c) )\qquad\mbox{fof}(\mbox{prove\_th}_{113},\mbox{conjecture})$

\medskip\textbf{SET133-6.p} Corollary 1 to membership in unordered triple






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$u\in x\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_member\_of\_triple}_1,\mbox{negated\_conjecture})$

$v\in x\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_member\_of\_triple}_2,\mbox{negated\_conjecture})$

$w\in x\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_member\_of\_triple}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{set\_builder}(u,\mbox{set\_builder}(v,\mbox{set\_builder}(w,\mbox{null\_class}))),x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_member\_of\_triple}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET624$\wedge$5.p} TPS problem BOOL-PROP-100

Trybulec's 100th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }( y@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) )\iff ( \exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} )\mbox{ or }\exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_100\_pme},\mbox{conjecture})$

\medskip\textbf{SET950+1.p} Basic properties of sets, theorem 103

~ ( subset(A,cart\_prod(B,C)) \& in(D,A) \& ~ ( in(E,B) \& in(F,C)
 \& D = ordered\_pair(E,F) ) )




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

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

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

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

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

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

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

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

$\forall a,b,c,d{:}\;\neg\,a\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }\mbox{in}(d,a)\mbox{ and }\forall e,f{:}\;\neg\,\mbox{in}(e,b)\mbox{ and }\mbox{in}(f,c)\mbox{ and }d = \mbox{ordered\_pair}(e,f)\qquad\mbox{fof}(\mbox{t103\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET017-6.p} Left cancellation for non-ordered pairs






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{unordered\_pair}(x,y) = \mbox{unordered\_pair}(x,z)\qquad\mbox{cnf}(\mbox{prove\_left\_cancellation}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(y,z)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_left\_cancellation}_2,\mbox{negated\_conjecture})$

$y\ne z\qquad\mbox{cnf}(\mbox{prove\_left\_cancellation}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET355-6.p} If X is in Y, then X is a subset of the sum class of Y






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in y\qquad\mbox{cnf}(\mbox{prove\_subclass\_of\_union}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,\mbox{sum\_class}(y))\qquad\mbox{cnf}(\mbox{prove\_subclass\_of\_union}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET692+4.p} A = A $\wedge$ B iff A (= B

A is a subset of B if and only if it is equal to the
 intersection of A and B.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;( \mbox{equal\_set}(a,\mbox{intersection}(a,b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{thI}_{19},\mbox{conjecture})$

\medskip\textbf{SET849-2.p} Problem about Zorn's lemma






$\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_refl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oequal\_\_succ\_\_Union}_1,\mbox{axiom})$

$\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\ne \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET563-6.p} Compatible function property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{compatible}(\mbox{xh}_1,\mbox{xf}_1,\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{compatible}(\mbox{xh}_2,\mbox{xf}_2,\mbox{xf}_3)\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{range\_of}(\mbox{xh}_1),\mbox{domain\_of}(\mbox{xh}_2))\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET046-5.p} No set of non-circular sets

A set is circular if it is a member of another set which
 in turn is a member of the orginal. Intuitively all sets are
 non-circular. Prove there is no set of non-circular sets.




$( \mbox{element}(x,a)\mbox{ and }\mbox{element}(x,y) ) \ \Rightarrow \ \neg\,\mbox{element}(y,x)\qquad\mbox{cnf}(\mbox{clause}_1,\mbox{negated\_conjecture})$

$\mbox{element}(x,f(x))\mbox{ or }\mbox{element}(x,a)\qquad\mbox{cnf}(\mbox{clause}_2,\mbox{negated\_conjecture})$

$\mbox{element}(f(x),x)\mbox{ or }\mbox{element}(x,a)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET672$\wedge$3.p} Y1 restricted to R (X to Y) is (X to Y1)

Y1 restricted to a relation R from X to Y is a relation from X
 to Y1.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall z{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{is\_rel\_on}@ r@ x@ y ) \ \Rightarrow \ ( \mbox{is\_rel\_on}@ ( \mbox{restrict\_rel\_codomain}@ r@ z )@ x@ z ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET358+4.p} Sum distributes over union

The union of sum(A) and sum(B) is equal to the sum of the union
 of A and B.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;\mbox{equal\_set}(\mbox{union}(\mbox{\mbox{sum}}(a),\mbox{\mbox{sum}}(b)),\mbox{\mbox{sum}}(\mbox{union}(a,b)))\qquad\mbox{fof}(\mbox{thI}_{37},\mbox{conjecture})$

\medskip\textbf{SET460-6.p} Range of identity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{range\_of}(\mbox{identity\_relation})\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_range\_of\_identity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET632$\wedge$5.p} TPS problem BOOL-PROP-114

Trybulec's 114th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) )\mbox{ and }\neg\,\exists \mbox{xx}{:}\;a{:}\;( y@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) ) \ \Rightarrow \ x = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_114\_pme},\mbox{conjecture})$

\medskip\textbf{SET082+1.p} If X is not a set, {X} = null class






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;( \neg\,x\in \mbox{universal\_class} \ \Rightarrow \ \mbox{singleton}(x) = \mbox{null\_class} )\qquad\mbox{fof}(\mbox{singleton\_is\_null\_class},\mbox{conjecture})$

\medskip\textbf{SET323-6.p} Range property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{range\_of}(x) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_range\_property7}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_range\_property7}_2,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_range\_property7}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET036-3.p} Properties of apply for functions, part 1 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(x)\subseteq \mbox{domain\_of}(y) \ \Rightarrow \ \mbox{domain\_of}(x) = \mbox{domain\_of}(y\circ x)\qquad\mbox{cnf}(\mbox{domain\_of\_composition},\mbox{axiom})$

$( \mbox{function}(x)\mbox{ and }\mbox{function}(y) ) \ \Rightarrow \ \mbox{function}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_function},\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},u,v)\mbox{ and }\mbox{maps}(\mbox{xg},v,w) ) \ \Rightarrow \ \mbox{maps}(\mbox{xg}\circ \mbox{xf},u,w)\qquad\mbox{cnf}(\mbox{maps\_for\_composition},\mbox{axiom})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{b\_little\_set},\mbox{hypothesis})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$\mbox{ordered\_pair}(a,b)\in \mbox{a\_function}\qquad\mbox{cnf}(\mbox{ordered\_pair\_in\_function},\mbox{hypothesis})$

$\mbox{apply}(\mbox{a\_function},a)\ne b\qquad\mbox{cnf}(\mbox{prove\_apply\_for\_functions}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET691+4.p} A set is a subset of empty set if and only if it is equal to it






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a{:}\;( a\subseteq \mbox{empty\_set}\iff \mbox{equal\_set}(a,\mbox{empty\_set}) )\qquad\mbox{fof}(\mbox{thI}_{16},\mbox{conjecture})$

\medskip\textbf{SET825-2.p} Problem about set theory






$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_a},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Relation\_OId},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Relation\_OIdI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_a},\mbox{v\_b},\mbox{t\_a},\mbox{t\_a}),\mbox{c\_Relation\_OId},\mbox{tc\_prod}(\mbox{t\_a},\mbox{t\_a})) \ \Rightarrow \ \mbox{v\_a} = \mbox{v\_b}\qquad\mbox{cnf}(\mbox{cls\_Relation\_Opair\_\_in\_\_Id\_\_conv\_\_iff1}_0,\mbox{axiom})$

$\mbox{v\_Q}(\mbox{v\_n})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\neg\,\mbox{v\_Q}(\mbox{v\_m})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Pair}(c_0,c_0,\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\mbox{ or }\mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_xa}(\mbox{v\_U}),\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$( \mbox{c\_in}(\mbox{c\_Pair}(\mbox{c\_Suc}(\mbox{v\_x}(\mbox{v\_U})),\mbox{c\_Suc}(\mbox{v\_xa}(\mbox{v\_U})),\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\mbox{ and }\mbox{c\_in}(\mbox{c\_Pair}(c_0,c_0,\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Pair}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_nat},\mbox{tc\_nat}),\mbox{v\_U},\mbox{tc\_prod}(\mbox{tc\_nat},\mbox{tc\_nat}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET175$\wedge$5.p} TPS problem BOOL-PROP-69

Trybulec's 69th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }( x@ \mbox{xz}\mbox{ and }y@ \mbox{xz} ) ) ) = x\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_69\_pme},\mbox{conjecture})$

\medskip\textbf{SET063$\wedge$5.p} TPS problem BOOL-PROP-30

Trybulec's 30th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ \mbox{\$false} ) \ \Rightarrow \ x = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_30\_pme},\mbox{conjecture})$

\medskip\textbf{SET281-6.p} Domain relation is a function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{function}(\mbox{domain\_relation})\qquad\mbox{cnf}(\mbox{prove\_domain\_relation\_is\_a\_function}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET465-6.p} Identity is one-to-one






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{one\_to\_one}(\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_identity\_is\_1\_to\_1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET210-6.p} Corollary 1 to cross product product monotonicity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,z),\mbox{cross\_product}(\mbox{union}(x,y),z))\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_X\_product\_monotonicity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET293-6.p} Inverse of universal class is the universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{universal\_class}'\ne \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_universal\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET075-6.p} Corollary to unordered pair axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_unordered\_pair\_axiom3}_1,\mbox{negated\_conjecture})$

$\mbox{unordered\_pair}(x,y) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_unordered\_pair\_axiom3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET065-7.p} The null class is a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{special\_classes\_lemma},\mbox{axiom})$

$\neg\,z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{existence\_of\_null\_class},\mbox{axiom})$

$\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{cnf}(\mbox{null\_class\_is\_subclass},\mbox{axiom})$

$\mbox{subclass}(x,\mbox{null\_class}) \ \Rightarrow \ x = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{corollary\_of\_null\_class\_is\_subclass},\mbox{axiom})$

$z = \mbox{null\_class}\mbox{ or }\mbox{not\_subclass\_element}(z,\mbox{null\_class})\in z\qquad\mbox{cnf}(\mbox{null\_class\_is\_unique},\mbox{axiom})$

$\neg\,\mbox{null\_class}\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_a\_set}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET925+1.p} difference(singleton(A),B) = empty\_set $\leq$$>$ in(A,B)






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

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

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

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

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{empty\_set}\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{t68\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{empty\_set}\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l36\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET012-1.p} Complement is an involution






$\mbox{include}(\mbox{'Axioms/SET002-0.ax'})$

$\neg\,\mbox{equal\_sets}(( a' )',a)\qquad\mbox{cnf}(\mbox{prove\_involution},\mbox{negated\_conjecture})$

\medskip\textbf{SET723+4.p} If FoG = FoH and F is injective, then G = H






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,b,c)\mbox{ and }\mbox{maps}(g,a,b)\mbox{ and }\mbox{maps}(h,a,b)\mbox{ and }\mbox{injective}(f,b,c)\mbox{ and }\mbox{equal\_maps}(\mbox{compose\_function}(f,g,a,b,c),\mbox{compose\_function}(f,h,a,b,c),a,c) ) \ \Rightarrow \ \mbox{equal\_maps}(g,h,a,b) )\qquad\mbox{fof}(\mbox{thII}_{14},\mbox{conjecture})$

\medskip\textbf{SET731+4.p} Property of restriction 2

If F is a mapping from A to B,and G equal to F on A and
 C =image2(F,A), then G is surjective.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }c\subseteq b\mbox{ and }\mbox{image}_2(f,a) = c\mbox{ and }\forall x,y{:}\;( ( x\in a\mbox{ and }y\in c ) \ \Rightarrow \ ( \mbox{apply}(g,x,y)\iff \mbox{apply}(f,x,y) ) ) ) \ \Rightarrow \ \mbox{surjective}(g,a,c) )\qquad\mbox{fof}(\mbox{thII}_{22},\mbox{conjecture})$

\medskip\textbf{SET513-6.p} Corollary 3 to special members of ordered pairs






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(\mbox{universal\_class},\mbox{universal\_class})\ne \mbox{unordered\_pair}(\mbox{null\_class},\mbox{singleton}(\mbox{null\_class}))\qquad\mbox{cnf}(\mbox{prove\_corollary\_3\_to\_property\_1\_of\_ordered\_pair}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET410-6.p} Composition of element relation 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{sum\_class}(y)\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_element\_relation2}_1,\mbox{negated\_conjecture})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_element\_relation2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(x,y)\in \mbox{element\_relation}\circ \mbox{element\_relation}\qquad\mbox{cnf}(\mbox{prove\_composition\_of\_element\_relation2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET976+1.p} in(o\_pair(A,B),cart\_prod(C,sgtn(D))) $\leq$$>$ ( in(A,C) \& B = D )






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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,\mbox{singleton}(d)))\iff ( \mbox{in}(a,c)\mbox{ and }b = d ) )\qquad\mbox{fof}(\mbox{t129\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET010-1.p} X \ Y $\wedge$ Z = (X \ Y) U (X \ Z)






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-3.ax'})$

$\mbox{intersection}(a,b,\mbox{aIb})\qquad\mbox{cnf}(\mbox{a\_intersection\_b},\mbox{hypothesis})$

$c\setminus a{=}\mbox{cDa}\qquad\mbox{cnf}(\mbox{c\_minus\_a},\mbox{hypothesis})$

$c\setminus b{=}\mbox{cDb}\qquad\mbox{cnf}(\mbox{c\_minus\_b},\mbox{hypothesis})$

$c\setminus \mbox{aIb}{=}\mbox{cD\_aIb}\qquad\mbox{cnf}(\mbox{c\_minus\_aIb},\mbox{hypothesis})$

$\neg\,\mbox{union}(\mbox{cDa},\mbox{cDb},\mbox{cD\_aIb})\qquad\mbox{cnf}(\mbox{prove\_cDa\_union\_cDb\_is\_cD\_aIb},\mbox{negated\_conjecture})$

\medskip\textbf{SET701+4.p} Property of intersection and difference 3

A is a subset of B if and only if the intersection of A and
 of the difference of B is a subset of the intersection of C
 and of the difference of C.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,c,e{:}\;( ( a\subseteq e\mbox{ and }b\subseteq e ) \ \Rightarrow \ ( a\subseteq b\iff \mbox{intersection}(a,e\setminus b)\subseteq \mbox{intersection}(c,e\setminus c) ) )\qquad\mbox{fof}(\mbox{thI}_{35},\mbox{conjecture})$

\medskip\textbf{SET770+4.p} Two equivalence classes are equal or disjoint






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall e,r,a,b{:}\;( ( \mbox{equivalence}(r,e)\mbox{ and }a\in e\mbox{ and }b\in e ) \ \Rightarrow \ ( \mbox{equal\_set}(\mbox{equivalence\_class}(a,e,r),\mbox{equivalence\_class}(b,e,r))\mbox{ or }\mbox{disjoint}(\mbox{equivalence\_class}(a,e,r),\mbox{equivalence\_class}(b,e,r)) ) )\qquad\mbox{fof}(\mbox{thIII}_{06},\mbox{conjecture})$

\medskip\textbf{SET493-6.p} Diagonalization corollary






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{domain\_of}(\mbox{intersection}(y,\mbox{identity\_relation}))\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_corollary}_1,\mbox{negated\_conjecture})$

$\mbox{function}(y)\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_corollary}_2,\mbox{negated\_conjecture})$

$\mbox{apply}(y,x)\ne x\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_corollary}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET902+1.p} Basic properties of sets, theorem 43

~ ( singleton(A) = set\_union2(B,C) \& ~ ( B = singleton(A) \&
 C = singleton(A) ) \& ~ ( B = empty\_set \& C = singleton(A) ) \&
 ~ ( B = singleton(A) \& C = empty\_set ) )




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

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

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

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

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

$\forall a{:}\;\mbox{singleton}(a)\ne \mbox{empty\_set}\qquad\mbox{fof}(\mbox{l1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq \mbox{singleton}(b)\iff ( a = \mbox{empty\_set}\mbox{ or }a = \mbox{singleton}(b) ) )\qquad\mbox{fof}(\mbox{l4\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b,c{:}\;\neg\,\mbox{singleton}(a) = \mbox{set\_union}_2(b,c)\mbox{ and }\neg\,b = \mbox{singleton}(a)\mbox{ and }c = \mbox{singleton}(a)\mbox{ and }\neg\,b = \mbox{empty\_set}\mbox{ and }c = \mbox{singleton}(a)\mbox{ and }\neg\,b = \mbox{singleton}(a)\mbox{ and }c = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t43\_zfmisc}_1,\mbox{conjecture})$

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

\medskip\textbf{SET030-3.p} Function values are little sets






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$\neg\,\mbox{little\_set}(\mbox{apply}(\mbox{a\_function},b))\qquad\mbox{cnf}(\mbox{prove\_function\_values\_are\_small},\mbox{negated\_conjecture})$

\medskip\textbf{SET110-6.p} 2nd is the ordered pair, first condition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{ordered\_pair}(\mbox{first}(x),\mbox{second}(x))\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_1st\_and\_2nd\_3}_1,\mbox{negated\_conjecture})$

$\mbox{second}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_1st\_and\_2nd\_3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET890+1.p} union(singleton(A)) = A






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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ a\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{l50\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a{:}\;\mbox{union}(\mbox{singleton}(a)) = a\qquad\mbox{fof}(\mbox{t31\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET600$\wedge$5.p} TPS problem BOOL-PROP-59

Trybulec's 59th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( ( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }y@ \mbox{xz} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} )\iff ( x = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} )\mbox{ and }y = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_59\_pme},\mbox{conjecture})$

\medskip\textbf{SET193-6.p} Subclass property 10






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(y',x')\qquad\mbox{cnf}(\mbox{prove\_subclass\_property10}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_property10}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET445-6.p} Corollary to function property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_function\_property1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{union}(x,\mbox{singleton}(\mbox{ordered\_pair}(y,z))))\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_function\_property1}_2,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_function\_property1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET067$\wedge$1.p} If one argument is a proper class, pair contains only the other






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \subseteq @ ( \mbox{unord\_pair}@ x@ y )@ ( \mbox{unord\_pair}@ y@ x ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET216-6.p} Corollary 7 to cross product product monotonicity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,\mbox{intersection}(y,z)),\mbox{cross\_product}(x,y))\qquad\mbox{cnf}(\mbox{prove\_corollary\_7\_to\_X\_product\_monotonicity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET009$\wedge$5.p} TPS problem BOOL-PROP-47

Trybulec's 47th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( z@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_47\_pme},\mbox{conjecture})$

\medskip\textbf{SET952+1.p} subset(cartesian\_product(A,B),powerset(powerset(union(A,B))))






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

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

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

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

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

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

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

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

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

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

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

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

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

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

$\forall a,b{:}\;\mbox{cartesian\_product}_2(a,b)\subseteq \mbox{powerset}(\mbox{powerset}(\mbox{set\_union}_2(a,b)))\qquad\mbox{fof}(\mbox{t105\_zfmisc}_1,\mbox{conjecture})$

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

$\forall a,b,c{:}\;( \mbox{unordered\_pair}(a,b)\subseteq c\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,c) ) )\qquad\mbox{fof}(\mbox{t38\_zfmisc}_1,\mbox{axiom})$

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

\medskip\textbf{SET002-1.p} Idempotency of union






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-1.ax'})$

$\mbox{union}(a,a,\mbox{aUa})\qquad\mbox{cnf}(\mbox{a\_union\_a\_is\_aUa},\mbox{hypothesis})$

$\neg\,\mbox{equal\_sets}(\mbox{aUa},a)\qquad\mbox{cnf}(\mbox{prove\_a\_equals\_aUa},\mbox{negated\_conjecture})$

\medskip\textbf{SET147-6.p} Universal class is identity for intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{universal\_class},x)\ne x\qquad\mbox{cnf}(\mbox{prove\_identity\_for\_intersection}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET174-6.p} Corollary to absorbtion for intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x,\mbox{intersection}(y,\mbox{union}(x,z)))\ne \mbox{intersection}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_absorbtion\_for\_intersection}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET366+4.p} The empty set is in every power set






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a{:}\;\mbox{empty\_set}\in \mbox{power\_set}(a)\qquad\mbox{fof}(\mbox{thI}_{47},\mbox{conjecture})$

\medskip\textbf{SET076-6.p} Unorderd pair is a subset

If both members of an unordered pair belong to a set, the
 pair is a subset.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$x\in z\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_is\_subset}_1,\mbox{negated\_conjecture})$

$y\in z\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_is\_subset}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{unordered\_pair}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_is\_subset}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET144-6.p} Commutativity of intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x,y)\ne \mbox{intersection}(y,x)\qquad\mbox{cnf}(\mbox{prove\_commutativity\_of\_intersection}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET529-6.p} Corollary 2 to ordered pair components are sets






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(\mbox{first}(\mbox{ordered\_pair}(x,y)),\mbox{second}(\mbox{ordered\_pair}(x,y))) = \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET420-6.p} Composition property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_composition\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(\mbox{image}(\mbox{image}(\mbox{composition\_function},\mbox{singleton}(x)),y))\ne x\circ \mbox{sum\_class}(y)\qquad\mbox{cnf}(\mbox{prove\_composition\_property4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET586$\wedge$5.p} TPS problem BOOL-PROP-40

Trybulec's 40th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_40\_pme},\mbox{conjecture})$

\medskip\textbf{SET522-6.p} Ordered pair determines components 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{second}(\mbox{ordered\_pair}(x,y)) = \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_determines\_components2}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_determines\_components2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET063+3.p} If X is a subset of the empty set, then X is the empty set






$\forall b{:}\;\mbox{empty\_set}\subseteq b\qquad\mbox{fof}(\mbox{empty\_set\_subset},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b{:}\;( b\subseteq \mbox{empty\_set} \ \Rightarrow \ b = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{prove\_subset\_of\_empty\_set\_is\_empty\_set},\mbox{conjecture})$

\medskip\textbf{SET981+1.p} Basic properties of sets, theorem 135






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

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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

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

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

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

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

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

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

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ a\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{l50\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a{:}\;\exists b{:}\;\forall c{:}\;( \mbox{in}(c,b)\iff ( \mbox{in}(c,\mbox{union}(a))\mbox{ and }\exists d{:}\;( c = d\mbox{ and }\exists e{:}\;( \mbox{in}(e,d)\mbox{ and }\mbox{in}(e,a) ) ) ) )\qquad\mbox{fof}(\mbox{s1\_xboole\_0\_\_e2\_121\_2\_\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,\mbox{cartesian\_product}_2(b,c))\mbox{ and }\forall d,e{:}\;\mbox{ordered\_pair}(d,e)\ne a\qquad\mbox{fof}(\mbox{t102\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( ( a\subseteq \mbox{cartesian\_product}_2(a,b)\mbox{ or }a\subseteq \mbox{cartesian\_product}_2(b,a) ) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t135\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{set\_union}_2(a,b) = \mbox{empty\_set} \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t15\_xboole}_1,\mbox{axiom})$

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

\medskip\textbf{SET065-6.p} The null class is a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{null\_class}\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_a\_set}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET217-6.p} Corollary 8 to cross product product monotonicity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,\mbox{intersection}(y,z)),\mbox{cross\_product}(x,z))\qquad\mbox{cnf}(\mbox{prove\_corollary\_8\_to\_X\_product\_monotonicity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET288-6.p} Domain property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_domain\_property9}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{intersection}(y',x)) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_domain\_property9}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_domain\_property9}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET207-6.p} Cross product property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{intersection}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})),\mbox{cross\_product}(\mbox{domain\_of}(x),\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET395-6.p} Composition relates to image






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(\mbox{xr}\circ \mbox{yr},x)\ne \mbox{image}(\mbox{xr},\mbox{image}(\mbox{yr},x))\qquad\mbox{cnf}(\mbox{prove\_compositions\_relates\_to\_image}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET928+1.p} diff(uno\_pair(A,B),C) = uno\_pair(A,B) $\leq$$>$ (~in(A,C) \& ~in(B,C))






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

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

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

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

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

$\forall a,b,c{:}\;\neg\,\mbox{disjoint}(\mbox{unordered\_pair}(a,b),c)\mbox{ and }\mbox{in}(a,c)\qquad\mbox{fof}(\mbox{t55\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\neg\,\mbox{in}(a,b)\mbox{ and }\neg\,\mbox{in}(c,b)\mbox{ and }\neg\,\mbox{disjoint}(\mbox{unordered\_pair}(a,c),b)\qquad\mbox{fof}(\mbox{t57\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c) = \mbox{unordered\_pair}(a,b)\iff ( \neg\,\mbox{in}(a,c)\mbox{ and }\neg\,\mbox{in}(b,c) ) )\qquad\mbox{fof}(\mbox{t72\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{disjoint}(a,b)\iff \mbox{set\_difference}(a,b) = a )\qquad\mbox{fof}(\mbox{t83\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SET742+4.p} Problem on composition of mappings 7

Consider three mappings F from A to B,G from B to C,H from
 C to A. If GoF and HoG are one-to-one, then F is one-to-one.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,c,a)\mbox{ and }\mbox{one\_to\_one}(\mbox{compose\_function}(g,f,a,b,c),a,c)\mbox{ and }\mbox{one\_to\_one}(\mbox{compose\_function}(h,g,b,c,a),b,a) ) \ \Rightarrow \ \mbox{one\_to\_one}(f,a,b) )\qquad\mbox{fof}(\mbox{thII}_{33},\mbox{conjecture})$

\medskip\textbf{SET194-6.p} X is a subset of the union of X and Y






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(x,\mbox{union}(x,y))\qquad\mbox{cnf}(\mbox{prove\_lattice\_upper\_bound1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET962+1.p} cartesian\_product(A,A) = cartesian\_product(B,B) =$>$ A = B






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

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

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b{:}\;( \mbox{cartesian\_product}_2(a,a) = \mbox{cartesian\_product}_2(b,b) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t115\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

\medskip\textbf{SET530-6.p} Corollary 3 to ordered pair components are sets






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(\mbox{first}(\mbox{ordered\_pair}(x,y)),\mbox{second}(\mbox{ordered\_pair}(x,y))) = \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary}_1,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET140-6.p} Triple reduction 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$\mbox{set\_builder}(x,\mbox{set\_builder}(y,\mbox{set\_builder}(x,\mbox{null\_class})))\ne \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_triple\_reduction2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET272-6.p} Domain property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{xr},\mbox{intersection}(\mbox{domain\_of}(\mbox{xr}),x),y)\ne \mbox{restrict}(\mbox{xr},x,y)\qquad\mbox{cnf}(\mbox{prove\_domain\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET094-6.p} Property 1 of singleton sets






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{singleton}(\mbox{member\_of}(x)) = x\qquad\mbox{cnf}(\mbox{prove\_property\_of\_singletons1}_1,\mbox{negated\_conjecture})$

$y\in x\qquad\mbox{cnf}(\mbox{prove\_property\_of\_singletons1}_2,\mbox{negated\_conjecture})$

$\mbox{member\_of}(x)\ne y\qquad\mbox{cnf}(\mbox{prove\_property\_of\_singletons1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET102-6.p} Ordered pair member of ordered pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{unordered\_pair}(x,\mbox{singleton}(y))\in \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_member\_of\_ordered\_pair}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET098-6.p} Corollary 1 to a class contains 0, 1, or at least 2 members






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{not\_subclass\_element}(\mbox{intersection}(\mbox{singleton}(\mbox{not\_subclass\_element}(x,\mbox{null\_class}))',x),\mbox{null\_class})\in x\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_number\_of\_elements\_in\_class}_1,\mbox{negated\_conjecture})$

$\mbox{singleton}(\mbox{not\_subclass\_element}(x,\mbox{null\_class}))\ne x\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_number\_of\_elements\_in\_class}_2,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_number\_of\_elements\_in\_class}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET178-6.p} Corollary 1 to distribution property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,\mbox{union}(y,\mbox{intersection}(x',z)))\ne \mbox{union}(x,\mbox{union}(y,z))\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_distribution\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET543-6.p} Corollary to application property 10






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_application\_property10}_1,\mbox{negated\_conjecture})$

$x\in y\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_application\_property10}_2,\mbox{negated\_conjecture})$

$x\in \mbox{domain\_of}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_application\_property10}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{apply}(\mbox{xf},x)\in \mbox{image}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_application\_property10}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET014+3.p} If X (= Z and Y (= Z, then X U Y (= Z

If X is a subset of Z and Y is a subset of Z, then the union of
 X and Y is a subset of Z.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }d\subseteq c ) \ \Rightarrow \ \mbox{union}(b,d)\subseteq c )\qquad\mbox{fof}(\mbox{prove\_union\_subset},\mbox{conjecture})$

\medskip\textbf{SET169$\wedge$5.p} TPS problem BOOL-PROP-70

Trybulec's 70th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }( y@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }y@ \mbox{xz} )\mbox{ or }( x@ \mbox{xz}\mbox{ and }z@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_70\_pme},\mbox{conjecture})$

\medskip\textbf{SET649$\wedge$3.p} Domain R a subset of X \& range R a subset of Y =$>$ R is (X to Y)

If the domain of a relation R from X to Y is a subset of X and
 the range of R is a subset of Y, R is a relation from X to Y.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \subseteq @ ( \mbox{rel\_domain}@ r )@ x\mbox{ and }\subseteq @ ( \mbox{rel\_codomain}@ r )@ y ) \ \Rightarrow \ ( \mbox{sub\_rel}@ r@ ( \mbox{cartesian\_product}@ x@ y ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET014$\wedge$4.p} Union of subsets is a subset

If A and B are contained in C then the union of A and B is also.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},a{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \subseteq @ x@ a\mbox{ and }\subseteq @ y@ a ) \ \Rightarrow \ ( \subseteq @ ( \mbox{union}@ x@ y )@ a ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET236-6.p} Cross product property 11






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property11}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(\mbox{first}(\mbox{not\_subclass\_element}(x,y)),\mbox{second}(\mbox{not\_subclass\_element}(x,y)))\in y\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property11}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property11}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET025-8.p} Ordered pairs are little sets






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(a)\qquad\mbox{cnf}(\mbox{an\_ordered\_pair\_predicate},\mbox{hypothesis})$

$\neg\,\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{prove\_predicate\_is\_small},\mbox{negated\_conjecture})$

\medskip\textbf{SET175-6.p} Absorbtion for union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,\mbox{intersection}(x,y))\ne x\qquad\mbox{cnf}(\mbox{prove\_absorbtion\_for\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET979+1.p} Basic properties of sets, theorem 132






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

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

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

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

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

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

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

$\forall a,b,c{:}\;( \mbox{cartesian\_product}_2(\mbox{set\_union}_2(a,b),c) = \mbox{set\_union}_2(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,c))\mbox{ and }\mbox{cartesian\_product}_2(c,\mbox{set\_union}_2(a,b)) = \mbox{set\_union}_2(\mbox{cartesian\_product}_2(c,a),\mbox{cartesian\_product}_2(c,b)) )\qquad\mbox{fof}(\mbox{t120\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{cartesian\_product}_2(\mbox{unordered\_pair}(a,b),c) = \mbox{set\_union}_2(\mbox{cartesian\_product}_2(\mbox{singleton}(a),c),\mbox{cartesian\_product}_2(\mbox{singleton}(b),c))\mbox{ and }\mbox{cartesian\_product}_2(c,\mbox{unordered\_pair}(a,b)) = \mbox{set\_union}_2(\mbox{cartesian\_product}_2(c,\mbox{singleton}(a)),\mbox{cartesian\_product}_2(c,\mbox{singleton}(b))) )\qquad\mbox{fof}(\mbox{t132\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{set\_union}_2(\mbox{singleton}(a),\mbox{singleton}(b))\qquad\mbox{fof}(\mbox{t41\_enumset}_1,\mbox{axiom})$

\medskip\textbf{SET697+4.p} Property of intersection and difference

A is a subset of B if and only if the intersection of A and
 of the difference of B is empty.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,e{:}\;( ( a\subseteq e\mbox{ and }b\subseteq e ) \ \Rightarrow \ ( a\subseteq b\iff \mbox{equal\_set}(\mbox{intersection}(a,e\setminus b),\mbox{empty\_set}) ) )\qquad\mbox{fof}(\mbox{thI}_{31},\mbox{conjecture})$

\medskip\textbf{SET603$\wedge$5.p} TPS problem BOOL-PROP-74

Trybulec's 74th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,\mbox{\$false} ) ) = x\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_74\_pme},\mbox{conjecture})$

\medskip\textbf{SET810+4.p} Ordinal numbers do not contain each other

If a and b are ordinal numbers, it is not possible that a belongs
 to b and b belongs to a





$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+4.ax'})$

$\forall a,b{:}\;( ( a\in \mbox{on}\mbox{ and }b\in \mbox{on} ) \ \Rightarrow \ \neg\,a\in b\mbox{ and }b\in a )\qquad\mbox{fof}(\mbox{thV}_3,\mbox{conjecture})$

\medskip\textbf{SET422-6.p} Compose class and apply






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_compose\_class\_and\_apply}_1,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{compose\_class}(x),y)\ne x\circ y\qquad\mbox{cnf}(\mbox{prove\_compose\_class\_and\_apply}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET450-6.p} Condition 2 for one function to be a subset of another






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_function\_subset\_function2}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{domain\_of}(x),\mbox{domain\_of}(y))\qquad\mbox{cnf}(\mbox{prove\_function\_subset\_function2}_2,\mbox{negated\_conjecture})$

$\mbox{apply}(x,\mbox{not\_subfunction}(x,y)) = \mbox{apply}(y,\mbox{not\_subfunction}(x,y))\qquad\mbox{cnf}(\mbox{prove\_function\_subset\_function2}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_function\_subset\_function2}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET228-6.p} Cross product right cancellation 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(u,v) = \mbox{cross\_product}(w,x)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_right\_cancellation1}_1,\mbox{negated\_conjecture})$

$v\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_right\_cancellation1}_2,\mbox{negated\_conjecture})$

$u\ne w\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_right\_cancellation1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET941+1.p} ( in(C,A) =$>$ subset(C,B) ) =$>$ subset(union(A),B)






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

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

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

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

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

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

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ c\subseteq b ) \ \Rightarrow \ \mbox{union}(a)\subseteq b )\qquad\mbox{fof}(\mbox{t94\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET812+4.p} An ordinal A is equal to its intersection with its power-set






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+4.ax'})$

$\forall a{:}\;( a\in \mbox{on} \ \Rightarrow \ \mbox{equal\_set}(a,\mbox{intersection}(a,\mbox{power\_set}(a))) )\qquad\mbox{fof}(\mbox{thV}_{10},\mbox{conjecture})$

\medskip\textbf{SET047-5.p} Set equality is symmetric

Define set equality as having exactly the same members. Prove
 set equality is symmetric.




$( \mbox{set\_equal}(x,y)\mbox{ and }\mbox{element}(z,x) ) \ \Rightarrow \ \mbox{element}(z,y)\qquad\mbox{cnf}(\mbox{element\_substitution}_1,\mbox{axiom})$

$( \mbox{set\_equal}(x,y)\mbox{ and }\mbox{element}(z,y) ) \ \Rightarrow \ \mbox{element}(z,x)\qquad\mbox{cnf}(\mbox{element\_substitution}_2,\mbox{axiom})$

$\mbox{element}(f(x,y),x)\mbox{ or }\mbox{element}(f(x,y),y)\mbox{ or }\mbox{set\_equal}(x,y)\qquad\mbox{cnf}(\mbox{clause}_3,\mbox{axiom})$

$( \mbox{element}(f(x,y),y)\mbox{ and }\mbox{element}(f(x,y),x) ) \ \Rightarrow \ \mbox{set\_equal}(x,y)\qquad\mbox{cnf}(\mbox{clause}_4,\mbox{axiom})$

$\mbox{set\_equal}(a,b)\mbox{ or }\mbox{set\_equal}(b,a)\qquad\mbox{cnf}(\mbox{prove\_symmetry}_1,\mbox{negated\_conjecture})$

$\mbox{set\_equal}(b,a) \ \Rightarrow \ \neg\,\mbox{set\_equal}(a,b)\qquad\mbox{cnf}(\mbox{prove\_symmetry}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET582$\wedge$5.p} TPS problem BOOL-PROP-25

Trybulec's 25th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( \neg\,x@ \mbox{xx}\iff ( y@ \mbox{xx} ) )\iff ( z@ \mbox{xx} ) ) \ \Rightarrow \ x = ( \lambda \mbox{xz}{:}\;a{:}\;( ( y@ \mbox{xz}\mbox{ and }\neg\,z@ \mbox{xz} )\mbox{ or }( z@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} ) ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_25\_pme},\mbox{conjecture})$

\medskip\textbf{SET843-2.p} Problem about Zorn's lemma






$\mbox{c\_in}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_upper}_0,\mbox{axiom})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_refl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_OsuccI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oeq\_\_succ\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{v\_m}\ne \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\mbox{ or }\mbox{v\_m} = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET986+1.p} in(A,B) =$>$ union(difference(B,singleton(A)),singleton(A)) = B






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

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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

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

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

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

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

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

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

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

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

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_union}_2(\mbox{set\_difference}(b,\mbox{singleton}(a)),\mbox{singleton}(a)) = b )\qquad\mbox{fof}(\mbox{t140\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( \mbox{in}(a,\mbox{set\_difference}(b,\mbox{singleton}(c)))\iff ( \mbox{in}(a,b)\mbox{ and }a\ne c ) )\qquad\mbox{fof}(\mbox{t64\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET607+3.p} X U (Y \ X) = X U Y

The union of X and the difference of Y and X is the union of X
 and Y.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c\setminus b) = \mbox{union}(b,c)\qquad\mbox{fof}(\mbox{prove\_th}_{79},\mbox{conjecture})$

\medskip\textbf{SET062$\wedge$5.p} TPS problem BOOL-PROP-27

Trybulec's 27th Boolean property of sets.




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

$\forall x{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( \mbox{\$false} \ \Rightarrow \ ( x@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_27\_pme},\mbox{conjecture})$

\medskip\textbf{SET834-2.p} Problem about set theory






$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnCI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnCI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) )\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnE}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET576+3.p} Trybulec's 17th Boolean property of sets






$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{disjoint}(b,c)\iff \neg\,\mbox{intersect}(b,c) )\qquad\mbox{fof}(\mbox{disjoint\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b,c{:}\;( \forall d{:}\;( d\in b \ \Rightarrow \ \neg\,d\in c ) \ \Rightarrow \ \mbox{disjoint}(b,c) )\qquad\mbox{fof}(\mbox{prove\_th}_{17},\mbox{conjecture})$

\medskip\textbf{SET412-6.p} Compose condition for singleton membership 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{singleton\_relation}\qquad\mbox{cnf}(\mbox{prove\_compose\_condition\_for\_singleton\_membership2}_1,\mbox{negated\_conjecture})$

$\mbox{singleton}(x)\ne y\qquad\mbox{cnf}(\mbox{prove\_compose\_condition\_for\_singleton\_membership2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET614+3.p} X \ Y \ Z = X \ (Y U Z)

The difference of X and the difference of Y and Z is the
 difference of X and (the union of Y and Z).




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( b\setminus c )\setminus d = b\setminus \mbox{union}(c,d)\qquad\mbox{fof}(\mbox{prove\_difference\_difference\_union},\mbox{conjecture})$

\medskip\textbf{SET232-6.p} Cross product property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property7}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{first}(\mbox{not\_subclass\_element}(\mbox{cross\_product}(x,y),z))\in x\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET114-6.p} 2nd is unique if x is not an ordered pair of sets, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{ordered\_pair}(\mbox{first}_1(x),\mbox{second}_1(x))\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_unique\_1st\_and\_2nd\_in\_pair\_of\_non\_sets2}_1,\mbox{negated\_conjecture})$

$\mbox{second}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_unique\_2nd\_in\_pair\_of\_non\_sets},\mbox{negated\_conjecture})$

\medskip\textbf{SET919+1.p} in(A,B) =$>$ ((in(C,B) \& A!=C) | intsctn(uno\_pair(A,C),B) = sgtn(A))






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

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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

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

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

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

$\forall a,b,c{:}\;( \mbox{in}(a,b) \ \Rightarrow \ ( ( \mbox{in}(c,b)\mbox{ and }a\ne c )\mbox{ or }\mbox{set\_intersection}_2(\mbox{unordered\_pair}(a,c),b) = \mbox{singleton}(a) ) )\qquad\mbox{fof}(\mbox{t60\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET538-6.p} Application property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(\mbox{sum\_class}(\mbox{xf}),\mbox{image}(\mbox{xf},\mbox{singleton}(x)))\ne \mbox{apply}(\mbox{xf},x)\qquad\mbox{cnf}(\mbox{prove\_application\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET712+4.p} The inverse of a one-to-one mapping is a mapping






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{one\_to\_one}(f,a,b) ) \ \Rightarrow \ \mbox{maps}(\mbox{inverse\_function}(f,a,b),b,a) )\qquad\mbox{fof}(\mbox{thII}_{03},\mbox{conjecture})$

\medskip\textbf{SET104+1.p} Special member 2 of an ordered pair






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( \mbox{unordered\_pair}(\mbox{null\_class},\mbox{singleton}(\mbox{singleton}(y))) = \mbox{ordered\_pair}(x,y)\mbox{ or }x\in \mbox{universal\_class} )\qquad\mbox{fof}(\mbox{property\_2\_of\_ordered\_pair},\mbox{conjecture})$

\medskip\textbf{SET651$\wedge$3.p} Domain of R (X to Y) a subset of X1 =$>$ R is (X1 to Y)

If the domain of a relation R from X to Y is a subset of X1
 then R is a relation from X1 to Y.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

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

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \subseteq @ ( \mbox{rel\_domain}@ r )@ a ) \ \Rightarrow \ ( \mbox{sub\_rel}@ r@ ( \mbox{cartesian\_product}@ a@ \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$true} ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET511-6.p} Corollary 1 to special members of ordered pairs






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{unordered\_pair}(\mbox{singleton}(x),\mbox{unordered\_pair}(x,\mbox{null\_class}))\ne \mbox{ordered\_pair}(x,\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_property\_1\_of\_ordered\_pair}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET169-6.p} Intersection distributes over union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{intersection}(x,y),\mbox{intersection}(x,z))\ne \mbox{intersection}(x,\mbox{union}(y,z))\qquad\mbox{cnf}(\mbox{prove\_intersection\_over\_union1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET025-6.p} Ordered pairs are little sets






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{ordered\_pair}(x,y)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_is\_set}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET143$\wedge$3.p} Associativity of intersection

The intersection of (the intersection of X and Y) and Z is the
 intersection of X and (the intersection of Y and Z).




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},z{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{intersection}@ ( \mbox{intersection}@ x@ y )@ z ) = ( \mbox{intersection}@ x@ ( \mbox{intersection}@ y@ z ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET754+4.p} C is a subset of the inverse image of the image of C






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }c\subseteq a ) \ \Rightarrow \ c\subseteq \mbox{inverse\_image}_2(f,\mbox{image}_2(f,c)) )\qquad\mbox{fof}(\mbox{thIIa}_{04},\mbox{conjecture})$

\medskip\textbf{SET725+4.p} If GoF and FoH are identities, then F is one-to-one






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,a)\mbox{ and }\mbox{maps}(h,b,a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(g,f,a,b,a),a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(f,h,b,a,b),b) ) \ \Rightarrow \ \mbox{one\_to\_one}(f,a,b) )\qquad\mbox{fof}(\mbox{thII}_{16},\mbox{conjecture})$

\medskip\textbf{SET843-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Main\_OUnion\_\_least\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_least}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Main\_OUnion\_\_least\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_least}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_OsuccI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oeq\_\_succ\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{v\_m}\ne \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\mbox{ or }\mbox{v\_m} = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET041-3.p} Properties of apply for composition of functions, 3 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(x)\subseteq \mbox{domain\_of}(y) \ \Rightarrow \ \mbox{domain\_of}(x) = \mbox{domain\_of}(y\circ x)\qquad\mbox{cnf}(\mbox{domain\_of\_composition},\mbox{axiom})$

$( \mbox{function}(x)\mbox{ and }\mbox{function}(y) ) \ \Rightarrow \ \mbox{function}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_function},\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},u,v)\mbox{ and }\mbox{maps}(\mbox{xg},v,w) ) \ \Rightarrow \ \mbox{maps}(\mbox{xg}\circ \mbox{xf},u,w)\qquad\mbox{cnf}(\mbox{maps\_for\_composition},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{function}(\mbox{xf})\mbox{ and }\mbox{ordered\_pair}(x,y)\in \mbox{xf} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x) = y\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_1,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf})\mbox{ and }\mbox{apply}(\mbox{xf},x) = y ) \ \Rightarrow \ \mbox{ordered\_pair}(x,y)\in \mbox{xf}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_2,\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},\mbox{xd},\mbox{xr})\mbox{ and }x\in \mbox{xd} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_3,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf}) ) \ \Rightarrow \ \mbox{apply}(\mbox{xg},\mbox{apply}(\mbox{xf},x))\subseteq \mbox{apply}(\mbox{xg}\circ \mbox{xf},x)\qquad\mbox{cnf}(\mbox{apply\_for\_composition}_1,\mbox{axiom})$

$\mbox{function}(\mbox{xf}) \ \Rightarrow \ \mbox{apply}(\mbox{xg}\circ \mbox{xf},x)\subseteq \mbox{apply}(\mbox{xg},\mbox{apply}(\mbox{xf},x))\qquad\mbox{cnf}(\mbox{apply\_for\_composition}_2,\mbox{axiom})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$a\in \mbox{domain\_of}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{member\_of\_domain},\mbox{hypothesis})$

$\mbox{apply}(\mbox{another\_function},\mbox{apply}(\mbox{a\_function},a))\ne \mbox{apply}(\mbox{another\_function}\circ \mbox{a\_function},a)\qquad\mbox{cnf}(\mbox{prove\_apply\_for\_composition}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET607$\wedge$3.p} X U (Y \ X) = X U Y

The union of X and the difference of Y and X is the union of X
 and Y.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{union}@ x@ ( \mbox{setminus}@ y@ x ) ) = ( \mbox{union}@ x@ y )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET200-6.p} Union is monotonic






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_union\_is\_monotonic}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(z,w)\qquad\mbox{cnf}(\mbox{prove\_union\_is\_monotonic}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{union}(x,z),\mbox{union}(y,w))\qquad\mbox{cnf}(\mbox{prove\_union\_is\_monotonic}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET616+3.p} If X \ Y = Y \ X, then X = Y

If the difference of X and Y is the difference of Y and X, then
 X is Y.




$\forall b,c{:}\;( \forall d{:}\;( d\in b\iff d\in c ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{member\_equal},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b\setminus c = c\setminus b \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{prove\_th}_{90},\mbox{conjecture})$

\medskip\textbf{SET146+3.p} The intersection of X and the empty set is the empty set






$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b{:}\;\mbox{intersection}(b,\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{prove\_th}_{61},\mbox{conjecture})$

\medskip\textbf{SET793+4.p} If the order is total, a maximal element is the greatest element






$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e,m{:}\;( ( \mbox{total\_order}(r,e)\mbox{ and }\mbox{max}(m,r,e) ) \ \Rightarrow \ \mbox{greatest}(m,r,e) )\qquad\mbox{fof}(\mbox{thIV}_5,\mbox{conjecture})$

\medskip\textbf{SET784+1.p} Mapping axioms for the SET006+0 set theory axioms






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

\medskip\textbf{SET482-6.p} Replacement property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(\mbox{domain\_of}(x),\mbox{range\_of}(x))\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_replacement\_property4}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_replacement\_property4}_2,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property4}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET508-6.p} Corollary 1 to singleton in unordered pair axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{unordered\_pair}(x,\mbox{universal\_class})\ne \mbox{singleton}(x)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_singleton\_in\_unordered\_pair}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET552-6.p} Application property 20






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_application\_property17}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(x,\mbox{domain\_of}(\mbox{intersection}(\mbox{xf},\mbox{identity\_relation})))\qquad\mbox{cnf}(\mbox{prove\_application\_property17}_2,\mbox{negated\_conjecture})$

$\mbox{image}(\mbox{xf},x)\ne x\qquad\mbox{cnf}(\mbox{prove\_application\_property17}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET250-6.p} Corollary to restriction property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{restrict}(x,y,y),\mbox{restrict}(x,y,y)')\ne \mbox{restrict}(\mbox{union}(x,x'),y,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_restriction\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET017+1.p} Left cancellation for unordered pairs






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y,z{:}\;( ( y\in \mbox{universal\_class}\mbox{ and }z\in \mbox{universal\_class}\mbox{ and }\mbox{unordered\_pair}(x,y) = \mbox{unordered\_pair}(x,z) ) \ \Rightarrow \ y = z )\qquad\mbox{fof}(\mbox{left\_cancellation},\mbox{conjecture})$

\medskip\textbf{SET095-6.p} If X is in Y, then the singleton containing X is a subset of Y






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$x\in y\qquad\mbox{cnf}(\mbox{prove\_property\_of\_singletons2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{singleton}(x),y)\qquad\mbox{cnf}(\mbox{prove\_property\_of\_singletons2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET565-6.p} Corollary 2 to compatible function property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{compatible}(\mbox{xh}_1,\mbox{xf}_1,\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_compatible\_function\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{compatible}(\mbox{xh}_2,\mbox{xf}_2,\mbox{xf}_3)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_compatible\_function\_property3}_2,\mbox{negated\_conjecture})$

$\mbox{cross\_product}(\mbox{domain\_of}(\mbox{xh}_2\circ \mbox{xh}_1),\mbox{domain\_of}(\mbox{xh}_2\circ \mbox{xh}_1))\ne \mbox{cross\_product}(\mbox{domain\_of}(\mbox{xh}_1),\mbox{domain\_of}(\mbox{xh}_1))\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_compatible\_function\_property3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET008$\wedge$5.p} TPS problem BOOL-PROP-78

Trybulec's 78th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_78\_pme},\mbox{conjecture})$

\medskip\textbf{SET007-1.p} Intersection distributes over union






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-1.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-2.ax'})$

$\mbox{union}(b,c,\mbox{bUc})\qquad\mbox{cnf}(\mbox{b\_union\_c},\mbox{axiom})$

$\mbox{intersection}(a,b,\mbox{aIb})\qquad\mbox{cnf}(\mbox{a\_intersection\_b},\mbox{axiom})$

$\mbox{intersection}(a,c,\mbox{aIc})\qquad\mbox{cnf}(\mbox{a\_intersection\_c},\mbox{axiom})$

$\mbox{intersection}(a,\mbox{bUc},\mbox{aI\_bUc})\qquad\mbox{cnf}(\mbox{a\_intersection\_bUc},\mbox{axiom})$

$\neg\,\mbox{union}(\mbox{aIb},\mbox{aIc},\mbox{aI\_bUc})\qquad\mbox{cnf}(\mbox{prove\_aIb\_union\_aIc\_is\_aI\_bUc},\mbox{negated\_conjecture})$

\medskip\textbf{SET794+4.p} If the order is total, a minimal element is the least element






$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e,m{:}\;( ( \mbox{total\_order}(r,e)\mbox{ and }\mbox{min}(m,r,e) ) \ \Rightarrow \ \mbox{least}(m,r,e) )\qquad\mbox{fof}(\mbox{thIV}_6,\mbox{conjecture})$

\medskip\textbf{SET828-2.p} Problem about set theory






$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET341-6.p} Image under universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(\mbox{universal\_class},x)\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_image\_under\_universal\_class}_1,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_image\_under\_universal\_class}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET286-6.p} Corollary to domain property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_domain\_property7}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{union}(x,\mbox{singleton}(\mbox{ordered\_pair}(\mbox{domain\_of}(x),y))))\ne \mbox{successor}(\mbox{domain\_of}(x))\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_domain\_property7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET002+3.p} Idempotency of union






$\forall b,c{:}\;( b\subseteq c \ \Rightarrow \ \mbox{union}(b,c) = c )\qquad\mbox{fof}(\mbox{subset\_union},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b{:}\;\mbox{union}(b,b) = b\qquad\mbox{fof}(\mbox{prove\_idempotency\_of\_union},\mbox{conjecture})$

\medskip\textbf{SET159+4.p} Associativity of union






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,c{:}\;\mbox{equal\_set}(\mbox{union}(\mbox{union}(a,b),c),\mbox{union}(a,\mbox{union}(b,c)))\qquad\mbox{fof}(\mbox{thI}_{09},\mbox{conjecture})$

\medskip\textbf{SET382-6.p} Corollary 1 to relation property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_relation\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{restrict}(x,\mbox{universal\_class},\mbox{universal\_class})\ne x\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_relation\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET601+3.p} X $\wedge$ Y U Y $\wedge$ Z U Z $\wedge$ X = (X U Y) $\wedge$ (Y U Z) $\wedge$ (Z U X)

The intersection of X and the union of Y and the intersection
 of Y and the union of Z and the intersection of Z and X is the
 intersection of (the union of X and Y) and the intersection of
 (the union of Y and Z) and (the union of Z and X).




$\forall b,c,d{:}\;\mbox{union}(\mbox{union}(b,c),d) = \mbox{union}(b,\mbox{union}(c,d))\qquad\mbox{fof}(\mbox{associativity\_of\_union},\mbox{axiom})$

$\forall b{:}\;\mbox{intersection}(b,b) = b\qquad\mbox{fof}(\mbox{idempotency\_of\_intersection},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{intersection}(\mbox{intersection}(b,c),d) = \mbox{intersection}(b,\mbox{intersection}(c,d))\qquad\mbox{fof}(\mbox{associativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,\mbox{intersection}(b,c)) = b\qquad\mbox{fof}(\mbox{union\_intersection},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{union}(b,\mbox{intersection}(c,d)) = \mbox{intersection}(\mbox{union}(b,c),\mbox{union}(b,d))\qquad\mbox{fof}(\mbox{union\_distributes\_over\_intersection},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{union}(\mbox{union}(\mbox{intersection}(b,c),\mbox{intersection}(c,d)),\mbox{intersection}(d,b)) = \mbox{intersection}(\mbox{intersection}(\mbox{union}(b,c),\mbox{union}(c,d)),\mbox{union}(d,b))\qquad\mbox{fof}(\mbox{prove\_th}_{72},\mbox{conjecture})$

\medskip\textbf{SET014-6.p} If X (= Z and Y (= Z, then X U Y (= Z

If A and B are contained in C then the union of A and B is also.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(y,z)\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{union}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_least\_upper\_bound}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET121-6.p} Corollary 3 to components of equal ordered pairs being equal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_OP\_determines\_components2}_1,\mbox{negated\_conjecture})$

$\mbox{first}(\mbox{ordered\_pair}(x,y))\ne \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_OP\_determines\_components2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET851-2.p} Problem about Zorn's lemma






$\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_refl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_y},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_x},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_U},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_U} = \mbox{v\_m} )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET781+3.p} Set theory axioms based on NBG set theory






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

\medskip\textbf{SET926+1.p} difference(sgtn(A),B) = empty | difference(sgtn(A),B) = sgtn(A)






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

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

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{singleton}(a)\iff \neg\,\mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l34\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{empty\_set}\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l36\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{empty\_set}\mbox{ or }\mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{singleton}(a) )\qquad\mbox{fof}(\mbox{t69\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET009+3.p} If X is a subset of Y, then Z \ Y is a subset of Z \ X

If X is a subset of Y, then the difference of Z and Y is a
 subset of the difference of Z and X.




$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq c \ \Rightarrow \ ( d\setminus c )\subseteq ( d\setminus b ) )\qquad\mbox{fof}(\mbox{prove\_subset\_difference},\mbox{conjecture})$

\medskip\textbf{SET117+1.p} Corollary 1 to every ordered pair being a set






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;( \mbox{ordered\_pair}(\mbox{first}(x),\mbox{second}(x)) = x \ \Rightarrow \ x\in \mbox{universal\_class} )\qquad\mbox{fof}(\mbox{corollary\_1\_to\_ordered\_pairs\_are\_sets},\mbox{conjecture})$

\medskip\textbf{SET623+3.p} (X sym\ Y) sym\ Z = X sym\ (Y sym\ Z)

The symmetric difference of (the symmetric difference of X and Y)
 and Z is the symmetric difference of X and (the symmetric
 difference of Y and Z).




$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{union}(\mbox{union}(b,c),d) = \mbox{union}(b,\mbox{union}(c,d))\qquad\mbox{fof}(\mbox{associativity\_of\_union},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{intersection}(\mbox{intersection}(b,c),d) = \mbox{intersection}(b,\mbox{intersection}(c,d))\qquad\mbox{fof}(\mbox{associativity\_of\_intersection},\mbox{axiom})$

$\forall b,c,d{:}\;b\setminus ( c\setminus d ) = \mbox{union}(b\setminus c,\mbox{intersection}(b,d))\qquad\mbox{fof}(\mbox{difference\_difference\_union}_1,\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c)\setminus \mbox{intersection}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{difference\_union\_intersection},\mbox{axiom})$

$\forall b,c,d{:}\;( b\setminus c )\setminus d = b\setminus \mbox{union}(c,d)\qquad\mbox{fof}(\mbox{difference\_difference\_union}_2,\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{union}(b,c)\setminus d = \mbox{union}(b\setminus d,c\setminus d)\qquad\mbox{fof}(\mbox{difference\_distributes\_over\_union},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{symmetric\_difference}(\mbox{symmetric\_difference}(b,c),d) = \mbox{symmetric\_difference}(b,\mbox{symmetric\_difference}(c,d))\qquad\mbox{fof}(\mbox{prove\_th}_{99},\mbox{conjecture})$

\medskip\textbf{SET767+4.p} Equivalence classes on E are power\_set E






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall e,r,a{:}\;( \mbox{equivalence}(r,e) \ \Rightarrow \ \mbox{equivalence\_class}(a,e,r)\subseteq e )\qquad\mbox{fof}(\mbox{thIII}_{03},\mbox{conjecture})$

\medskip\textbf{SET177-6.p} Distribution property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,\mbox{intersection}(x',z))\ne \mbox{union}(x,z)\qquad\mbox{cnf}(\mbox{prove\_distribution\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET332-6.p} Corollary to range is image of domain






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(\mbox{xr},\mbox{universal\_class})\ne \mbox{range\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_image\_of\_domain}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET257-6.p} Restriction property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{cross\_product}(x,y),x,y)\ne \mbox{cross\_product}(x,y)\qquad\mbox{cnf}(\mbox{prove\_restriction\_property5}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET581+3.p} Trybulec's 24th Boolean property of sets






$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{not\_equal}(b,c)\iff b\ne c )\qquad\mbox{fof}(\mbox{not\_equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\in c\mbox{ and }b\in d ) \ \Rightarrow \ \mbox{not\_equal}(\mbox{intersection}(c,d),\mbox{empty\_set}) )\qquad\mbox{fof}(\mbox{prove\_th}_{24},\mbox{conjecture})$

\medskip\textbf{SET022-3.p} The first component of an ordered pair is a little set






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(a)\qquad\mbox{cnf}(\mbox{an\_ordered\_pair\_predicate},\mbox{hypothesis})$

$\neg\,\mbox{little\_set}(\mbox{first}(a))\qquad\mbox{cnf}(\mbox{prove\_first\_component\_is\_small},\mbox{negated\_conjecture})$

\medskip\textbf{SET929+1.p} diff(uno\_pair(A,B),C) = empty $\leq$$>$ ( in(A,C) \& in(B,C) )






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

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

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

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

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

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

$\forall a,b{:}\;( \mbox{set\_difference}(a,b) = \mbox{empty\_set}\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t37\_xboole}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{unordered\_pair}(a,b)\subseteq c\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,c) ) )\qquad\mbox{fof}(\mbox{t38\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c) = \mbox{empty\_set}\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,c) ) )\qquad\mbox{fof}(\mbox{t73\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET402-6.p} Right compose with universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\circ \mbox{universal\_class}\ne \mbox{cross\_product}(\mbox{universal\_class},\mbox{range\_of}(x))\qquad\mbox{cnf}(\mbox{prove\_right\_compose\_with\_universal\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET188-6.p} Subclass property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(y',x) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_subclass\_property6}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_property6}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET067-6.p} Proper class in an unordered pair, part 1

If one argument of an unordered pair is a proper class, the
 pair contains only the other.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{singleton}(x),\mbox{unordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{prove\_singleton\_in\_unordered\_pair1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET309-6.p} Monotonicity of range 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{range\_of}(\mbox{intersection}(x,y)),\mbox{range\_of}(y))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_range2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET765+4.p} The restriction of an equivalence relation is an equivalence






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall e,r,x{:}\;( ( \mbox{equivalence}(r,e)\mbox{ and }x\subseteq e ) \ \Rightarrow \ \mbox{equivalence}(r,x) )\qquad\mbox{fof}(\mbox{thIII}_{01},\mbox{conjecture})$

\medskip\textbf{SET449-6.p} Condition 1 for one function to be a subset of another






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_function\_subset\_function1}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(\mbox{domain\_of}(x),\mbox{domain\_of}(y))\qquad\mbox{cnf}(\mbox{prove\_function\_subset\_function1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_function\_subset\_function1}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subfunction}(x,y)\in \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_function\_subset\_function1}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET008+3.p} (X \ Y) $\wedge$ Y = the empty set

The intersection of (the difference of X and Y) and Y is the
 empty set.




$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b\setminus c,c) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{prove\_intersection\_difference\_empty\_set},\mbox{conjecture})$

\medskip\textbf{SET201-6.p} Intersection is monotonic






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_intersection\_is\_monotonic}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(z,w)\qquad\mbox{cnf}(\mbox{prove\_intersection\_is\_monotonic}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{intersection}(x,z),\mbox{intersection}(y,w))\qquad\mbox{cnf}(\mbox{prove\_intersection\_is\_monotonic}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET816+4.p} The sum of a collection of ordinal numbers is a collection






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+4.ax'})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{thI}_3,\mbox{axiom})$

$\forall a{:}\;( a\subseteq \mbox{on} \ \Rightarrow \ \mbox{\mbox{sum}}(a)\subseteq \mbox{on} )\qquad\mbox{fof}(\mbox{thV}_{16},\mbox{conjecture})$

\medskip\textbf{SET108-6.p} 1st and 2nd make the ordered pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(y,z)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_1st\_and\_2nd\_1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(\mbox{first}(\mbox{ordered\_pair}(y,z)),\mbox{second}(\mbox{ordered\_pair}(y,z)))\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_1st\_and\_2nd\_1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET103+1.p} Special member 1 of an ordered pair






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( \mbox{unordered\_pair}(\mbox{singleton}(x),\mbox{unordered\_pair}(x,\mbox{null\_class})) = \mbox{ordered\_pair}(x,y)\mbox{ or }y\in \mbox{universal\_class} )\qquad\mbox{fof}(\mbox{property\_1\_of\_ordered\_pair},\mbox{conjecture})$

\medskip\textbf{SET059-6.p} Expanded equality definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{not\_subclass\_element}(x,y)\in y\qquad\mbox{cnf}(\mbox{prove\_equality4}_1,\mbox{negated\_conjecture})$

$\mbox{not\_subclass\_element}(y,x)\in x\qquad\mbox{cnf}(\mbox{prove\_equality4}_2,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_equality4}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET457-6.p} Identity is a function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{function}(\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_identity\_is\_a\_function}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET190-6.p} Subclass property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_property7}_1,\mbox{negated\_conjecture})$

$\mbox{union}(x',y)\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_subclass\_property7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET634+3.p} X $\wedge$ (Y \ Z) = X $\wedge$ Y \ Z

The intersection of X and the difference of Y and Z is the
 intersection of X and the difference of Y and Z.




$\forall b,c{:}\;( \forall d{:}\;( d\in b\iff d\in c ) \ \Rightarrow \ b = c )\qquad\mbox{fof}(\mbox{member\_equal},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{intersection}(b,c\setminus d) = \mbox{intersection}(b,c)\setminus d\qquad\mbox{fof}(\mbox{prove\_difference\_and\_intersection},\mbox{conjecture})$

\medskip\textbf{SET636+3.p} X is disjoint from Y iff X $\wedge$ Y = the empty set

X is disjoint from Y iff the intersection of X and Y is the
 empty set.




$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{disjoint}(b,c)\iff \neg\,\mbox{intersect}(b,c) )\qquad\mbox{fof}(\mbox{disjoint\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{disjoint}(b,c)\iff \mbox{intersection}(b,c) = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{prove\_th}_{118},\mbox{conjecture})$

\medskip\textbf{SET062-6.p} The empty set is a subset of X






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_subclass}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET736+4.p} Problem on composition of mappings 1

Consider three mappings F from A to B,G from B to C,H from
 C to A. If HoGoF and FoHoG are injective and GoFoH surjective,
 then F is one-to-one.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,c,a)\mbox{ and }\mbox{injective}(\mbox{compose\_function}(h,\mbox{compose\_function}(g,f,a,b,c),a,c,a),a,a)\mbox{ and }\mbox{injective}(\mbox{compose\_function}(f,\mbox{compose\_function}(h,g,b,c,a),b,a,b),b,b)\mbox{ and }\mbox{surjective}(\mbox{compose\_function}(g,\mbox{compose\_function}(f,h,c,a,b),c,b,c),c,c) ) \ \Rightarrow \ \mbox{one\_to\_one}(f,a,b) )\qquad\mbox{fof}(\mbox{thII}_{27},\mbox{conjecture})$

\medskip\textbf{SET013-3.p} The intersection of sets is commutative






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{intersection}(\mbox{as},\mbox{bs}) = \mbox{cs}\qquad\mbox{cnf}(\mbox{intersection\_of\_a\_and\_b\_is\_c},\mbox{hypothesis})$

$\mbox{intersection}(\mbox{bs},\mbox{as}) = \mbox{ds}\qquad\mbox{cnf}(\mbox{intersection\_of\_b\_and\_a\_is\_d},\mbox{hypothesis})$

$\mbox{cs}\ne \mbox{ds}\qquad\mbox{cnf}(\mbox{prove\_c\_equals\_d},\mbox{negated\_conjecture})$

\medskip\textbf{SET673$\wedge$3.p} Y a subset of Y1 =$>$ Y1 restricted to R (X to Y) is R

If Y is a subset of Y1 then Y1 restricted to a relation R from
 X to Y is R.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall z{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{is\_rel\_on}@ r@ x@ y\mbox{ and }\subseteq @ y@ z ) \ \Rightarrow \ ( \mbox{restrict\_rel\_codomain}@ r@ z ) = r )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET276-6.p} Corollary 2 to domain property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(\mbox{domain\_of}(\mbox{restrict}(x,y,z)),u),\mbox{cross\_product}(\mbox{intersection}(\mbox{domain\_of}(x),y),u))\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_domain\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET135-6.p} Corollary 3 to membership in unordered triple






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_3\_to\_member\_of\_triple}_1,\mbox{negated\_conjecture})$

$\mbox{set\_builder}(x,\mbox{set\_builder}(u,\mbox{set\_builder}(z,\mbox{null\_class}))) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_3\_to\_member\_of\_triple}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET557$\wedge$6.p} TPS problem THM43

Restatement of Cantor's theorem.




$\forall s{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;\neg\,\exists g{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;\forall f{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( f@ \mbox{xx} ) \ \Rightarrow \ ( s@ \mbox{xx} ) ) \ \Rightarrow \ \exists j{:}\;\mbox{\$i}{:}\;( s@ j\mbox{ and }( g@ j ) = f ) )\qquad\mbox{thf}(\mbox{cTHM43\_pme},\mbox{conjecture})$

\medskip\textbf{SET224-6.p} Cross product double distribution for intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{cross\_product}(w,x),\mbox{cross\_product}(y,z))\ne \mbox{cross\_product}(\mbox{intersection}(w,y),\mbox{intersection}(x,z))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_double\_distribution}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET025-3.p} Ordered pairs are little sets






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\neg\,\mbox{little\_set}(\mbox{ordered\_pair}(a,b))\qquad\mbox{cnf}(\mbox{prove\_ordered\_pairs\_are\_small},\mbox{negated\_conjecture})$

\medskip\textbf{SET013-4.p} The intersection of sets is commutative






$x\in y \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(a_2,\mbox{axiom})$

$\mbox{little\_set}(f_1(x,y))\mbox{ or }x = y\qquad\mbox{cnf}(\mbox{extensionality}_1,\mbox{axiom})$

$f_1(x,y)\in x\mbox{ or }f_1(x,y)\in y\mbox{ or }x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$( f_1(x,y)\in x\mbox{ and }f_1(x,y)\in y ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_3,\mbox{axiom})$

$z\in \mbox{intersection}(x,y) \ \Rightarrow \ z\in x\qquad\mbox{cnf}(\mbox{intersection}_1,\mbox{axiom})$

$z\in \mbox{intersection}(x,y) \ \Rightarrow \ z\in y\qquad\mbox{cnf}(\mbox{intersection}_2,\mbox{axiom})$

$( z\in x\mbox{ and }z\in y ) \ \Rightarrow \ z\in \mbox{intersection}(x,y)\qquad\mbox{cnf}(\mbox{intersection}_3,\mbox{axiom})$

$\neg\,z\in \mbox{empty\_set}\qquad\mbox{cnf}(\mbox{empty\_set},\mbox{axiom})$

$\mbox{little\_set}(z) \ \Rightarrow \ z\in \mbox{universal\_set}\qquad\mbox{cnf}(\mbox{universal\_set},\mbox{axiom})$

$\mbox{intersection}(\mbox{as},\mbox{bs}) = \mbox{cs}\qquad\mbox{cnf}(\mbox{intersection\_of\_a\_and\_b\_is\_c},\mbox{hypothesis})$

$\mbox{intersection}(\mbox{bs},\mbox{as}) = \mbox{ds}\qquad\mbox{cnf}(\mbox{intersection\_of\_b\_and\_a\_is\_d},\mbox{hypothesis})$

$\mbox{cs}\ne \mbox{ds}\qquad\mbox{cnf}(\mbox{prove\_c\_equals\_d},\mbox{negated\_conjecture})$

\medskip\textbf{SET544-6.p} Corollary to application property 11






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(\mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_application\_property11}_1,\mbox{negated\_conjecture})$

$x\in \mbox{domain\_of}(\mbox{yf}\circ \mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_application\_property11}_2,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{yf}\circ \mbox{xf},x)\ne \mbox{apply}(\mbox{yf},\mbox{apply}(\mbox{xf},x))\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_application\_property11}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET528-6.p} Corollary 1 to ordered pair components are sets






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(\mbox{first}(x),\mbox{second}(x)) = x\qquad\mbox{cnf}(\mbox{prove\_corollary}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET054-7.p} Subclass is reflexive






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\neg\,\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{prove\_subclass\_is\_reflexive}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET401-6.p} Left compose with universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{universal\_class}\circ x\ne \mbox{cross\_product}(\mbox{domain\_of}(x),\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_left\_compose\_with\_universal\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET863-2.p} Problem about Zorn's lemma






$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_z},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_y},\mbox{v\_x},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_xa},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_xa},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_7,\mbox{negated\_conjecture})$

\medskip\textbf{SET766+4.p} A member belongs to its equivalence class






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall e,r,a{:}\;( ( \mbox{equivalence}(r,e)\mbox{ and }a\in e ) \ \Rightarrow \ a\in \mbox{equivalence\_class}(a,e,r) )\qquad\mbox{fof}(\mbox{thIII}_{02},\mbox{conjecture})$

\medskip\textbf{SET835-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET852-2.p} Problem about Zorn's lemma






$\neg\,\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_U},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_1,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_2,\mbox{axiom})$

\medskip\textbf{SET970+1.p} Basic properties of sets, theorem 123






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

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

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

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

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

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

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

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

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

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

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

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

$\forall a,b,c,d,e{:}\;\neg\,\mbox{in}(a,\mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(b,c),\mbox{cartesian\_product}_2(d,e)))\mbox{ and }\forall f,g{:}\;\neg\,a = \mbox{ordered\_pair}(f,g)\mbox{ and }\mbox{in}(f,\mbox{set\_intersection}_2(b,d))\mbox{ and }\mbox{in}(g,\mbox{set\_intersection}_2(c,e))\qquad\mbox{fof}(\mbox{t104\_zfmisc}_1,\mbox{axiom})$

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

$\forall a,b,c,d{:}\;\mbox{cartesian\_product}_2(\mbox{set\_intersection}_2(a,b),\mbox{set\_intersection}_2(c,d)) = \mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,d))\qquad\mbox{fof}(\mbox{t123\_zfmisc}_1,\mbox{conjecture})$

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

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }a\subseteq c ) \ \Rightarrow \ a\subseteq \mbox{set\_intersection}_2(b,c) )\qquad\mbox{fof}(\mbox{t19\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SET263-6.p} Domain preserves union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{domain\_of}(x),\mbox{domain\_of}(y))\ne \mbox{domain\_of}(\mbox{union}(x,y))\qquad\mbox{cnf}(\mbox{prove\_domain\_preserves\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET826-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_V},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_V},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_V},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET169+3.p} Intersection distributes over union

The intersection of X and (the union of Y and Z) is the union
 of (the intersection of X and Y) and (the intersection of X
 and Z).




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{intersection}(b,\mbox{union}(c,d)) = \mbox{union}(\mbox{intersection}(b,c),\mbox{intersection}(b,d))\qquad\mbox{fof}(\mbox{prove\_intersection\_distributes\_over\_union},\mbox{conjecture})$

\medskip\textbf{SET015-2.p} The union of sets is commutative






$\mbox{include}(\mbox{'Axioms/SET002-0.ax'})$

$\mbox{equal\_sets}(\mbox{union}(\mbox{as},\mbox{bs}),\mbox{cs})\qquad\mbox{cnf}(\mbox{a\_union\_b\_is\_c},\mbox{hypothesis})$

$\mbox{equal\_sets}(\mbox{union}(\mbox{bs},\mbox{as}),\mbox{ds})\qquad\mbox{cnf}(\mbox{b\_union\_a\_is\_d},\mbox{hypothesis})$

$\neg\,\mbox{equal\_sets}(\mbox{cs},\mbox{ds})\qquad\mbox{cnf}(\mbox{prove\_c\_equals\_d},\mbox{negated\_conjecture})$

\medskip\textbf{SET064-7.p} The null class is unique






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{special\_classes\_lemma},\mbox{axiom})$

$\neg\,z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{existence\_of\_null\_class},\mbox{axiom})$

$\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{cnf}(\mbox{null\_class\_is\_subclass},\mbox{axiom})$

$\mbox{subclass}(x,\mbox{null\_class}) \ \Rightarrow \ x = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{corollary\_of\_null\_class\_is\_subclass},\mbox{axiom})$

$z\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_unique}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(z,\mbox{null\_class})\in z\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_unique}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET406-6.p} Corollary 1 monotonicity of composition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{xr},\mbox{yr})\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_monotonicity\_of\_composition}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{zr}\circ ( \mbox{xr}\circ \mbox{ur} ),\mbox{zr}\circ ( \mbox{yr}\circ \mbox{ur} ))\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_monotonicity\_of\_composition}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET172-6.p} Distribution of union over intersection 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{union}(x,z),\mbox{union}(y,z))\ne \mbox{union}(\mbox{intersection}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_union\_over\_intersection2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET609$\wedge$5.p} TPS problem BOOL-PROP-81

Trybulec's 81st Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} )\mbox{ or }( x@ \mbox{xz}\mbox{ and }z@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_81\_pme},\mbox{conjecture})$

\medskip\textbf{SET803+4.p} Two different maximal elements implies no greatest element






$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e{:}\;( \mbox{order}(r,e) \ \Rightarrow \ \forall m_1,m_2{:}\;( ( \mbox{max}(m_1,r,e)\mbox{ and }\mbox{max}(m_2,r,e)\mbox{ and }m_1\ne m_2 ) \ \Rightarrow \ \neg\,\exists m{:}\;\mbox{greatest}(m,r,e) ) )\qquad\mbox{fof}(\mbox{thIV}_{15},\mbox{conjecture})$

\medskip\textbf{SET327-6.p} Image alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{image}(\mbox{xr},x)\qquad\mbox{cnf}(\mbox{prove\_image\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\neg\,1_{\text{dom}(\mbox{xr})}\in x\qquad\mbox{cnf}(\mbox{prove\_image\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET915+1.p} ~ in(A,B) =$>$ disjoint(singleton(A),B)






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

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

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

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

$\forall a,b{:}\;( \neg\,\mbox{in}(a,b) \ \Rightarrow \ \mbox{disjoint}(\mbox{singleton}(a),b) )\qquad\mbox{fof}(\mbox{t56\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \neg\,\mbox{in}(a,b) \ \Rightarrow \ \mbox{disjoint}(\mbox{singleton}(a),b) )\qquad\mbox{fof}(\mbox{l28\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET083-6.p} A singleton set depends on its element, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{singleton}(x) = \mbox{singleton}(y)\qquad\mbox{cnf}(\mbox{prove\_singleton\_identified\_by\_element1}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_singleton\_identified\_by\_element1}_2,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_singleton\_identified\_by\_element1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET132-6.p} Membership in unordered triple, part 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple3}_1,\mbox{negated\_conjecture})$

$\neg\,u\in \mbox{set\_builder}(x,\mbox{set\_builder}(y,\mbox{set\_builder}(u,\mbox{null\_class})))\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET289-6.p} Proof of Goedel's axiom B6, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(x',\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_inverse\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET183+3.p} If X is a subset of  Y, then the intersection of X and Y is X






$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c)\subseteq b\qquad\mbox{fof}(\mbox{intersection\_is\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c \ \Rightarrow \ \mbox{intersection}(b,c) = b )\qquad\mbox{fof}(\mbox{prove\_subset\_intersection},\mbox{conjecture})$

\medskip\textbf{SET585+3.p} The intersection of X and Y is a subset of the union of X and Z






$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }c\subseteq d ) \ \Rightarrow \ b\subseteq d )\qquad\mbox{fof}(\mbox{transitivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;b\subseteq \mbox{union}(b,c)\qquad\mbox{fof}(\mbox{subset\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c)\subseteq b\qquad\mbox{fof}(\mbox{intersection\_is\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{intersection}(b,c)\subseteq \mbox{union}(b,d)\qquad\mbox{fof}(\mbox{prove\_intersection\_subset\_of\_union},\mbox{conjecture})$

\medskip\textbf{SET280-6.p} Domain property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(x) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_domain\_property6}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_domain\_property6}_2,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_domain\_property6}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET063+4.p} The intersection of a set and empty set is empty






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a{:}\;\mbox{equal\_set}(\mbox{intersection}(a,\mbox{empty\_set}),\mbox{empty\_set})\qquad\mbox{fof}(\mbox{thI}_{17},\mbox{conjecture})$

\medskip\textbf{SET608$\wedge$5.p} TPS problem BOOL-PROP-80

Trybulec's 80th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }y@ \mbox{xz} )\mbox{ or }( x@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} ) ) ) = x\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_80\_pme},\mbox{conjecture})$

\medskip\textbf{SET549-6.p} Application property 17






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{application\_function},\mbox{singleton}(x)),\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_application\_property14}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET719+4.p} Inverse of composition

The inverse of the composition of mappings is equal to the
 composition of the inverse mappings.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{one\_to\_one}(f,a,b)\mbox{ and }\mbox{one\_to\_one}(g,b,c) ) \ \Rightarrow \ \mbox{equal\_maps}(\mbox{inverse\_function}(\mbox{compose\_function}(g,f,a,b,c),a,c),\mbox{compose\_function}(\mbox{inverse\_function}(f,a,b),\mbox{inverse\_function}(g,b,c),c,b,a),c,a) )\qquad\mbox{fof}(\mbox{thII}_{10},\mbox{conjecture})$

\medskip\textbf{SET296-6.p} Domain of inverse






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(x')\ne \mbox{range\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_domain\_of\_inverse}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET446-6.p} Function property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_function\_property2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{cross\_product}(x,\mbox{singleton}(y)))\qquad\mbox{cnf}(\mbox{prove\_function\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET091-6.p} Member\_of(X) is unique if X is not a singleton, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{member\_of}_1(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_of\_non\_singleton\_unique1}_1,\mbox{negated\_conjecture})$

$\mbox{member\_of}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_member\_of\_non\_singleton\_unique1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET377-6.p} Corollary 1 to power class set builder






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{intersection}(\mbox{power\_class}(z),z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_power\_class\_set\_builder}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{successor}(x)\in \mbox{power\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_power\_class\_set\_builder}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET415-6.p} Composition with singleton function 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{domain\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_composition\_with\_singleton\_function1}_1,\mbox{negated\_conjecture})$

$z\circ \mbox{singleton}(\mbox{ordered\_pair}(x,y))\ne \mbox{singleton}(\mbox{ordered\_pair}(x,\mbox{apply}(z,y)))\qquad\mbox{cnf}(\mbox{prove\_composition\_with\_singleton\_function1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET199+4.p} If Z (= X and Z (= Y, then Z (= X $\wedge$ Y

If Z is a subset of X and Y, then Z is a subset the intersection.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,x,y{:}\;( ( a\subseteq x\mbox{ and }a\subseteq y )\iff a\subseteq \mbox{intersection}(x,y) )\qquad\mbox{fof}(\mbox{thI}_{46},\mbox{conjecture})$

\medskip\textbf{SET589+3.p} If X (= Y and Z (= V, then X \ V (= Y \ Z

If X is a subset of Y and Z is a subset of V, then the
 difference of X and V is a subset of the difference of Y and Z.




$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }c\subseteq d ) \ \Rightarrow \ b\subseteq d )\qquad\mbox{fof}(\mbox{transitivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq c \ \Rightarrow \ ( b\setminus d )\subseteq ( c\setminus d ) )\qquad\mbox{fof}(\mbox{difference\_subset}_1,\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq c \ \Rightarrow \ ( d\setminus c )\subseteq ( d\setminus b ) )\qquad\mbox{fof}(\mbox{difference\_subset}_2,\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d,e{:}\;( ( b\subseteq c\mbox{ and }d\subseteq e ) \ \Rightarrow \ ( b\setminus e )\subseteq ( c\setminus d ) )\qquad\mbox{fof}(\mbox{prove\_th}_{48},\mbox{conjecture})$

\medskip\textbf{SET657$\wedge$3.p} The field of a relation R from X to Y is a subset of X union Y






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( \subseteq @ ( \mbox{rel\_field}@ r )@ ( \mbox{union}@ \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$true}@ \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$true} ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET349-6.p} Sum class of singleton null is null class 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{sum\_class}(\mbox{singleton}(\mbox{null\_class}))\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_of\_singleton\_null}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET622+3.p} X \ (Y sym\ Z) = (X \ (Y U Z)) U X $\wedge$ Y $\wedge$ Z

The difference of X and (the symmetric difference of Y and Z)
 is the union of (the difference of X and (the union of Y and Z))
 and the intersection of X and the intersection of Y and Z.




$\forall b,c,d{:}\;\mbox{intersection}(\mbox{intersection}(b,c),d) = \mbox{intersection}(b,\mbox{intersection}(c,d))\qquad\mbox{fof}(\mbox{associativity\_of\_intersection},\mbox{axiom})$

$\forall b,c,d{:}\;b\setminus ( c\setminus d ) = \mbox{union}(b\setminus c,\mbox{intersection}(b,d))\qquad\mbox{fof}(\mbox{difference\_difference\_union}_2,\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b,c)\setminus \mbox{intersection}(b,c)\qquad\mbox{fof}(\mbox{symmetric\_difference\_and\_difference},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;b\setminus \mbox{symmetric\_difference}(c,d) = \mbox{union}(b\setminus \mbox{union}(c,d),\mbox{intersection}(\mbox{intersection}(b,c),d))\qquad\mbox{fof}(\mbox{prove\_th}_{98},\mbox{conjecture})$

\medskip\textbf{SET346-6.p} Sum class alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in y\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$y\in x\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

$\neg\,z\in \mbox{sum\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_alternate\_defn3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET885+1.p} subset(unordered\_pair(A,B),singleton(C)) =$>$ A = C






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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

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

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

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

$\forall a,b,c{:}\;( \mbox{unordered\_pair}(a,b)\subseteq \mbox{singleton}(c) \ \Rightarrow \ a = c )\qquad\mbox{fof}(\mbox{t26\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET212-6.p} Corollary 3 to cross product product monotonicity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,y),\mbox{cross\_product}(x,\mbox{union}(y,z)))\qquad\mbox{cnf}(\mbox{prove\_corollary\_3\_to\_X\_product\_monotonicity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET515-6.p} No class belongs to itself






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in x\qquad\mbox{cnf}(\mbox{prove\_no\_class\_belongs\_to\_itself}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET203-6.p} Corollary to cross product product property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$u\in x\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_X\_product\_property1}_1,\mbox{negated\_conjecture})$

$v\in y\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_X\_product\_property1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_X\_product\_property1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET122+1.p} Corollary 4 to components of equal ordered pairs being equal






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( y\in \mbox{universal\_class}\mbox{ or }\mbox{second}(\mbox{ordered\_pair}(x,y)) = \mbox{ordered\_pair}(x,y) )\qquad\mbox{fof}(\mbox{corollary\_2\_to\_OP\_determines\_components}_2,\mbox{conjecture})$

\medskip\textbf{SET768+4.p} Equality of equivalence classes 1

Two equivalence classes are equal if and only if the members
 are equivalent.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall e,r,a,b{:}\;( ( \mbox{equivalence}(r,e)\mbox{ and }a\in e\mbox{ and }b\in e ) \ \Rightarrow \ ( \mbox{equal\_set}(\mbox{equivalence\_class}(a,e,r),\mbox{equivalence\_class}(b,e,r))\iff \mbox{apply}(r,a,b) ) )\qquad\mbox{fof}(\mbox{thIII}_{04},\mbox{conjecture})$

\medskip\textbf{SET040-3.p} Properties of apply for composition of functions, 2 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(x)\subseteq \mbox{domain\_of}(y) \ \Rightarrow \ \mbox{domain\_of}(x) = \mbox{domain\_of}(y\circ x)\qquad\mbox{cnf}(\mbox{domain\_of\_composition},\mbox{axiom})$

$( \mbox{function}(x)\mbox{ and }\mbox{function}(y) ) \ \Rightarrow \ \mbox{function}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_function},\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},u,v)\mbox{ and }\mbox{maps}(\mbox{xg},v,w) ) \ \Rightarrow \ \mbox{maps}(\mbox{xg}\circ \mbox{xf},u,w)\qquad\mbox{cnf}(\mbox{maps\_for\_composition},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{function}(\mbox{xf})\mbox{ and }\mbox{ordered\_pair}(x,y)\in \mbox{xf} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x) = y\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_1,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf})\mbox{ and }\mbox{apply}(\mbox{xf},x) = y ) \ \Rightarrow \ \mbox{ordered\_pair}(x,y)\in \mbox{xf}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_2,\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},\mbox{xd},\mbox{xr})\mbox{ and }x\in \mbox{xd} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_3,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf}) ) \ \Rightarrow \ \mbox{apply}(\mbox{xg},\mbox{apply}(\mbox{xf},x))\subseteq \mbox{apply}(\mbox{xg}\circ \mbox{xf},x)\qquad\mbox{cnf}(\mbox{apply\_for\_composition}_1,\mbox{axiom})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$\neg\,\mbox{apply}(\mbox{another\_function}\circ \mbox{a\_function},\mbox{element})\subseteq \mbox{apply}(\mbox{another\_function},\mbox{apply}(\mbox{a\_function},\mbox{element}))\qquad\mbox{cnf}(\mbox{prove\_apply\_for\_composition}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET831-2.p} Problem about set theory






$\mbox{v\_X} = \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_X} = \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{v\_X} = \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{v\_X} = \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_X}\ne \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ ( \mbox{v\_X} = \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OIntE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OIntE}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_inter}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OIntI}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

\medskip\textbf{SET789+4.p} The greatest element, if it existes, is unique






$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e,m{:}\;( ( \mbox{order}(r,e)\mbox{ and }\mbox{greatest}(m,r,e) ) \ \Rightarrow \ \forall x{:}\;( \mbox{greatest}(x,r,e) \ \Rightarrow \ m = x ) )\qquad\mbox{fof}(\mbox{thIV}_1,\mbox{conjecture})$

\medskip\textbf{SET669$\wedge$3.p} Id on Y subset of R  =$>$ Y subset of domain R \& Y is range R

If the identity relation on Y is a subset of a relation R from X
 to Y then Y is a subset of the domain of R and Y is the range of R.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{sub\_rel}@ ( \mbox{id\_rel}@ \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$true} )@ r ) \ \Rightarrow \ ( \subseteq @ \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$true}@ ( \mbox{rel\_domain}@ r )\mbox{ and }( \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$true} ) = ( \mbox{rel\_codomain}@ r ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET880+1.p} difference(singleton(A),singleton(B)) = empty\_set =$>$ A = B






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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{empty\_set}\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l36\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),\mbox{singleton}(b)) = \mbox{empty\_set} \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t21\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET292-6.p} Inverse of null class is the null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{null\_class}'\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET014-4.p} Union of subsets is a subset

If A and B are contained in C then the union of A and B is also.




$x\in y \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(a_2,\mbox{axiom})$

$\mbox{little\_set}(f_1(x,y))\mbox{ or }x = y\qquad\mbox{cnf}(\mbox{extensionality}_1,\mbox{axiom})$

$f_1(x,y)\in x\mbox{ or }f_1(x,y)\in y\mbox{ or }x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$( f_1(x,y)\in x\mbox{ and }f_1(x,y)\in y ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_3,\mbox{axiom})$

$z\in \mbox{intersection}(x,y) \ \Rightarrow \ z\in x\qquad\mbox{cnf}(\mbox{intersection}_1,\mbox{axiom})$

$z\in \mbox{intersection}(x,y) \ \Rightarrow \ z\in y\qquad\mbox{cnf}(\mbox{intersection}_2,\mbox{axiom})$

$( z\in x\mbox{ and }z\in y ) \ \Rightarrow \ z\in \mbox{intersection}(x,y)\qquad\mbox{cnf}(\mbox{intersection}_3,\mbox{axiom})$

$z\in x' \ \Rightarrow \ \neg\,z\in x\qquad\mbox{cnf}(\mbox{complement}_1,\mbox{axiom})$

$\mbox{little\_set}(z) \ \Rightarrow \ ( z\in x'\mbox{ or }z\in x )\qquad\mbox{cnf}(\mbox{complement}_2,\mbox{axiom})$

$\mbox{union}(x,y) = \mbox{intersection}(x',y')'\qquad\mbox{cnf}(\mbox{union},\mbox{axiom})$

$\neg\,z\in \mbox{empty\_set}\qquad\mbox{cnf}(\mbox{empty\_set},\mbox{axiom})$

$\mbox{little\_set}(z) \ \Rightarrow \ z\in \mbox{universal\_set}\qquad\mbox{cnf}(\mbox{universal\_set},\mbox{axiom})$

$( x\subseteq y\mbox{ and }u\in x ) \ \Rightarrow \ u\in y\qquad\mbox{cnf}(\mbox{subset}_1,\mbox{axiom})$

$x\subseteq y\mbox{ or }f_{17}(x,y)\in x\qquad\mbox{cnf}(\mbox{subset}_2,\mbox{axiom})$

$f_{17}(x,y)\in y \ \Rightarrow \ x\subseteq y\qquad\mbox{cnf}(\mbox{subset}_3,\mbox{axiom})$

$\mbox{as}\subseteq \mbox{cs}\qquad\mbox{cnf}(\mbox{a\_subset\_of\_c},\mbox{hypothesis})$

$\mbox{bs}\subseteq \mbox{cs}\qquad\mbox{cnf}(\mbox{b\_subset\_of\_c},\mbox{hypothesis})$

$\neg\,\mbox{union}(\mbox{as},\mbox{bs})\subseteq \mbox{cs}\qquad\mbox{cnf}(\mbox{prove\_a\_union\_b\_subset\_of\_c},\mbox{negated\_conjecture})$

\medskip\textbf{SET744+4.p} Problem on composition of mappings 9

Consider three mappings F from A to B,G from B to C,H from
 C to A. If GoF and HoG are one-to-one, then H is one-to-one.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,c,a)\mbox{ and }\mbox{one\_to\_one}(\mbox{compose\_function}(g,f,a,b,c),a,c)\mbox{ and }\mbox{one\_to\_one}(\mbox{compose\_function}(h,g,b,c,a),b,a) ) \ \Rightarrow \ \mbox{one\_to\_one}(h,c,a) )\qquad\mbox{fof}(\mbox{thII}_{35},\mbox{conjecture})$

\medskip\textbf{SET295-6.p} Inverse distributes over intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x',y')\ne \mbox{intersection}(x,y)'\qquad\mbox{cnf}(\mbox{prove\_inverse\_over\_intersection}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET670$\wedge$3.p} R (X to Y) restricted to X1 is (X1 to Y)

A relation R from X to Y restricted to X1 is a relation from X1
 to Y.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall z{:}\;\mbox{\$i}\to \mbox{\$o},r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{is\_rel\_on}@ r@ x@ y ) \ \Rightarrow \ ( \mbox{is\_rel\_on}@ ( \mbox{restrict\_rel\_domain}@ r@ z )@ z@ y ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET014-3.p} Union of subsets is a subset

If A and B are contained in C then the union of A and B is also.




$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{as}\subseteq \mbox{cs}\qquad\mbox{cnf}(\mbox{a\_subset\_of\_c},\mbox{hypothesis})$

$\mbox{bs}\subseteq \mbox{cs}\qquad\mbox{cnf}(\mbox{b\_subset\_of\_c},\mbox{hypothesis})$

$\neg\,\mbox{union}(\mbox{as},\mbox{bs})\subseteq \mbox{cs}\qquad\mbox{cnf}(\mbox{prove\_a\_union\_b\_subset\_of\_c},\mbox{negated\_conjecture})$

\medskip\textbf{SET836-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET152-6.p} Complement of universal class is null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{universal\_class}'\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_complement\_of\_universal\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET251-6.p} Restriction of element relation, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_restriction\_of\_element\_relation}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{restrict}(\mbox{element\_relation},x,\mbox{singleton}(y)))\ne \mbox{intersection}(x,y)\qquad\mbox{cnf}(\mbox{prove\_restriction\_of\_element\_relation}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET240-6.p} Restriction alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{restrict}(\mbox{xr},x,y)\qquad\mbox{cnf}(\mbox{prove\_restriction\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\neg\,u\in x\qquad\mbox{cnf}(\mbox{prove\_restriction\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET822-2.p} Problem about set theory






$\neg\,\mbox{c\_in}(\mbox{v\_a},\mbox{c\_emptyset},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OemptyE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{c\_insert}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{v\_a} = \mbox{v\_b} )\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertE}_0,\mbox{axiom})$

$\mbox{v\_P}(\mbox{v\_f}(\mbox{v\_b}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_f}(\mbox{v\_U}),\mbox{v\_V},\mbox{tc\_IntDef\_Oint}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U},\mbox{v\_V}),\mbox{v\_V},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_f}(\mbox{v\_U}),\mbox{v\_V},\mbox{tc\_IntDef\_Oint}) \ \Rightarrow \ \neg\,\mbox{v\_P}(\mbox{v\_x}(\mbox{v\_U},\mbox{v\_V}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET447-6.p} Function property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(y)\qquad\mbox{cnf}(\mbox{prove\_function\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_function\_property3}_2,\mbox{negated\_conjecture})$

$\mbox{restrict}(y,\mbox{domain\_of}(x),\mbox{universal\_class})\ne x\qquad\mbox{cnf}(\mbox{prove\_function\_property3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET143-6.p} Associativity of intersection

The intersection of (the intersection of X and Y) and Z is the
 intersection of X and (the intersection of Y and Z).




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{intersection}(x,y),z)\ne \mbox{intersection}(x,\mbox{intersection}(y,z))\qquad\mbox{cnf}(\mbox{prove\_associativity\_of\_intersection}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET025+1.p} An ordered pair is a set






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;\mbox{ordered\_pair}(x,y)\in \mbox{universal\_class}\qquad\mbox{fof}(\mbox{ordered\_pair\_is\_set},\mbox{conjecture})$

\medskip\textbf{SET957+1.p} Basic properties of sets, theorem 110






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

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

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

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

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

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

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

$\forall a,b,c,d{:}\;\neg\,a\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }\mbox{in}(d,a)\mbox{ and }\forall e,f{:}\;\neg\,\mbox{in}(e,b)\mbox{ and }\mbox{in}(f,c)\mbox{ and }d = \mbox{ordered\_pair}(e,f)\qquad\mbox{fof}(\mbox{t103\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d,e,f{:}\;( ( a\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }d\subseteq \mbox{cartesian\_product}_2(e,f)\mbox{ and }\forall g,h{:}\;( \mbox{in}(\mbox{ordered\_pair}(g,h),a)\iff \mbox{in}(\mbox{ordered\_pair}(g,h),d) ) ) \ \Rightarrow \ a = d )\qquad\mbox{fof}(\mbox{t110\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

\medskip\textbf{SET615$\wedge$5.p} TPS problem BOOL-PROP-89

Trybulec's 89th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} )\mbox{ and }\neg\,z@ \mbox{xx} ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,z@ \mbox{xz} )\mbox{ or }( y@ \mbox{xz}\mbox{ and }\neg\,z@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_89\_pme},\mbox{conjecture})$

\medskip\textbf{SET256-6.p} Monotonicity of restriction 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(z_1,z_2)\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_restriction3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{restrict}(x,y,z_1),\mbox{restrict}(x,y,z_2))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_restriction3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET372+4.p} Power set distributes over intersection

The power\_set of the intersection of A and B is equal to the
 intersection of the power\_set of A and the power\_set of B.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;\mbox{equal\_set}(\mbox{power\_set}(\mbox{intersection}(a,b)),\mbox{intersection}(\mbox{power\_set}(a),\mbox{power\_set}(b)))\qquad\mbox{fof}(\mbox{thI}_{21},\mbox{conjecture})$

\medskip\textbf{SET832-2.p} Problem about set theory






$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_V},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_V},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET598$\wedge$5.p} TPS problem BOOL-PROP-57

Trybulec's 57th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( x = ( \lambda \mbox{xx}{:}\;a{:}\;( y@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) )\iff ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) )\mbox{ and }\forall v{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( v@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( v@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( v@ \mbox{xx} ) \ \Rightarrow \ ( x@ \mbox{xx} ) ) ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_57\_pme},\mbox{conjecture})$

\medskip\textbf{SET408-6.p} Inverse of composition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$( \mbox{xr}\circ \mbox{yr} )'\ne \mbox{yr}'\circ \mbox{xr}'\qquad\mbox{cnf}(\mbox{prove\_inverse\_of\_composition}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET120+1.p} Corollary 2 to components of equal ordered pairs being equal






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( x\in \mbox{universal\_class}\mbox{ or }\mbox{second}(\mbox{ordered\_pair}(x,y)) = \mbox{ordered\_pair}(x,y) )\qquad\mbox{fof}(\mbox{corollary\_2\_to\_OP\_determines\_components}_1,\mbox{conjecture})$

\medskip\textbf{SET010$\wedge$5.p} TPS problem BOOL-PROP-86

Trybulec's 86th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} )\mbox{ or }( x@ \mbox{xz}\mbox{ and }\neg\,z@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_86\_pme},\mbox{conjecture})$

\medskip\textbf{SET948+1.p} Basic properties of sets, theorem 101

((in(C,union(A,B)) \& in(D,union(A,B)) ) =$>$ (C=D | disjoint(C,D)) )
 =$>$ union(intersection(A,B)) = intersection(union(A),union(B))




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

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

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

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

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

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

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

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

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

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

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

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

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

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

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

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

$\forall a,b{:}\;( \forall c,d{:}\;( ( \mbox{in}(c,\mbox{set\_union}_2(a,b))\mbox{ and }\mbox{in}(d,\mbox{set\_union}_2(a,b)) ) \ \Rightarrow \ ( c = d\mbox{ or }\mbox{disjoint}(c,d) ) ) \ \Rightarrow \ \mbox{union}(\mbox{set\_intersection}_2(a,b)) = \mbox{set\_intersection}_2(\mbox{union}(a),\mbox{union}(b)) )\qquad\mbox{fof}(\mbox{t101\_zfmisc}_1,\mbox{conjecture})$

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

$\forall a,b{:}\;\mbox{union}(\mbox{set\_intersection}_2(a,b))\subseteq \mbox{set\_intersection}_2(\mbox{union}(a),\mbox{union}(b))\qquad\mbox{fof}(\mbox{t97\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET496-6.p} Diagonalization alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\neg\,z\in \mbox{diagonalise}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(z,z)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_alternate\_defn3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET431-6.p} The composition of single-valued classes is single-valued






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_compose\_single\_valued\_classes}_1,\mbox{negated\_conjecture})$

$\mbox{single\_valued\_class}(\mbox{yr})\qquad\mbox{cnf}(\mbox{prove\_compose\_single\_valued\_classes}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{single\_valued\_class}(\mbox{xr}\circ \mbox{yr})\qquad\mbox{cnf}(\mbox{prove\_compose\_single\_valued\_classes}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET028-4.p} Relationship between apply and image, part 1 of 2






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\neg\,\mbox{apply}(\mbox{a\_function},\mbox{element})\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(\mbox{element}),\mbox{a\_function}))\qquad\mbox{cnf}(\mbox{prove\_property\_of\_image\_and\_apply}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET008-1.p} (X \ Y) $\wedge$ Y = the empty set

The difference of two sets contains no members of the
 subtracted set.




$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-3.ax'})$

$b\setminus a{=}\mbox{bDa}\qquad\mbox{cnf}(\mbox{b\_minus\_a},\mbox{hypothesis})$

$\neg\,\mbox{intersection}(a,\mbox{bDa},\mbox{aI\_bDa})\qquad\mbox{cnf}(\mbox{a\_intersection\_bDa},\mbox{negated\_conjecture})$

$\neg\,a\in \mbox{aI\_bDa}\qquad\mbox{cnf}(\mbox{prove\_aI\_bDa\_is\_empty},\mbox{negated\_conjecture})$

\medskip\textbf{SET171$\wedge$5.p} TPS problem BOOL-PROP-71

Trybulec's 71st Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }( y@ \mbox{xz}\mbox{ and }z@ \mbox{xz} ) ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} )\mbox{ and }( x@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_71\_pme},\mbox{conjecture})$

\medskip\textbf{SET799+4.p} Least upper bounds of set in total order

If an order is total, the least upper bound of a set is less
 than the least upper bound of a subset of it.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e{:}\;( \mbox{order}(r,e) \ \Rightarrow \ \forall x_1,x_2{:}\;( ( x_1\subseteq e\mbox{ and }x_2\subseteq e\mbox{ and }x_1\subseteq x_2 ) \ \Rightarrow \ \forall m_1,m_2{:}\;( ( \mbox{least\_upper\_bound}(m_1,x_1,r,e)\mbox{ and }\mbox{least\_upper\_bound}(m_2,x_2,r,e) ) \ \Rightarrow \ \mbox{apply}(r,m_1,m_2) ) ) )\qquad\mbox{fof}(\mbox{thIV}_{11},\mbox{conjecture})$

\medskip\textbf{SET796+4.p} If R(a,b) then a is the greatest lower bound of unordered\_pair(a,b)






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e,a,b{:}\;( ( \mbox{order}(r,e)\mbox{ and }a\in e\mbox{ and }b\in e\mbox{ and }\mbox{apply}(r,a,b) ) \ \Rightarrow \ \mbox{greatest\_lower\_bound}(a,\mbox{unordered\_pair}(a,b),r,e) )\qquad\mbox{fof}(\mbox{thIV}_8,\mbox{conjecture})$

\medskip\textbf{SET534-6.p} The range of Z is the class of applications of Z to Z's domain 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_class\_of\_applications\_to\_domain2}_1,\mbox{negated\_conjecture})$

$y\in \mbox{range\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_class\_of\_applications\_to\_domain2}_2,\mbox{negated\_conjecture})$

$\mbox{apply}(z,1_{\text{dom}(z)})\ne y\qquad\mbox{cnf}(\mbox{prove\_class\_of\_applications\_to\_domain2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET698+4.p} Property of union and difference

A is a subset of B if and only if the union of A and of the
 difference of B in E is equal to E.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,e{:}\;( ( a\subseteq e\mbox{ and }b\subseteq e ) \ \Rightarrow \ ( a\subseteq b\iff \mbox{equal\_set}(\mbox{union}(e\setminus a,b),e) ) )\qquad\mbox{fof}(\mbox{thI}_{32},\mbox{conjecture})$

\medskip\textbf{SET081-6.p} Only the element can belong to its singleton






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$y\in \mbox{singleton}(x)\qquad\mbox{cnf}(\mbox{prove\_only\_member\_in\_singleton}_1,\mbox{negated\_conjecture})$

$y\ne x\qquad\mbox{cnf}(\mbox{prove\_only\_member\_in\_singleton}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET600+3.p} X U Y = empty set iff X = empty set \& Y = empty set

The union of X and Y is the empty set iff X is the empty set
 and Y is the empty set.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{union}(b,c) = \mbox{empty\_set}\iff ( b = \mbox{empty\_set}\mbox{ and }c = \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{prove\_th}_{59},\mbox{conjecture})$

\medskip\textbf{SET721+4.p} If the composition of F and G is injective, then F is injective






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{injective}(\mbox{compose\_function}(g,f,a,b,c),a,c) ) \ \Rightarrow \ \mbox{injective}(f,a,b) )\qquad\mbox{fof}(\mbox{thII}_{12},\mbox{conjecture})$

\medskip\textbf{SET023-3.p} The second component of an ordered pair is a little set






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(a)\qquad\mbox{cnf}(\mbox{an\_ordered\_pair\_predicate},\mbox{hypothesis})$

$\neg\,\mbox{little\_set}(\mbox{second}(a))\qquad\mbox{cnf}(\mbox{prove\_second\_component\_is\_small},\mbox{negated\_conjecture})$

\medskip\textbf{SET705+4.p} A is a member of power\_set(A)






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a{:}\;a\in \mbox{power\_set}(a)\qquad\mbox{fof}(\mbox{thI}_{48},\mbox{conjecture})$

\medskip\textbf{SET042-4.p} Ordered pairs are in cross products






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$a\in \mbox{set\_a}\qquad\mbox{cnf}(\mbox{member\_of\_set\_a},\mbox{hypothesis})$

$b\in \mbox{set\_b}\qquad\mbox{cnf}(\mbox{member\_of\_set\_b},\mbox{hypothesis})$

$\neg\,\mbox{ordered\_pair}(a,b)\in \mbox{cross\_product}(\mbox{set\_a},\mbox{set\_b})\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_is\_in\_cross\_product},\mbox{negated\_conjecture})$

\medskip\textbf{SET920+1.p} intersection(uno\_pair(A,B),C) = uno\_pair(A,B) =$>$ in(A,C)






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

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

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

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

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

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

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

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

$\forall a,b,c{:}\;( \mbox{set\_intersection}_2(\mbox{unordered\_pair}(a,b),c) = \mbox{unordered\_pair}(a,b) \ \Rightarrow \ \mbox{in}(a,c) )\qquad\mbox{fof}(\mbox{t63\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET148+3.p} Idempotency of intersection






$\forall b,c{:}\;( b\subseteq c \ \Rightarrow \ \mbox{intersection}(b,c) = b )\qquad\mbox{fof}(\mbox{subset\_intersection},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b{:}\;\mbox{intersection}(b,b) = b\qquad\mbox{fof}(\mbox{prove\_idempotency\_of\_intersection},\mbox{conjecture})$

\medskip\textbf{SET930+1.p} Basic properties of sets, theorem 74

~ ( difference(unordered\_pair(A,B),C) != empty\_set
 \& difference(unordered\_pair(A,B),C) != singleton(A)
 \& difference(unordered\_pair(A,B),C) != singleton(B)
 \& difference(unordered\_pair(A,B),C) != unordered\_pair(A,B) )




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

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

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

$\forall a,b,c{:}\;( \mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c) = \mbox{singleton}(a)\iff ( \neg\,\mbox{in}(a,c)\mbox{ and }( \mbox{in}(b,c)\mbox{ or }a = b ) ) )\qquad\mbox{fof}(\mbox{l39\_zfmisc}_1,\mbox{axiom})$

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

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

$\forall a,b,c{:}\;( \mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c) = \mbox{unordered\_pair}(a,b)\iff ( \neg\,\mbox{in}(a,c)\mbox{ and }\neg\,\mbox{in}(b,c) ) )\qquad\mbox{fof}(\mbox{t72\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c) = \mbox{empty\_set}\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,c) ) )\qquad\mbox{fof}(\mbox{t73\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c)\ne \mbox{empty\_set}\mbox{ and }\mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c)\ne \mbox{singleton}(a)\mbox{ and }\mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c)\ne \mbox{singleton}(b)\mbox{ and }\mbox{set\_difference}(\mbox{unordered\_pair}(a,b),c)\ne \mbox{unordered\_pair}(a,b)\qquad\mbox{fof}(\mbox{t74\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET758+4.p} The image of the inverse image of Y is a subset of Y






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,y{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }y\subseteq b ) \ \Rightarrow \ \mbox{image}_3(f,\mbox{inverse\_image}_3(f,y,a),b)\subseteq y )\qquad\mbox{fof}(\mbox{thIIa}_{08},\mbox{conjecture})$

\medskip\textbf{SET814+4.p} The sum of an ordinal number is a subset of itself






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+4.ax'})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{thI}_3,\mbox{axiom})$

$\forall a{:}\;( a\in \mbox{on} \ \Rightarrow \ \mbox{\mbox{sum}}(a)\subseteq a )\qquad\mbox{fof}(\mbox{thV}_{14},\mbox{conjecture})$

\medskip\textbf{SET014-2.p} Union of subsets is a subset

If A and B are contained in C then the union of A and B is also.




$\mbox{include}(\mbox{'Axioms/SET002-0.ax'})$

$\mbox{as}\subseteq \mbox{cs}\qquad\mbox{cnf}(\mbox{a\_subset\_of\_c},\mbox{hypothesis})$

$\mbox{bs}\subseteq \mbox{cs}\qquad\mbox{cnf}(\mbox{b\_subset\_of\_c},\mbox{hypothesis})$

$\neg\,\mbox{union}(\mbox{as},\mbox{bs})\subseteq \mbox{cs}\qquad\mbox{cnf}(\mbox{prove\_a\_union\_b\_subset\_of\_c},\mbox{negated\_conjecture})$

\medskip\textbf{SET451-6.p} Subset relation alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{subset\_relation},\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_subset\_relation\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET328-6.p} Corollary to image alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{image}(\mbox{xr},\mbox{singleton}(x))\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_image\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_image\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(x,y)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_image\_alternate\_defn3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET801+4.p} M is the greatest element iff it is a member and a LUB






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e{:}\;( \mbox{order}(r,e) \ \Rightarrow \ \forall x{:}\;( x\subseteq e \ \Rightarrow \ \forall m{:}\;( \mbox{greatest}(m,r,x)\iff ( m\in x\mbox{ and }\mbox{least\_upper\_bound}(m,x,r,e) ) ) ) )\qquad\mbox{fof}(\mbox{thIV}_{13},\mbox{conjecture})$

\medskip\textbf{SET024-6.p} A set belongs to its singleton






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_set\_in\_its\_singleton}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{singleton}(x)\qquad\mbox{cnf}(\mbox{prove\_set\_in\_its\_singleton}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET072-6.p} Right cancellation for unordered pairs






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{unordered\_pair}(x,z) = \mbox{unordered\_pair}(y,z)\qquad\mbox{cnf}(\mbox{prove\_right\_cancellation}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_right\_cancellation}_2,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_right\_cancellation}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET969+1.p} Basic properties of sets, theorem 122






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

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

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

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

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

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d)) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(b,a),\mbox{cartesian\_product}_2(d,c)) )\qquad\mbox{fof}(\mbox{t107\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d,e,f{:}\;( ( a\subseteq \mbox{cartesian\_product}_2(b,c)\mbox{ and }d\subseteq \mbox{cartesian\_product}_2(e,f)\mbox{ and }\forall g,h{:}\;( \mbox{in}(\mbox{ordered\_pair}(g,h),a)\iff \mbox{in}(\mbox{ordered\_pair}(g,h),d) ) ) \ \Rightarrow \ a = d )\qquad\mbox{fof}(\mbox{t110\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{cartesian\_product}_2(\mbox{set\_intersection}_2(a,b),c) = \mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,c))\mbox{ and }\mbox{cartesian\_product}_2(c,\mbox{set\_intersection}_2(a,b)) = \mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(c,a),\mbox{cartesian\_product}_2(c,b)) )\qquad\mbox{fof}(\mbox{t122\_zfmisc}_1,\mbox{conjecture})$

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

\medskip\textbf{SET194$\wedge$5.p} TPS problem BOOL-PROP-31

Trybulec's 31st Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_31\_pme},\mbox{conjecture})$

\medskip\textbf{SET923+1.p} ~ ( difference(A,sgtn(B)) = empty \& A != empty \& A != sgtn(B) )






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

$\forall a,b{:}\;( a\subseteq \mbox{singleton}(b)\iff ( a = \mbox{empty\_set}\mbox{ or }a = \mbox{singleton}(b) ) )\qquad\mbox{fof}(\mbox{l4\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b{:}\;( \mbox{set\_difference}(a,b) = \mbox{empty\_set}\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t37\_xboole}_1,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{set\_difference}(a,\mbox{singleton}(b)) = \mbox{empty\_set}\mbox{ and }a\ne \mbox{empty\_set}\mbox{ and }a\ne \mbox{singleton}(b)\qquad\mbox{fof}(\mbox{t66\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET826-2.p} Problem about set theory






$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_V},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_V},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET848-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_OsuccI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_chain\_\_lemma4}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{v\_m} = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oequal\_\_succ\_\_Union}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oequal\_\_succ\_\_Union}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_minus}(\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_c},\mbox{t\_a})\ne \mbox{v\_c}\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_not\_\_equals}_0,\mbox{axiom})$

$\neg\,\mbox{c\_in}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}) = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET619$\wedge$5.p} TPS problem BOOL-PROP-95

Trybulec's 95th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }y@ \mbox{xz} ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} )\mbox{ or }( y@ \mbox{xz}\mbox{ and }\neg\,x@ \mbox{xz} )\mbox{ or }( x@ \mbox{xz}\mbox{ and }y@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_95\_pme},\mbox{conjecture})$

\medskip\textbf{SET388-6.p} Composition alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{xf}\circ \mbox{yf}\qquad\mbox{cnf}(\mbox{prove\_composition\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,1_{\text{dom}(\mbox{xf})})\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_composition\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET747+4.p} If F is increasing and G decreasing, then GoF is decreasing






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c,r,s,t{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{increasing}(f,a,r,b,s)\mbox{ and }\mbox{decreasing}(g,b,s,c,t) ) \ \Rightarrow \ \mbox{decreasing}(\mbox{compose\_function}(g,f,a,b,c),a,r,c,t) )\qquad\mbox{fof}(\mbox{thII}_{38},\mbox{conjecture})$

\medskip\textbf{SET181-6.p} Corollary to distribution property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x',\mbox{union}(y,\mbox{intersection}(x,z)))\ne \mbox{union}(x',\mbox{union}(y,z))\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_distribution\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET059-7.p} Expanded equality definition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y\qquad\mbox{cnf}(\mbox{prove\_equality4}_1,\mbox{negated\_conjecture})$

$\mbox{not\_subclass\_element}(y,x)\in x\qquad\mbox{cnf}(\mbox{prove\_equality4}_2,\mbox{negated\_conjecture})$

$x\ne y\qquad\mbox{cnf}(\mbox{prove\_equality4}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET757+4.p} Inverse image intersection equals intersection inverse images






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,x,y{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }x\subseteq b\mbox{ and }y\subseteq b ) \ \Rightarrow \ \mbox{equal\_set}(\mbox{inverse\_image}_3(f,\mbox{intersection}(x,y),a),\mbox{intersection}(\mbox{inverse\_image}_3(f,x,a),\mbox{inverse\_image}_3(f,y,a))) )\qquad\mbox{fof}(\mbox{thIIa}_{07},\mbox{conjecture})$

\medskip\textbf{SET626+3.p} If X intersects the intersection of Y and Z, then X intersects Y






$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( \mbox{intersect}(b,\mbox{intersection}(c,d)) \ \Rightarrow \ \mbox{intersect}(b,c) )\qquad\mbox{fof}(\mbox{prove\_th}_{102},\mbox{conjecture})$

\medskip\textbf{SET070-7.p} Proper class in an unordered pair, part 4

If one argument of an unordered pair is a proper class, the
 pair contains only the other.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{special\_classes\_lemma},\mbox{axiom})$

$\neg\,z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{existence\_of\_null\_class},\mbox{axiom})$

$\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{cnf}(\mbox{null\_class\_is\_subclass},\mbox{axiom})$

$\mbox{subclass}(x,\mbox{null\_class}) \ \Rightarrow \ x = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{corollary\_of\_null\_class\_is\_subclass},\mbox{axiom})$

$z = \mbox{null\_class}\mbox{ or }\mbox{not\_subclass\_element}(z,\mbox{null\_class})\in z\qquad\mbox{cnf}(\mbox{null\_class\_is\_unique},\mbox{axiom})$

$\mbox{null\_class}\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{null\_class\_is\_a\_set},\mbox{axiom})$

$\mbox{unordered\_pair}(x,y) = \mbox{unordered\_pair}(y,x)\qquad\mbox{cnf}(\mbox{commutativity\_of\_unordered\_pair},\mbox{axiom})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_equals\_singleton2}_1,\mbox{negated\_conjecture})$

$\mbox{unordered\_pair}(x,y)\ne \mbox{singleton}(y)\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_equals\_singleton2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET510-6.p} Corollary to singleton is null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{singleton}(\mbox{universal\_class})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_singleton\_is\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET044+1.p} Anti-Russell Sets

If there were an anti-Russell set (a set that contains exactly
 those sets that are members of themselves), then not every set
 has a complement.




$\exists y{:}\;\forall x{:}\;( \mbox{element}(x,y)\iff \mbox{element}(x,x) ) \ \Rightarrow \ \neg\,\forall x_1{:}\;\exists y_1{:}\;\forall z{:}\;( \mbox{element}(z,y_1)\iff \neg\,\mbox{element}(z,x_1) )\qquad\mbox{fof}(\mbox{pel}_{40},\mbox{conjecture})$

\medskip\textbf{SET157-6.p} Complement is unique






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x,y) = \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_complement\_is\_unique}_1,\mbox{negated\_conjecture})$

$\mbox{intersection}(x,y) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_complement\_is\_unique}_2,\mbox{negated\_conjecture})$

$x'\ne y\qquad\mbox{cnf}(\mbox{prove\_complement\_is\_unique}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET827-2.p} Problem about set theory






$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_V},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_V},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET213-6.p} Corollary 4 to cross product product monotonicity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,z),\mbox{cross\_product}(x,\mbox{union}(y,z)))\qquad\mbox{cnf}(\mbox{prove\_corollary\_4\_to\_X\_product\_monotonicity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET968+1.p} Basic properties of sets, theorem 121






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

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

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

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

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

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

$\forall a,b,c{:}\;( \mbox{cartesian\_product}_2(\mbox{set\_union}_2(a,b),c) = \mbox{set\_union}_2(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,c))\mbox{ and }\mbox{cartesian\_product}_2(c,\mbox{set\_union}_2(a,b)) = \mbox{set\_union}_2(\mbox{cartesian\_product}_2(c,a),\mbox{cartesian\_product}_2(c,b)) )\qquad\mbox{fof}(\mbox{t120\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;\mbox{cartesian\_product}_2(\mbox{set\_union}_2(a,b),\mbox{set\_union}_2(c,d)) = \mbox{set\_union}_2(\mbox{set\_union}_2(\mbox{set\_union}_2(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(a,d)),\mbox{cartesian\_product}_2(b,c)),\mbox{cartesian\_product}_2(b,d))\qquad\mbox{fof}(\mbox{t121\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;\mbox{set\_union}_2(\mbox{set\_union}_2(a,b),c) = \mbox{set\_union}_2(a,\mbox{set\_union}_2(b,c))\qquad\mbox{fof}(\mbox{t4\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SET396-6.p} Domain of composition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{domain\_of}(\mbox{xr}\circ \mbox{yr}),\mbox{domain\_of}(\mbox{yr}))\qquad\mbox{cnf}(\mbox{prove\_domain\_of\_composition}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET724$\wedge$4.p} If GoF = HoF and F is surjective, then G = H






$\mbox{include}(\mbox{'Axioms/SET008\string^1.ax'})$

$\forall f{:}\;\mbox{\$i}\to \mbox{\$i},g{:}\;\mbox{\$i}\to \mbox{\$i},h{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( ( \mbox{fun\_composition}@ f@ g ) = ( \mbox{fun\_composition}@ f@ h )\mbox{ and }\mbox{fun\_surjective}@ f ) \ \Rightarrow \ g = h )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET182-6.p} Distribution property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{intersection}(x',y),\mbox{intersection}(x,y))\ne y\qquad\mbox{cnf}(\mbox{prove\_distribution\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET752+4.p} The image of union is equal to the union of images






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,x,y{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }x\subseteq a\mbox{ and }y\subseteq a ) \ \Rightarrow \ \mbox{equal\_set}(\mbox{image}_2(f,\mbox{union}(x,y)),\mbox{union}(\mbox{image}_2(f,x),\mbox{image}_2(f,y))) )\qquad\mbox{fof}(\mbox{thIIa}_{02},\mbox{conjecture})$

\medskip\textbf{SET037-4.p} Properties of apply for functions, part 2 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$a\in \mbox{domain\_of}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{member\_of\_function\_domain},\mbox{hypothesis})$

$\mbox{apply}(\mbox{a\_function},a) = b\qquad\mbox{cnf}(\mbox{applying\_the\_function},\mbox{hypothesis})$

$\neg\,\mbox{ordered\_pair}(a,b)\in \mbox{a\_function}\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_in\_function},\mbox{negated\_conjecture})$

\medskip\textbf{SET066-6.p} Unordered pair is commutative






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{unordered\_pair}(x,y)\ne \mbox{unordered\_pair}(y,x)\qquad\mbox{cnf}(\mbox{prove\_commutativity\_of\_unordered\_pair}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET498-6.p} Special case of the Russell class, without the regularity axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_russell\_class2}_1,\mbox{negated\_conjecture})$

$\neg\,z\in \mbox{diagonalise}(\mbox{element\_relation})\qquad\mbox{cnf}(\mbox{prove\_russell\_class2}_2,\mbox{negated\_conjecture})$

$\neg\,z\in z\qquad\mbox{cnf}(\mbox{prove\_russell\_class2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET340-6.p} Subclass alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_subclass\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{image}(\mbox{element\_relation},y')'\qquad\mbox{cnf}(\mbox{prove\_subclass\_alternate\_defn2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET125-6.p} Alternative definition of set builder, part 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$x\in z\qquad\mbox{cnf}(\mbox{prove\_set\_builder\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{set\_builder}(y,z)\qquad\mbox{cnf}(\mbox{prove\_set\_builder\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET739+4.p} Problem on composition of mappings 4

Consider three mappings F from A to B,G from B to C,H from
 C to A. If HoGoF is injective and FoHoG and GoFoH surjective,
 then F is one-to-one.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,c,a)\mbox{ and }\mbox{injective}(\mbox{compose\_function}(h,\mbox{compose\_function}(g,f,a,b,c),a,c,a),a,a)\mbox{ and }\mbox{surjective}(\mbox{compose\_function}(f,\mbox{compose\_function}(h,g,b,c,a),b,a,b),b,b)\mbox{ and }\mbox{surjective}(\mbox{compose\_function}(g,\mbox{compose\_function}(f,h,c,a,b),c,b,c),c,c) ) \ \Rightarrow \ \mbox{one\_to\_one}(f,a,b) )\qquad\mbox{fof}(\mbox{thII}_{30},\mbox{conjecture})$

\medskip\textbf{SET097-6.p} A class contains 0, 1, or at least 2 members






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{not\_subclass\_element}(\mbox{intersection}(\mbox{singleton}(\mbox{not\_subclass\_element}(x,\mbox{null\_class}))',x),\mbox{null\_class})\in \mbox{intersection}(\mbox{singleton}(\mbox{not\_subclass\_element}(x,\mbox{null\_class}))',x)\qquad\mbox{cnf}(\mbox{prove\_number\_of\_elements\_in\_class}_1,\mbox{negated\_conjecture})$

$\mbox{singleton}(\mbox{not\_subclass\_element}(x,\mbox{null\_class}))\ne x\qquad\mbox{cnf}(\mbox{prove\_number\_of\_elements\_in\_class}_2,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_number\_of\_elements\_in\_class}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET375-6.p} Power class is closed under intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{power\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_under\_intersection}_1,\mbox{negated\_conjecture})$

$y\in \mbox{power\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_under\_intersection}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{intersection}(x,y)\in \mbox{power\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_under\_intersection}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET503-6.p} The universal class not set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{universal\_class}\in x\qquad\mbox{cnf}(\mbox{prove\_universal\_class\_not\_set}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET833-2.p} Problem about set theory






$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_V},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_V},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET202-6.p} Cross product property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,y),\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET027-3.p} Transitivity of subset






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$a\subseteq b\qquad\mbox{cnf}(\mbox{a\_subset\_b},\mbox{hypothesis})$

$b\subseteq c\qquad\mbox{cnf}(\mbox{b\_subset\_c},\mbox{hypothesis})$

$\neg\,a\subseteq c\qquad\mbox{cnf}(\mbox{prove\_a\_subset\_c},\mbox{negated\_conjecture})$

\medskip\textbf{SET269-6.p} Domain only considers ordered pairs






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(\mbox{intersection}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})))\ne \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_domain\_does\_ordered\_pairs}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET937+1.p} subset(pset(diff(A,B)),union(sgtn(empty),diff(pset(A),pset(B))))






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

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

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

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

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

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

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

$\forall a,b,c{:}\;( c = \mbox{set\_difference}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\neg\,\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d4\_xboole}_0,\mbox{axiom})$

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

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

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

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

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

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

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

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

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

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

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

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_intersection}_2(a,b) = a )\qquad\mbox{fof}(\mbox{t28\_xboole}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_difference}(a,b)\subseteq a\qquad\mbox{fof}(\mbox{t36\_xboole}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }\mbox{disjoint}(b,c) ) \ \Rightarrow \ \mbox{disjoint}(a,c) )\qquad\mbox{fof}(\mbox{t63\_xboole}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{disjoint}(\mbox{set\_difference}(a,b),b)\qquad\mbox{fof}(\mbox{t79\_xboole}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{powerset}(\mbox{set\_difference}(a,b))\subseteq \mbox{set\_union}_2(\mbox{singleton}(\mbox{empty\_set}),\mbox{set\_difference}(\mbox{powerset}(a),\mbox{powerset}(b)))\qquad\mbox{fof}(\mbox{t84\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET785+1.p} Equivalence relation axioms for the SET006+0 set theory axioms






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

\medskip\textbf{SET716$\wedge$4.p} The composition of injective mappings is injective






$\mbox{include}(\mbox{'Axioms/SET008\string^1.ax'})$

$\forall f{:}\;\mbox{\$i}\to \mbox{\$i},g{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( ( \mbox{fun\_injective}@ f\mbox{ and }\mbox{fun\_injective}@ g ) \ \Rightarrow \ ( \mbox{fun\_injective}@ ( \mbox{fun\_composition}@ f@ g ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET043$\wedge$5.p} TPS problem RUSSELL1

One form of Russell's Paradox.




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

$\neg\,\exists u{:}\;\mbox{\$i}{:}\;\forall v{:}\;\mbox{\$i}{:}\;( ( \mbox{cE}@ v@ u )\iff \neg\,\mbox{cE}@ v@ v )\qquad\mbox{thf}(\mbox{cRUSSELL}_1,\mbox{conjecture})$

\medskip\textbf{SET821-2.p} Problem about set theory






$\mbox{class\_Orderings\_Oorder}(\mbox{t\_a}) \ \Rightarrow \ \neg\,\mbox{c\_less}(\mbox{v\_x},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Orderings\_Oorder\_\_less\_\_irrefl\_\_iff1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_c},\mbox{c\_uminus}(\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OComplD\_\_dest}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_uminus}(\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OComplI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_a},\mbox{c\_insert}(\mbox{v\_b},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_insert}(\mbox{v\_x},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertCI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_a},\mbox{c\_insert}(\mbox{v\_b},\mbox{v\_A},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_a},\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{v\_a} = \mbox{v\_b} )\qquad\mbox{cnf}(\mbox{cls\_Set\_OinsertE}_0,\mbox{axiom})$

$\mbox{c\_less}(\mbox{v\_a},\mbox{v\_b},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_less}(\mbox{v\_b},\mbox{v\_c},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_b},\mbox{v\_U},\mbox{tc\_IntDef\_Oint}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_U},\mbox{tc\_IntDef\_Oint})\mbox{ or }\mbox{c\_in}(\mbox{v\_a},\mbox{v\_U},\mbox{tc\_IntDef\_Oint}) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{class\_Orderings\_Oorder}(\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{clsarity\_IntDef\_\_Oint}_{31},\mbox{axiom})$

\medskip\textbf{SET372-6.p} Power set distributes over intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(\mbox{power\_class}(x),\mbox{power\_class}(y))\ne \mbox{power\_class}(\mbox{intersection}(x,y))\qquad\mbox{cnf}(\mbox{prove\_power\_class\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET611$\wedge$3.p} X $\wedge$ Y = the empty set iff X \ Y = X

The intersection of X and Y is the empty set iff the difference
 of X and Y is X.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall a{:}\;\mbox{\$i}\to \mbox{\$o},b{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{intersection}@ a@ b ) = \mbox{emptyset}\iff ( \mbox{setminus}@ a@ b ) = a )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET749+4.p} If F and G and decreasing, then GoF is increasing






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c,r,s,t{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{decreasing}(f,a,r,b,s)\mbox{ and }\mbox{decreasing}(g,b,s,c,t) ) \ \Rightarrow \ \mbox{increasing}(\mbox{compose\_function}(g,f,a,b,c),a,r,c,t) )\qquad\mbox{fof}(\mbox{thII}_{40},\mbox{conjecture})$

\medskip\textbf{SET933+1.p} subset(singleton(A),powerset(A))






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

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

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

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

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

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

$\forall a{:}\;\mbox{singleton}(a)\subseteq \mbox{powerset}(a)\qquad\mbox{fof}(\mbox{t80\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET294-6.p} Inverse distributes over union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x',y')\ne \mbox{union}(x,y)'\qquad\mbox{cnf}(\mbox{prove\_inverse\_over\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET043+1.p} Russell's Paradox

Russell's paradox : there is no Russell set (a set which
 contains exactly those sets which are not members
 of themselves).




$\neg\,\exists x{:}\;\forall y{:}\;( \mbox{element}(y,x)\iff \neg\,\mbox{element}(y,y) )\qquad\mbox{fof}(\mbox{pel}_{39},\mbox{conjecture})$

\medskip\textbf{SET262-6.p} Domain of universal class is the universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(\mbox{universal\_class})\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_domain\_of\_universal\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET360-6.p} Domain is sum squared






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{domain\_of}(x),\mbox{sum\_class}(\mbox{sum\_class}(x)))\qquad\mbox{cnf}(\mbox{prove\_domain\_is\_sum\_squared}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET303-6.p} Range alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{range\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_range\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(1_{\text{dom}(\mbox{xr})},y)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_range\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET740+4.p} Problem on composition of mappings 5

Consider three mappings F from A to B,G from B to C,H from
 C to A. If HoGoF is injective and FoHoG and GoFoH surjective,
 then G is one-to-one.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,c,a)\mbox{ and }\mbox{injective}(\mbox{compose\_function}(h,\mbox{compose\_function}(g,f,a,b,c),a,c,a),a,a)\mbox{ and }\mbox{surjective}(\mbox{compose\_function}(f,\mbox{compose\_function}(h,g,b,c,a),b,a,b),b,b)\mbox{ and }\mbox{surjective}(\mbox{compose\_function}(g,\mbox{compose\_function}(f,h,c,a,b),c,b,c),c,c) ) \ \Rightarrow \ \mbox{one\_to\_one}(g,b,c) )\qquad\mbox{fof}(\mbox{thII}_{31},\mbox{conjecture})$

\medskip\textbf{SET562-6.p} Compatible function property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{operation}(\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property2}_0,\mbox{negated\_conjecture})$

$\mbox{compatible}(\mbox{xh},\mbox{xf}_1,\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property2}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{domain\_of}(\mbox{xh}),\mbox{domain\_of}(\mbox{xh}))\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(\mbox{apply}(\mbox{xh},x),\mbox{apply}(\mbox{xh},y))\in \mbox{domain\_of}(\mbox{xf}_2)\qquad\mbox{cnf}(\mbox{prove\_compatible\_function\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET394-6.p} Composition property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{restrict}(\mbox{identity\_relation},\mbox{domain\_of}(x),\mbox{universal\_class}),x'\circ x)\qquad\mbox{cnf}(\mbox{prove\_composition\_property2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET735+4.p} Property of mappings

If GoF1 and GoF2 are identities,and the images of A by F1
 and F2 are equal, then F1 and F2 are equal.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f_1,f_2,g,a,b{:}\;( ( \mbox{maps}(f_1,a,b)\mbox{ and }\mbox{maps}(f_2,a,b)\mbox{ and }\mbox{maps}(g,b,a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(g,f_1,a,b,a),a)\mbox{ and }\mbox{identity}(\mbox{compose\_function}(g,f_2,a,b,a),a)\mbox{ and }\mbox{equal\_set}(\mbox{image}_3(f_1,a,b),\mbox{image}_3(f_2,a,b)) ) \ \Rightarrow \ \mbox{equal\_maps}(f_1,f_2,a,b) )\qquad\mbox{fof}(\mbox{thII}_{26},\mbox{conjecture})$

\medskip\textbf{SET594+3.p} If X $\wedge$ Y U X $\wedge$ Z = X, then X (= Y U Z

If the intersection of X and the union of Y and the
 intersection of X and Z is X, then X is a subset of the union
 of Y and Z.




$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( \mbox{union}(\mbox{intersection}(b,c),\mbox{intersection}(b,d)) = b \ \Rightarrow \ b\subseteq \mbox{union}(c,d) )\qquad\mbox{fof}(\mbox{prove\_th}_{53},\mbox{conjecture})$

\medskip\textbf{SET599$\wedge$5.p} TPS problem BOOL-PROP-58

Trybulec's 58th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},\mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) \ \Rightarrow \ ( ( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} )\mbox{ or }( y@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_58\_pme},\mbox{conjecture})$

\medskip\textbf{SET795+4.p} If R(a,b) then b is the least upper bound of unordered\_pair(a,b)






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e,a,b{:}\;( ( \mbox{order}(r,e)\mbox{ and }a\in e\mbox{ and }b\in e\mbox{ and }\mbox{apply}(r,a,b) ) \ \Rightarrow \ \mbox{least\_upper\_bound}(b,\mbox{unordered\_pair}(a,b),r,e) )\qquad\mbox{fof}(\mbox{thIV}_7,\mbox{conjecture})$

\medskip\textbf{SET285-6.p} Domain property 7






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_domain\_property7}_1,\mbox{negated\_conjecture})$

$\mbox{domain\_of}(\mbox{singleton}(\mbox{ordered\_pair}(x,y)))\ne \mbox{singleton}(x)\qquad\mbox{cnf}(\mbox{prove\_domain\_property7}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET170-6.p} Distribution of intersection over union 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{intersection}(x,z),\mbox{intersection}(y,z))\ne \mbox{intersection}(\mbox{union}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_intersection\_over\_union2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET162+4.p} The union of a set and empty set is equal to the set






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a{:}\;\mbox{equal\_set}(\mbox{union}(a,\mbox{empty\_set}),a)\qquad\mbox{fof}(\mbox{thI}_{18},\mbox{conjecture})$

\medskip\textbf{SET192-6.p} Subclass property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_property9}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(y',x')\qquad\mbox{cnf}(\mbox{prove\_subclass\_property9}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET781-2.p} Set theory axioms based on Godel set theory






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

\medskip\textbf{SET029-3.p} Relationship between apply and image, part 2 of 2






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\neg\,\mbox{image}(\mbox{singleton\_set}(\mbox{element}),\mbox{a\_function})\subseteq \mbox{apply}(\mbox{a\_function},\mbox{element})\qquad\mbox{cnf}(\mbox{prove\_image\_and\_apply}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET435-6.p} Function alternate definition 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn4}_1,\mbox{negated\_conjecture})$

$\mbox{single\_valued}_1(x) = \mbox{single\_valued}_2(x)\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn4}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn4}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET297-6.p} Range of inverse






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{range\_of}(x')\ne \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_range\_of\_inverse}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET233-6.p} Cross product property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property8}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{second}(\mbox{not\_subclass\_element}(\mbox{cross\_product}(x,y),z))\in y\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property8}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET695+4.p} Difference of subsets

A is a subset of B if and only if the difference of B is
 a subset of the difference of A.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,e{:}\;( ( a\subseteq e\mbox{ and }b\subseteq e ) \ \Rightarrow \ ( a\subseteq b\iff ( e\setminus b )\subseteq ( e\setminus a ) ) )\qquad\mbox{fof}(\mbox{thI}_{24},\mbox{conjecture})$

\medskip\textbf{SET437-6.p} Subclass of function is a function, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_subclass\_of\_function2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{intersection}(y,x))\qquad\mbox{cnf}(\mbox{prove\_subclass\_of\_function2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET578+3.p} Trybulec's 19th Boolean property of sets






$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( \forall e{:}\;( e\in b\iff ( e\in c\mbox{ and }e\in d ) ) \ \Rightarrow \ b = \mbox{intersection}(c,d) )\qquad\mbox{fof}(\mbox{prove\_th}_{19},\mbox{conjecture})$

\medskip\textbf{SET380-6.p} Relation property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{restrict}(\mbox{xf},x,y),\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_relation\_property1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET471-6.p} Sets with more than one member 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$u\in x\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_many\_members1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{not\_subclass\_element}(\mbox{intersection}(x,\mbox{singleton}(u)'),\mbox{null\_class})\in x\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_many\_members1}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,x),\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_many\_members1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET020+1.p} Uniqueness of 1st and 2nd when X is an ordered pair of sets






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall u,v,x{:}\;( ( u\in \mbox{universal\_class}\mbox{ and }v\in \mbox{universal\_class}\mbox{ and }x = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ ( \mbox{first}(x) = u\mbox{ and }\mbox{second}(x) = v ) )\qquad\mbox{fof}(\mbox{unique\_1st\_and\_2nd\_in\_pair\_of\_sets}_1,\mbox{conjecture})$

\medskip\textbf{SET433-6.p} Function alternate definition 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(\mbox{single\_valued}_3(x),\mbox{single\_valued}_1(x))\in x\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET013+4.p} Commutativity of intersection






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;\mbox{equal\_set}(\mbox{intersection}(a,b),\mbox{intersection}(b,a))\qquad\mbox{fof}(\mbox{thI}_{06},\mbox{conjecture})$

\medskip\textbf{SET492-6.p} Diagonalization lemma 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{domain\_of}(\mbox{intersection}(y,\mbox{identity\_relation}))\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_lemma2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(x,x)\in y\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_lemma2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET012-4.p} Complement is an involution






$x\in y \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(a_2,\mbox{axiom})$

$\mbox{little\_set}(f_1(x,y))\mbox{ or }x = y\qquad\mbox{cnf}(\mbox{extensionality}_1,\mbox{axiom})$

$f_1(x,y)\in x\mbox{ or }f_1(x,y)\in y\mbox{ or }x = y\qquad\mbox{cnf}(\mbox{extensionality}_2,\mbox{axiom})$

$( f_1(x,y)\in x\mbox{ and }f_1(x,y)\in y ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{extensionality}_3,\mbox{axiom})$

$z\in x' \ \Rightarrow \ \neg\,z\in x\qquad\mbox{cnf}(\mbox{complement}_1,\mbox{axiom})$

$\mbox{little\_set}(z) \ \Rightarrow \ ( z\in x'\mbox{ or }z\in x )\qquad\mbox{cnf}(\mbox{complement}_2,\mbox{axiom})$

$\neg\,z\in \mbox{empty\_set}\qquad\mbox{cnf}(\mbox{empty\_set},\mbox{axiom})$

$\mbox{little\_set}(z) \ \Rightarrow \ z\in \mbox{universal\_set}\qquad\mbox{cnf}(\mbox{universal\_set},\mbox{axiom})$

$\mbox{as}' = \mbox{bs}\qquad\mbox{cnf}(\mbox{complement\_of\_a\_is\_b},\mbox{hypothesis})$

$\mbox{bs}' = \mbox{cs}\qquad\mbox{cnf}(\mbox{complement\_of\_b\_is\_c},\mbox{hypothesis})$

$\mbox{as}\ne \mbox{cs}\qquad\mbox{cnf}(\mbox{prove\_a\_equals\_c},\mbox{negated\_conjecture})$

\medskip\textbf{SET974+1.p} Basic properties of sets, theorem 127






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

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

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

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

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

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

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

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

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

$\forall a,b,c,d,e{:}\;\neg\,\mbox{in}(a,\mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(b,c),\mbox{cartesian\_product}_2(d,e)))\mbox{ and }\forall f,g{:}\;\neg\,a = \mbox{ordered\_pair}(f,g)\mbox{ and }\mbox{in}(f,\mbox{set\_intersection}_2(b,d))\mbox{ and }\mbox{in}(g,\mbox{set\_intersection}_2(c,e))\qquad\mbox{fof}(\mbox{t104\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( \mbox{disjoint}(a,b)\mbox{ or }\mbox{disjoint}(c,d) ) \ \Rightarrow \ \mbox{disjoint}(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,d)) )\qquad\mbox{fof}(\mbox{t127\_zfmisc}_1,\mbox{conjecture})$

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

\medskip\textbf{SET591$\wedge$5.p} TPS problem BOOL-PROP-50

Trybulec's 50th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx} ) ) \ \Rightarrow \ x = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_50\_pme},\mbox{conjecture})$

\medskip\textbf{SET844-2.p} Problem about Zorn's lemma






$\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_refl}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oeq\_\_succ\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET414-6.p} Composition distributes over union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\circ \mbox{union}(y,z)\ne \mbox{union}(x\circ y,x\circ z)\qquad\mbox{cnf}(\mbox{prove\_composition\_over\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET708+4.p} The composition of mappings is unique






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h_1,h_2,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{compose\_predicate}(h_1,g,f,a,b,c)\mbox{ and }\mbox{compose\_predicate}(h_2,g,f,a,b,c) ) \ \Rightarrow \ \mbox{equal\_maps}(h_1,h_2,a,c) )\qquad\mbox{fof}(\mbox{thII01a},\mbox{conjecture})$

\medskip\textbf{SET838-2.p} Problem about set theory






$\mbox{v\_f}(\mbox{v\_g}(\mbox{v\_x})) = \mbox{v\_x}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_f}(\mbox{v\_g}(\mbox{v\_U})) = \mbox{v\_U} \ \Rightarrow \ \mbox{v\_U} = \mbox{v\_x}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_g}(\mbox{v\_f}(\mbox{v\_U})) = \mbox{v\_U} \ \Rightarrow \ \mbox{v\_g}(\mbox{v\_f}(\mbox{v\_xa}(\mbox{v\_U}))) = \mbox{v\_xa}(\mbox{v\_U})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_xa}(\mbox{v\_U}) = \mbox{v\_U} \ \Rightarrow \ \mbox{v\_g}(\mbox{v\_f}(\mbox{v\_U}))\ne \mbox{v\_U}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET702+4.p} Property of product and intersection

The intersection of product(A) and product(B) is a subset of
 the product of the intersection of A and B.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;\mbox{intersection}(\mbox{\mbox{product}}(a),\mbox{\mbox{product}}(b))\subseteq \mbox{\mbox{product}}(\mbox{intersection}(a,b))\qquad\mbox{fof}(\mbox{thI}_{36},\mbox{conjecture})$

\medskip\textbf{SET630$\wedge$3.p} X $\wedge$ Y is disjoint from X sym\ Y

The intersection of X and Y is disjoint from the symmetric
 difference of X and Y.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( \mbox{misses}@ ( \mbox{intersection}@ x@ y )@ ( \mbox{excl\_union}@ x@ y ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET647$\wedge$3.p} Domain of R (X to Y) a subset of X =$>$ R is (X to range of R)

If the domain of a relation R from X to Y is a subset of X,
 R is a relation from X to the range of R.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \subseteq @ ( \mbox{rel\_domain}@ r )@ x ) \ \Rightarrow \ ( \mbox{sub\_rel}@ r@ ( \mbox{cartesian\_product}@ x@ ( \mbox{rel\_codomain}@ r ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET786-1.p} Peter Andrews Problem THM25






$( \mbox{element}(b,a)\mbox{ and }\mbox{element}(a,b) ) \ \Rightarrow \ \neg\,\mbox{element}(a,\mbox{sk}_1)\qquad\mbox{cnf}(\mbox{thm25}_1,\mbox{negated\_conjecture})$

$\mbox{element}(a,\mbox{sk}_1)\mbox{ or }\mbox{element}(a,\mbox{sk}_2(a))\qquad\mbox{cnf}(\mbox{thm25}_2,\mbox{negated\_conjecture})$

$\mbox{element}(a,\mbox{sk}_1)\mbox{ or }\mbox{element}(\mbox{sk}_2(a),a)\qquad\mbox{cnf}(\mbox{thm25}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET842-2.p} Problem about Zorn's lemma






$\neg\,\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_A},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Main\_OUnionE\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnionE}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_A},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_A},\mbox{c\_Main\_OUnionE\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnionE}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

\medskip\textbf{SET086$\wedge$1.p} A singleton set has a member






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\forall x{:}\;\mbox{\$i}{:}\;\exists y{:}\;\mbox{\$i}{:}\;( \mbox{singleton}@ x@ y )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET788+1.p} Symmetry of equality from set membership






$\forall x,y{:}\;( x{=}y\iff \forall z{:}\;( \mbox{a\_member\_of}(z,x)\iff \mbox{a\_member\_of}(z,y) ) ) \ \Rightarrow \ \forall x,y{:}\;( x{=}y\iff y{=}x )\qquad\mbox{fof}(\mbox{prove\_this},\mbox{conjecture})$

\medskip\textbf{SET987+1.p} ~ in(A,B) =$>$ difference(union(B,singleton(A)),singleton(A)) = B






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

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

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

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

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

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

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

$\forall a,b{:}\;( \neg\,\mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_difference}(\mbox{set\_union}_2(b,\mbox{singleton}(a)),\mbox{singleton}(a)) = b )\qquad\mbox{fof}(\mbox{t141\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;\mbox{set\_difference}(\mbox{set\_union}_2(a,b),b) = \mbox{set\_difference}(a,b)\qquad\mbox{fof}(\mbox{t40\_xboole}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{set\_difference}(a,\mbox{singleton}(b)) = a\iff \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{t65\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET594$\wedge$5.p} TPS problem BOOL-PROP-53

Trybulec's 53rd Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }y@ \mbox{xz} )\mbox{ or }( x@ \mbox{xz}\mbox{ and }z@ \mbox{xz} ) ) ) = x \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_53\_pme},\mbox{conjecture})$

\medskip\textbf{SET100-6.p} The relationship of singleton sets to ordered pairs






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{unordered\_pair}(x,y)\ne \mbox{union}(\mbox{singleton}(x),\mbox{singleton}(y))\qquad\mbox{cnf}(\mbox{prove\_unordered\_pairs\_and\_singletons}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET456-6.p} Identity alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_identity\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$x = y\qquad\mbox{cnf}(\mbox{prove\_identity\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(x,y)\in \mbox{identity\_relation}\qquad\mbox{cnf}(\mbox{prove\_identity\_alternate\_defn3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET763+4.p} If the image of X is empty then X is empty






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,x{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }x\subseteq a\mbox{ and }\mbox{equal\_set}(\mbox{image}_2(f,x),\mbox{empty\_set}) ) \ \Rightarrow \ \mbox{equal\_set}(x,\mbox{empty\_set}) )\qquad\mbox{fof}(\mbox{thIIa}_{13},\mbox{conjecture})$

\medskip\textbf{SET486-6.p} Replacement property 8






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{sum\_class}(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property8}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property8}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET748+4.p} If F is decreasing and G increasing, then GoF is decreasing






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c,r,s,t{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{decreasing}(f,a,r,b,s)\mbox{ and }\mbox{increasing}(g,b,s,c,t) ) \ \Rightarrow \ \mbox{decreasing}(\mbox{compose\_function}(g,f,a,b,c),a,r,c,t) )\qquad\mbox{fof}(\mbox{thII}_{39},\mbox{conjecture})$

\medskip\textbf{SET111-6.p} 1st is the ordered pair, second condition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(\mbox{first}(x),\mbox{second}(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_1st\_and\_2nd\_4}_1,\mbox{negated\_conjecture})$

$\mbox{first}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_existence\_of\_1st\_and\_2nd\_4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET149-6.p} Corollary to idempotency of intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x,\mbox{intersection}(x,y))\ne \mbox{intersection}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_idempotency\_of\_intersection}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET596$\wedge$5.p} TPS problem BOOL-PROP-55

Trybulec's 55th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) )\mbox{ and }( \lambda \mbox{xx}{:}\;a{:}\;( y@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) ) \ \Rightarrow \ ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_55\_pme},\mbox{conjecture})$

\medskip\textbf{SET917+1.p} disjoint(sgtn(A),B) | intersection(sgtn(A),B) = sgtn(A)






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

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

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

$\forall a,b{:}\;( \neg\,\mbox{in}(a,b) \ \Rightarrow \ \mbox{disjoint}(\mbox{singleton}(a),b) )\qquad\mbox{fof}(\mbox{l28\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_intersection}_2(b,\mbox{singleton}(a)) = \mbox{singleton}(a) )\qquad\mbox{fof}(\mbox{l32\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a,b{:}\;( \mbox{disjoint}(\mbox{singleton}(a),b)\mbox{ or }\mbox{set\_intersection}_2(\mbox{singleton}(a),b) = \mbox{singleton}(a) )\qquad\mbox{fof}(\mbox{t58\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET595+4.p} If X (= Y, then Y = X U (Y \ X)






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,e{:}\;( a\subseteq e \ \Rightarrow \ \mbox{equal\_set}(\mbox{union}(e\setminus a,a),e) )\qquad\mbox{fof}(\mbox{thI}_{27},\mbox{conjecture})$

\medskip\textbf{SET588$\wedge$5.p} TPS problem BOOL-PROP-46

Trybulec's 46th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_46\_pme},\mbox{conjecture})$

\medskip\textbf{SET598+3.p} X = Y $\wedge$ Z iff X (= Y, X (= Z, !V: V (= Y \& V (= Z, V (= X

X is the intersection of Y and Z if and only if the following
 conditions are satisfied: 1. X is a subset of Y, 2. X is a
 subset of Z, and 3. for every V such that V is a subset of Y
 and V is a subset of Z : V is a subset of X.




$\forall b,c{:}\;\mbox{intersection}(b,c)\subseteq b\qquad\mbox{fof}(\mbox{intersection\_is\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }b\subseteq d ) \ \Rightarrow \ b\subseteq \mbox{intersection}(c,d) )\qquad\mbox{fof}(\mbox{intersection\_of\_subsets},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( b = \mbox{intersection}(c,d)\iff ( b\subseteq c\mbox{ and }b\subseteq d\mbox{ and }\forall e{:}\;( ( e\subseteq c\mbox{ and }e\subseteq d ) \ \Rightarrow \ e\subseteq b ) ) )\qquad\mbox{fof}(\mbox{prove\_th}_{57},\mbox{conjecture})$

\medskip\textbf{SET841-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$( \mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_n},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_n} = \mbox{v\_m} )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_subsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET497-6.p} Special case of the Russell class, without the regularity axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{diagonalise}(\mbox{element\_relation})\qquad\mbox{cnf}(\mbox{prove\_russell\_class1}_1,\mbox{negated\_conjecture})$

$z\in z\qquad\mbox{cnf}(\mbox{prove\_russell\_class1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET275-6.p} Corollary 1 to domain property 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{domain\_of}(\mbox{restrict}(x,y,z)),\mbox{intersection}(\mbox{domain\_of}(x),y))\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_domain\_property4}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET131-6.p} Membership in unordered triple, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple2}_1,\mbox{negated\_conjecture})$

$\neg\,u\in \mbox{set\_builder}(x,\mbox{set\_builder}(u,\mbox{set\_builder}(z,\mbox{null\_class})))\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET811+4.p} A member of an ordinal number is an initial segment






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+4.ax'})$

$\forall a{:}\;( a\in \mbox{on} \ \Rightarrow \ \forall x{:}\;( x\in a \ \Rightarrow \ \mbox{equal\_set}(x,\mbox{initial\_segment}(x,\mbox{member\_predicate},a)) ) )\qquad\mbox{fof}(\mbox{thV}_5,\mbox{conjecture})$

\medskip\textbf{SET984+1.p} Basic properties of sets, theorem 138






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

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

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

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

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

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

$\forall a,b{:}\;( \mbox{cartesian\_product}_2(a,b) = \mbox{empty\_set}\iff ( a = \mbox{empty\_set}\mbox{ or }b = \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{t113\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;\mbox{cartesian\_product}_2(\mbox{set\_intersection}_2(a,b),\mbox{set\_intersection}_2(c,d)) = \mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,d))\qquad\mbox{fof}(\mbox{t123\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{cartesian\_product}_2(a,b) = \mbox{cartesian\_product}_2(c,d) \ \Rightarrow \ ( a = \mbox{empty\_set}\mbox{ or }b = \mbox{empty\_set}\mbox{ or }( a = c\mbox{ and }b = d ) ) )\qquad\mbox{fof}(\mbox{t134\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{cartesian\_product}_2(a,b)\subseteq \mbox{cartesian\_product}_2(c,d) \ \Rightarrow \ ( \mbox{cartesian\_product}_2(a,b) = \mbox{empty\_set}\mbox{ or }( a\subseteq c\mbox{ and }b\subseteq d ) ) )\qquad\mbox{fof}(\mbox{t138\_zfmisc}_1,\mbox{conjecture})$

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

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_intersection}_2(a,b) = a )\qquad\mbox{fof}(\mbox{t28\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SET845-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_in}(\mbox{c\_Main\_OUnion\_\_least\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_least}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Main\_OUnion\_\_least\_}_1(\mbox{v\_A},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_least}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_OsuccI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{v\_m} = \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Oeq\_\_succ\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_m} = \mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_m}\ne \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET502-6.p} Diagonalization property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{diagonalise}(\mbox{xr}\circ \mbox{xs}')\ne \mbox{range\_of}(\mbox{intersection}(\mbox{xr},\mbox{xs}))'\qquad\mbox{cnf}(\mbox{prove\_diagonalization\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET063-7.p} If X is a subset of the empty set, then X is the empty set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{special\_classes\_lemma},\mbox{axiom})$

$\neg\,z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{existence\_of\_null\_class},\mbox{axiom})$

$\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{cnf}(\mbox{null\_class\_is\_subclass},\mbox{axiom})$

$\mbox{subclass}(x,\mbox{null\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_of\_null\_class\_is\_subclass}_1,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_of\_null\_class\_is\_subclass}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET699+4.p} Property of intersection and difference 1

A is a subset of B if and only if the intersection of A and
 of the difference of B is a subset of the difference of A.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,e{:}\;( ( a\subseteq e\mbox{ and }b\subseteq e ) \ \Rightarrow \ ( a\subseteq b\iff \mbox{intersection}(a,e\setminus b)\subseteq ( e\setminus a ) ) )\qquad\mbox{fof}(\mbox{thI}_{33},\mbox{conjecture})$

\medskip\textbf{SET477-6.p} Axiom of subsets 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_axiom\_of\_subsets1}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_axiom\_of\_subsets1}_2,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_axiom\_of\_subsets1}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET229-6.p} Cross product right cancellation 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(u,v) = \mbox{cross\_product}(w,x)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_right\_cancellation2}_1,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_right\_cancellation2}_2,\mbox{negated\_conjecture})$

$u\ne w\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_right\_cancellation2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET304-6.p} Range alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_range\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_range\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{range\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_range\_alternate\_defn3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET089-6.p} A singleton set has a member, part 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{singleton}(\mbox{member\_of}(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_member\_exists4}_1,\mbox{negated\_conjecture})$

$\mbox{member\_of}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_member\_exists4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET595$\wedge$5.p} TPS problem BOOL-PROP-54

Trybulec's 54th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) \ \Rightarrow \ y = ( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }( y@ \mbox{xz}\mbox{ and }\neg\,x@ \mbox{xz} ) ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_54\_pme},\mbox{conjecture})$

\medskip\textbf{SET507-6.p} Universal class not subclass of null class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{universal\_class},\mbox{null\_class})\qquad\mbox{cnf}(\mbox{prove\_universal\_class\_not\_subclass\_of\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET607$\wedge$5.p} TPS problem BOOL-PROP-79

Trybulec's 79th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }( y@ \mbox{xz}\mbox{ and }\neg\,x@ \mbox{xz} ) ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( x@ \mbox{xz}\mbox{ or }y@ \mbox{xz} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_79\_pme},\mbox{conjecture})$

\medskip\textbf{SET209-6.p} Cross product is monotonic 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(y,z)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_is\_monotonic2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,y),\mbox{cross\_product}(x,z))\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_is\_monotonic2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET620+3.p} X sym\ Y = (X U Y) \ X $\wedge$ Y

The symmetric difference of X and Y is the difference of (the
 union of X and Y) and the intersection of X and Y.




$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;b\setminus \mbox{intersection}(b,c) = b\setminus c\qquad\mbox{fof}(\mbox{difference\_into\_intersection},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{union}(b,c)\setminus d = \mbox{union}(b\setminus d,c\setminus d)\qquad\mbox{fof}(\mbox{difference\_distributes\_over\_union},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b,c)\setminus \mbox{intersection}(b,c)\qquad\mbox{fof}(\mbox{prove\_th}_{96},\mbox{conjecture})$

\medskip\textbf{SET619+3.p} X U Y = (X sym\ Y) U X $\wedge$ Y

The union of X and Y is the union of (the symmetric difference
 of X and Y) and the intersection of X and Y.




$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{union}(\mbox{union}(b,c),d) = \mbox{union}(b,\mbox{union}(c,d))\qquad\mbox{fof}(\mbox{associativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,\mbox{intersection}(b,c)) = b\qquad\mbox{fof}(\mbox{union\_intersection},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(\mbox{intersection}(b,c),b\setminus c) = b\qquad\mbox{fof}(\mbox{union\_intersection\_difference},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(\mbox{symmetric\_difference}(b,c),\mbox{intersection}(b,c))\qquad\mbox{fof}(\mbox{prove\_th}_{95},\mbox{conjecture})$

\medskip\textbf{SET200+3.p} Union is monotonic

If X is a subset of Y and Z is a subset of V, then the union of
 X and Z is a subset of the union of Y and V.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d,e{:}\;( ( b\subseteq c\mbox{ and }d\subseteq e ) \ \Rightarrow \ \mbox{union}(b,d)\subseteq \mbox{union}(c,e) )\qquad\mbox{fof}(\mbox{prove\_th}_{34},\mbox{conjecture})$

\medskip\textbf{SET308-6.p} Monotonicity of range 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_range1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{range\_of}(x),\mbox{range\_of}(y))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_range1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET265-6.p} Domain is monotonic 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{domain\_of}(\mbox{intersection}(x,y)),\mbox{domain\_of}(x))\qquad\mbox{cnf}(\mbox{prove\_domain\_is\_monotonic2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET096-6.p} There are at most two subsets of a singleton set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{subclass}(x,\mbox{singleton}(y))\qquad\mbox{cnf}(\mbox{prove\_two\_subsets\_of\_singleton}_1,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_two\_subsets\_of\_singleton}_2,\mbox{negated\_conjecture})$

$\mbox{singleton}(y)\ne x\qquad\mbox{cnf}(\mbox{prove\_two\_subsets\_of\_singleton}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET120-6.p} Corollary 2 to components of equal ordered pairs being equal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_OP\_determines\_components1}_1,\mbox{negated\_conjecture})$

$\mbox{second}(\mbox{ordered\_pair}(x,y))\ne \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_OP\_determines\_components1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET711+4.p} The inverse of a mapping is unique






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{one\_to\_one}(f,a,b)\mbox{ and }\mbox{inverse\_predicate}(g,f,a,b)\mbox{ and }\mbox{inverse\_predicate}(h,f,a,b) ) \ \Rightarrow \ \mbox{equal\_maps}(g,h,b,a) )\qquad\mbox{fof}(\mbox{thII03a},\mbox{conjecture})$

\medskip\textbf{SET593+3.p} If X (= Y U Z, then X \ Y (= Z and X \ Z (= Y

If X is a subset of the union of Y and Z, then the difference
 of X and Y is a subset of Z and the difference of X and Z is
 a subset of Y.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq \mbox{union}(c,d) \ \Rightarrow \ ( ( b\setminus c )\subseteq d\mbox{ and }( b\setminus d )\subseteq c ) )\qquad\mbox{fof}(\mbox{prove\_th}_{52},\mbox{conjecture})$

\medskip\textbf{SET687+4.p} A set is a subset of itself






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{thI}_{01},\mbox{conjecture})$

\medskip\textbf{SET018-6.p} Second components of equal ordered pairs are equal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(w,x) = \mbox{ordered\_pair}(y,z)\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_determines\_components2}_1,\mbox{negated\_conjecture})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_determines\_components2}_2,\mbox{negated\_conjecture})$

$x\ne z\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_determines\_components2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET307-6.p} Range preserves union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(\mbox{range\_of}(x),\mbox{range\_of}(y))\ne \mbox{range\_of}(\mbox{union}(x,y))\qquad\mbox{cnf}(\mbox{prove\_range\_preserves\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET339-6.p} Subclass alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{image}(\mbox{element\_relation},y')'\qquad\mbox{cnf}(\mbox{prove\_subclass\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET021-3.p} 2nd is unique when x is an ordered pair of sets






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{b\_little\_set},\mbox{hypothesis})$

$\mbox{second}(\mbox{ordered\_pair}(a,b))\ne b\qquad\mbox{cnf}(\mbox{prove\_second\_is\_second},\mbox{negated\_conjecture})$

\medskip\textbf{SET714+4.p} The composition of inverse(F) and F is the identity






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{one\_to\_one}(f,a,b) ) \ \Rightarrow \ \mbox{identity}(\mbox{compose\_function}(\mbox{inverse\_function}(f,a,b),f,a,b,a),a) )\qquad\mbox{fof}(\mbox{thII}_{05},\mbox{conjecture})$

\medskip\textbf{SET376-6.p} Power class set builder






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in z\qquad\mbox{cnf}(\mbox{prove\_power\_class\_set\_builder}_1,\mbox{negated\_conjecture})$

$y\in \mbox{power\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_set\_builder}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{union}(\mbox{singleton}(x),y)\in \mbox{power\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_set\_builder}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET831-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_X} = \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_X} = \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{v\_X} = \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{v\_X} = \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_X}\ne \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ ( \mbox{v\_X} = \mbox{c\_inter}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET605$\wedge$5.p} TPS problem BOOL-PROP-76

Trybulec's 76th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx}\mbox{ or }y@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_76\_pme},\mbox{conjecture})$

\medskip\textbf{SET245-6.p} Restriction with null class 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{null\_class},x,y)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_restriction\_with\_null\_class1}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET083+1.p} A singleton set is determined by its element






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( ( \mbox{singleton}(x) = \mbox{singleton}(y)\mbox{ and }x\in \mbox{universal\_class} ) \ \Rightarrow \ x = y )\qquad\mbox{fof}(\mbox{singleton\_identified\_by\_element}_1,\mbox{conjecture})$

\medskip\textbf{SET753+4.p} Image of intersection is a subset of intersection of images






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,x,y{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }x\subseteq a\mbox{ and }y\subseteq a ) \ \Rightarrow \ \mbox{image}_2(f,\mbox{intersection}(x,y))\subseteq \mbox{intersection}(\mbox{image}_2(f,x),\mbox{image}_2(f,y)) )\qquad\mbox{fof}(\mbox{thIIa}_{03},\mbox{conjecture})$

\medskip\textbf{SET472-6.p} Sets with more than one member 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$u\in x\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_many\_members2}_1,\mbox{negated\_conjecture})$

$\mbox{not\_subclass\_element}(\mbox{intersection}(x,\mbox{singleton}(u)'),\mbox{null\_class}) = u\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_many\_members2}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,x),\mbox{identity\_relation})\qquad\mbox{cnf}(\mbox{prove\_sets\_with\_many\_members2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET630+3.p} X $\wedge$ Y is disjoint from X sym\ Y

The intersection of X and Y is disjoint from the symmetric
 difference of X and Y.




$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( \mbox{intersect}(b,\mbox{union}(c,d))\iff ( \mbox{intersect}(b,c)\mbox{ or }\mbox{intersect}(b,d) ) )\qquad\mbox{fof}(\mbox{intersect\_with\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{disjoint}(\mbox{intersection}(b,c),b\setminus c)\qquad\mbox{fof}(\mbox{intersection\_and\_union\_disjoint},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{disjoint}(b,c)\iff \neg\,\mbox{intersect}(b,c) )\qquad\mbox{fof}(\mbox{disjoint\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{disjoint}(\mbox{intersection}(b,c),\mbox{symmetric\_difference}(b,c))\qquad\mbox{fof}(\mbox{prove\_intersection\_and\_symmetric\_difference\_disjoint},\mbox{conjecture})$

\medskip\textbf{SET027$\wedge$5.p} TPS problem BOOL-PROP-29

Trybulec's 29th Boolean property of sets




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

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( y@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_29\_pme},\mbox{conjecture})$

\medskip\textbf{SET783+1.p} Naive set theory axioms based on Goedel's set theory






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

\medskip\textbf{SET747$\wedge$4.p} If F is increasing and G decreasing, then GoF is decreasing






$\mbox{include}(\mbox{'Axioms/SET008\string^1.ax'})$

$\forall f{:}\;\mbox{\$i}\to \mbox{\$i},g{:}\;\mbox{\$i}\to \mbox{\$i},\mbox{lESS}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{fun\_increasing}@ f@ \mbox{lESS}\mbox{ and }\mbox{fun\_decreasing}@ g@ \mbox{lESS} ) \ \Rightarrow \ ( \mbox{fun\_decreasing}@ ( \mbox{fun\_composition}@ f@ g )@ \mbox{lESS} ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET144+3.p} If X is a subset of Z, then X U Y $\wedge$ Z = (X U Y) $\wedge$ Z

If X is a subset of Z, then the union of X and the intersection
 of Y and Z is the intersection of (the union of X and Y) and Z.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq c \ \Rightarrow \ \mbox{union}(b,\mbox{intersection}(d,c)) = \mbox{intersection}(\mbox{union}(b,d),c) )\qquad\mbox{fof}(\mbox{prove\_th}_{44},\mbox{conjecture})$

\medskip\textbf{SET356-6.p} Corollary to subclass of union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(y,\mbox{power\_class}(\mbox{sum\_class}(y)))\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_subclass\_of\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET127-6.p} Relation to unordered pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$\mbox{set\_builder}(x,\mbox{singleton}(y))\ne \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_set\_builder\_and\_unordered\_pair}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET555-6.p} Cantor class alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{domain\_of}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_cantor\_class\_property2}_1,\mbox{negated\_conjecture})$

$\neg\,z\in \mbox{apply}(\mbox{xr},z)\qquad\mbox{cnf}(\mbox{prove\_cantor\_class\_property2}_2,\mbox{negated\_conjecture})$

$\neg\,z\in \mbox{cantor}(\mbox{xr})\qquad\mbox{cnf}(\mbox{prove\_cantor\_class\_property2}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET113+1.p} Uniqueness of 1st and 2nd when X is not an ordered pair of sets






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall u,v,x{:}\;( ( \exists y,z{:}\;( y\in \mbox{universal\_class}\mbox{ and }z\in \mbox{universal\_class}\mbox{ and }x = \mbox{ordered\_pair}(y,z) )\mbox{ and }x = u\mbox{ and }v = x )\mbox{ or }( \mbox{first}(x) = u\mbox{ and }\mbox{second}(x) = v ) )\qquad\mbox{fof}(\mbox{unique\_1st\_and\_2nd\_in\_pair\_of\_non\_sets}_1,\mbox{conjecture})$

\medskip\textbf{SET027+1.p} Transitivity of subset






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y,z{:}\;( ( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z) )\qquad\mbox{fof}(\mbox{transitivity},\mbox{conjecture})$

\medskip\textbf{SET036-4.p} Properties of apply for functions, part 1 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{b\_little\_set},\mbox{hypothesis})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$\mbox{ordered\_pair}(a,b)\in \mbox{a\_function}\qquad\mbox{cnf}(\mbox{ordered\_pair\_in\_function},\mbox{hypothesis})$

$\mbox{apply}(\mbox{a\_function},a)\ne b\qquad\mbox{cnf}(\mbox{prove\_apply\_for\_functions}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET093+1.p} Corollary to every singleton is a set






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;( \mbox{singleton}(\mbox{member\_of}(x)) = x \ \Rightarrow \ x\in \mbox{universal\_class} )\qquad\mbox{fof}(\mbox{corollary\_2\_to\_singletons\_are\_sets},\mbox{conjecture})$

\medskip\textbf{SET724+4.p} If GoF = HoF and F is surjective, then G = H






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,b,c)\mbox{ and }\mbox{surjective}(f,a,b)\mbox{ and }\mbox{equal\_maps}(\mbox{compose\_function}(g,f,a,b,c),\mbox{compose\_function}(h,f,a,b,c),a,c) ) \ \Rightarrow \ \mbox{equal\_maps}(g,h,b,c) )\qquad\mbox{fof}(\mbox{thII}_{15},\mbox{conjecture})$

\medskip\textbf{SET234-6.p} Cross product property 9






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(\mbox{first}(\mbox{not\_subclass\_element}(\mbox{cross\_product}(x,y),z)),\mbox{second}(\mbox{not\_subclass\_element}(\mbox{cross\_product}(x,y),z)))\in z\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property9}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{cross\_product}(x,y),z)\qquad\mbox{cnf}(\mbox{prove\_cross\_product\_property9}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET436-6.p} Subclass of function is a function, part 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_subclass\_of\_function1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(\mbox{intersection}(x,y))\qquad\mbox{cnf}(\mbox{prove\_subclass\_of\_function1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET809+4.p} An ordinal number is not a member of itself






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+4.ax'})$

$\forall a{:}\;( a\in \mbox{on} \ \Rightarrow \ \neg\,a\in a )\qquad\mbox{fof}(\mbox{thV}_2,\mbox{conjecture})$

\medskip\textbf{SET947+1.p} subset(A,powerset(union(A)))






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

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

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

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ a\subseteq \mbox{union}(b) )\qquad\mbox{fof}(\mbox{l50\_zfmisc}_1,\mbox{axiom})$

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

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

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

$\forall a{:}\;a\subseteq \mbox{powerset}(\mbox{union}(a))\qquad\mbox{fof}(\mbox{t100\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET584+3.p} If X (= Y, then X U Z (= Y U Z

If X is a subset of Y, then the union of X and Z is a subset of
 the union of Y and Z.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( b\subseteq c \ \Rightarrow \ \mbox{union}(b,d)\subseteq \mbox{union}(c,d) )\qquad\mbox{fof}(\mbox{prove\_th}_{33},\mbox{conjecture})$

\medskip\textbf{SET184-6.p} Subclass property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x,y) = x\qquad\mbox{cnf}(\mbox{prove\_subclass\_property2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET821-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_less}(\mbox{v\_a},\mbox{v\_b},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_less}(\mbox{v\_b},\mbox{v\_c},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_b},\mbox{v\_U},\mbox{tc\_IntDef\_Oint}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_U},\mbox{tc\_IntDef\_Oint})\mbox{ or }\mbox{c\_in}(\mbox{v\_a},\mbox{v\_U},\mbox{tc\_IntDef\_Oint}) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET004-1.p} A set is a subset of the union of itself and another set






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-1.ax'})$

$\mbox{union}(a,b,\mbox{aUb})\qquad\mbox{cnf}(\mbox{a\_union\_b\_is\_aUb},\mbox{hypothesis})$

$\neg\,a\subseteq \mbox{aUb}\qquad\mbox{cnf}(\mbox{prove\_a\_is\_a\_subset\_of\_aUb},\mbox{negated\_conjecture})$

\medskip\textbf{SET347+4.p} Sum of the empty set is the empty set






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{equal\_set}(\mbox{\mbox{sum}}(\mbox{empty\_set}),\mbox{empty\_set})\qquad\mbox{fof}(\mbox{thI}_{38},\mbox{conjecture})$

\medskip\textbf{SET284-6.p} Image of domain relation






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{sum\_class}(\mbox{image}(\mbox{domain\_relation},x))\ne \mbox{domain\_of}(\mbox{sum\_class}(x))\qquad\mbox{cnf}(\mbox{prove\_image\_of\_domain\_relation}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET173+3.p} Absorbtion for intersection

The intersection of X and the union of X and Y is X.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,\mbox{union}(b,c)) = b\qquad\mbox{fof}(\mbox{prove\_absorbtion\_for\_intersection},\mbox{conjecture})$

\medskip\textbf{SET099+1.p} Corollary 2 to a class contains 0, 1, or at least 2 members






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x{:}\;( \forall u,v{:}\;( ( u\in x\mbox{ and }v\in \mbox{intersection}(\mbox{singleton}(u)',x) ) \ \Rightarrow \ u = v ) \ \Rightarrow \ ( x = \mbox{null\_class}\mbox{ or }\exists y{:}\;\mbox{singleton}(y) = x ) )\qquad\mbox{fof}(\mbox{corollary\_2\_to\_number\_of\_elements\_in\_class},\mbox{conjecture})$

\medskip\textbf{SET985+1.p} Basic properties of sets, theorem 139






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

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

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

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

$\forall a,b{:}\;( \mbox{cartesian\_product}_2(a,b) = \mbox{empty\_set}\iff ( a = \mbox{empty\_set}\mbox{ or }b = \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{t113\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{cartesian\_product}_2(a,b)\subseteq \mbox{cartesian\_product}_2(c,d) \ \Rightarrow \ ( \mbox{cartesian\_product}_2(a,b) = \mbox{empty\_set}\mbox{ or }( a\subseteq c\mbox{ and }b\subseteq d ) ) )\qquad\mbox{fof}(\mbox{t138\_zfmisc}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \forall b,c,d{:}\;( ( \mbox{cartesian\_product}_2(a,b)\subseteq \mbox{cartesian\_product}_2(c,d)\mbox{ or }\mbox{cartesian\_product}_2(b,a)\subseteq \mbox{cartesian\_product}_2(d,c) ) \ \Rightarrow \ b\subseteq d ) )\qquad\mbox{fof}(\mbox{t139\_zfmisc}_1,\mbox{conjecture})$

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

\medskip\textbf{SET428-6.p} Single valued class alternate definition 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued}_1(x) = \mbox{single\_valued}_2(x)\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn4}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{single\_valued\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_single\_valued\_class\_alternate\_defn4}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET330-6.p} Corollary to image alternate definition 4






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_image\_alternate\_defn4}_1,\mbox{negated\_conjecture})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_image\_alternate\_defn4}_2,\mbox{negated\_conjecture})$

$\neg\,y\in \mbox{image}(\mbox{xr},\mbox{singleton}(x))\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_image\_alternate\_defn4}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET062-7.p} The empty set is a subset of X






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{special\_classes\_lemma},\mbox{axiom})$

$\neg\,z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{existence\_of\_null\_class},\mbox{axiom})$

$\neg\,\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{cnf}(\mbox{prove\_null\_class\_is\_subclass}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET913+1.p} ~ ( disjoint(singleton(A),B) \& in(A,B) )






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

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

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

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

$\forall a,b{:}\;\neg\,\mbox{disjoint}(\mbox{singleton}(a),b)\mbox{ and }\mbox{in}(a,b)\qquad\mbox{fof}(\mbox{t54\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;\neg\,\mbox{disjoint}(\mbox{singleton}(a),b)\mbox{ and }\mbox{in}(a,b)\qquad\mbox{fof}(\mbox{l25\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET038-3.p} Properties of apply for functions, part 3 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(x)\subseteq \mbox{domain\_of}(y) \ \Rightarrow \ \mbox{domain\_of}(x) = \mbox{domain\_of}(y\circ x)\qquad\mbox{cnf}(\mbox{domain\_of\_composition},\mbox{axiom})$

$( \mbox{function}(x)\mbox{ and }\mbox{function}(y) ) \ \Rightarrow \ \mbox{function}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_function},\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},u,v)\mbox{ and }\mbox{maps}(\mbox{xg},v,w) ) \ \Rightarrow \ \mbox{maps}(\mbox{xg}\circ \mbox{xf},u,w)\qquad\mbox{cnf}(\mbox{maps\_for\_composition},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{function}(\mbox{xf})\mbox{ and }\mbox{ordered\_pair}(x,y)\in \mbox{xf} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x) = y\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_1,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf})\mbox{ and }\mbox{apply}(\mbox{xf},x) = y ) \ \Rightarrow \ \mbox{ordered\_pair}(x,y)\in \mbox{xf}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_2,\mbox{axiom})$

$\mbox{maps}(\mbox{a\_function},\mbox{a\_domain},\mbox{a\_range})\qquad\mbox{cnf}(\mbox{a\_mapping},\mbox{hypothesis})$

$a\in \mbox{a\_domain}\qquad\mbox{cnf}(\mbox{member\_of\_domain},\mbox{hypothesis})$

$\neg\,\mbox{apply}(\mbox{a\_function},a)\in \mbox{a\_range}\qquad\mbox{cnf}(\mbox{prove\_mapping\_in\_range},\mbox{negated\_conjecture})$

\medskip\textbf{SET143+4.p} Associativity of intersection

The intersection of (the intersection of X and Y) and Z is the
 intersection of X and (the intersection of Y and Z).




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,c{:}\;\mbox{equal\_set}(\mbox{intersection}(\mbox{intersection}(a,b),c),\mbox{intersection}(a,\mbox{intersection}(b,c)))\qquad\mbox{fof}(\mbox{thI}_{08},\mbox{conjecture})$

\medskip\textbf{SET387-6.p} Composition alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{xf}\circ \mbox{yf}\qquad\mbox{cnf}(\mbox{prove\_composition\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,1_{\text{dom}(\mbox{xf})})\in \mbox{yf}\qquad\mbox{cnf}(\mbox{prove\_composition\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET242-6.p} Restriction alternate definition 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_restriction\_alternate\_defn5}_1,\mbox{negated\_conjecture})$

$z\in \mbox{cross\_product}(x,y)\qquad\mbox{cnf}(\mbox{prove\_restriction\_alternate\_defn5}_2,\mbox{negated\_conjecture})$

$\neg\,z\in \mbox{restrict}(\mbox{xr},x,y)\qquad\mbox{cnf}(\mbox{prove\_restriction\_alternate\_defn5}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET350-6.p} Sum class of singleton null is null class 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{sum\_class}(x) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_is\_null\_class}_1,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_is\_null\_class}_2,\mbox{negated\_conjecture})$

$x\ne \mbox{singleton}(\mbox{null\_class})\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_is\_null\_class}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET150-6.p} Complement is an involution






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$( x' )'\ne x\qquad\mbox{cnf}(\mbox{prove\_complement\_is\_self\_cancelling}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET878+1.p} intersection(singleton(A),unordered\_pair(A,B)) = singleton(A)






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_intersection}_2(b,\mbox{singleton}(a)) = \mbox{singleton}(a) )\qquad\mbox{fof}(\mbox{l32\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(\mbox{singleton}(a),\mbox{unordered\_pair}(a,b)) = \mbox{singleton}(a)\qquad\mbox{fof}(\mbox{t19\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET994+1.p} Functions and their basic properties, theorem 16






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{function}(a) )\qquad\mbox{fof}(\mbox{cc1\_funct}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ \mbox{relation}(a) )\qquad\mbox{fof}(\mbox{cc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;\mbox{element}(b,a)\qquad\mbox{fof}(\mbox{existence\_m1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\mbox{ and }\mbox{relation\_empty\_yielding}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc12\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\neg\,\mbox{empty}(\mbox{powerset}(a))\qquad\mbox{fof}(\mbox{fc1\_subset}_1,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\mbox{ and }\mbox{relation}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc4\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( ( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) ) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{relation\_dom}(a)) )\qquad\mbox{fof}(\mbox{fc5\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ ( \mbox{empty}(\mbox{relation\_dom}(a))\mbox{ and }\mbox{relation}(\mbox{relation\_dom}(a)) ) )\qquad\mbox{fof}(\mbox{fc7\_relat}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{function}(a) )\qquad\mbox{fof}(\mbox{rc1\_funct}_1,\mbox{axiom})$

$\exists a{:}\;( \mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc1\_relat}_1,\mbox{axiom})$

$\forall a{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\neg\,\mbox{empty}(b) ) )\qquad\mbox{fof}(\mbox{rc1\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \neg\,\mbox{empty}(a)\mbox{ and }\mbox{relation}(a) )\qquad\mbox{fof}(\mbox{rc2\_relat}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{element}(b,\mbox{powerset}(a))\mbox{ and }\mbox{empty}(b) )\qquad\mbox{fof}(\mbox{rc2\_subset}_1,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;( \mbox{relation}(a)\mbox{ and }\mbox{relation\_empty\_yielding}(a) )\qquad\mbox{fof}(\mbox{rc3\_relat}_1,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{relation}(b)\mbox{ and }\mbox{function}(b)\mbox{ and }\mbox{relation\_dom}(b) = a\mbox{ and }\forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{apply}(b,c) = n_0 ) )\qquad\mbox{fof}(\mbox{s3\_funct\_1\_\_e4\_14\_\_funct}_1,\mbox{axiom})$

$\forall a{:}\;\exists b{:}\;( \mbox{relation}(b)\mbox{ and }\mbox{function}(b)\mbox{ and }\mbox{relation\_dom}(b) = a\mbox{ and }\forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{apply}(b,c) = n_1 ) )\qquad\mbox{fof}(\mbox{s3\_funct\_1\_\_e7\_14\_\_funct}_1,\mbox{axiom})$

$\mbox{empty}(n_0)\qquad\mbox{fof}(\mbox{spc0\_boole},\mbox{axiom})$

$\neg\,\mbox{empty}(n_1)\qquad\mbox{fof}(\mbox{spc1\_boole},\mbox{axiom})$

$\forall a{:}\;( \forall b{:}\;( ( \mbox{relation}(b)\mbox{ and }\mbox{function}(b) ) \ \Rightarrow \ \forall c{:}\;( ( \mbox{relation}(c)\mbox{ and }\mbox{function}(c) ) \ \Rightarrow \ ( ( \mbox{relation\_dom}(b) = a\mbox{ and }\mbox{relation\_dom}(c) = a ) \ \Rightarrow \ b = c ) ) ) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t16\_funct}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{element}(a,b) )\qquad\mbox{fof}(\mbox{t1\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,b) \ \Rightarrow \ ( \mbox{empty}(b)\mbox{ or }\mbox{in}(a,b) ) )\qquad\mbox{fof}(\mbox{t2\_subset},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{element}(a,\mbox{powerset}(b))\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t3\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c)) ) \ \Rightarrow \ \mbox{element}(a,c) )\qquad\mbox{fof}(\mbox{t4\_subset},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{element}(b,\mbox{powerset}(c))\mbox{ and }\mbox{empty}(c)\qquad\mbox{fof}(\mbox{t5\_subset},\mbox{axiom})$

$\forall a{:}\;( \mbox{empty}(a) \ \Rightarrow \ a = \mbox{empty\_set} )\qquad\mbox{fof}(\mbox{t6\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{in}(a,b)\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t7\_boole},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(a)\mbox{ and }a\ne b\mbox{ and }\mbox{empty}(b)\qquad\mbox{fof}(\mbox{t8\_boole},\mbox{axiom})$

\medskip\textbf{SET971+1.p} Basic properties of sets, theorem 124






$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c,d{:}\;\mbox{cartesian\_product}_2(\mbox{set\_intersection}_2(a,b),\mbox{set\_intersection}_2(c,d)) = \mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(a,c),\mbox{cartesian\_product}_2(b,d))\qquad\mbox{fof}(\mbox{t123\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( ( a\subseteq b\mbox{ and }c\subseteq d ) \ \Rightarrow \ \mbox{set\_intersection}_2(\mbox{cartesian\_product}_2(a,d),\mbox{cartesian\_product}_2(b,c)) = \mbox{cartesian\_product}_2(a,c) )\qquad\mbox{fof}(\mbox{t124\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_intersection}_2(a,b) = a )\qquad\mbox{fof}(\mbox{t28\_xboole}_1,\mbox{axiom})$

\medskip\textbf{SET342-6.p} Image of union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{xr},\mbox{union}(y,z)),\mbox{union}(\mbox{image}(\mbox{xr},y),\mbox{image}(\mbox{xr},z)))\qquad\mbox{cnf}(\mbox{prove\_image\_of\_union}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET489-6.p} Replacement property 11






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(x,y)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property11}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property11}_2,\mbox{negated\_conjecture})$

$y\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property11}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET024-3.p} A set belongs to its singleton






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\neg\,a\in \mbox{singleton\_set}(a)\qquad\mbox{cnf}(\mbox{prove\_membership\_of\_singleton\_set},\mbox{negated\_conjecture})$

\medskip\textbf{SET045+1.p} No Universal Set

The restricted comprehension axiom says : given a set
 z, there is a set all of whose members are drawn from z and
 which satisfy some property. If there were a universal set,
 then the Russell set could be formed, using this axiom.
 So given the appropriate instance of this axiom, there
 is no universal set.




$\forall z{:}\;\exists y{:}\;\forall x{:}\;( \mbox{element}(x,y)\iff ( \mbox{element}(x,z)\mbox{ and }\neg\,\mbox{element}(x,x) ) )\qquad\mbox{fof}(\mbox{pel41}_1,\mbox{axiom})$

$\neg\,\exists z{:}\;\forall x{:}\;\mbox{element}(x,z)\qquad\mbox{fof}(\mbox{pel}_{41},\mbox{conjecture})$

\medskip\textbf{SET347-6.p} Sum class of the empty set is the empty set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{sum\_class}(\mbox{null\_class})\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_sum\_class\_of\_null\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET002+4.p} Idempotency of union






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a{:}\;\mbox{equal\_set}(\mbox{union}(a,a),a)\qquad\mbox{fof}(\mbox{thI}_{14},\mbox{conjecture})$

\medskip\textbf{SET606$\wedge$5.p} TPS problem BOOL-PROP-77

Trybulec's 77th Boolean property of sets




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_77\_pme},\mbox{conjecture})$

\medskip\textbf{SET820-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_P}(\mbox{v\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_V},\mbox{v\_U},\mbox{tc\_IntDef\_Oint}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_U},\mbox{tc\_IntDef\_Oint})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_W},\mbox{v\_U},\mbox{tc\_IntDef\_Oint}) \ \Rightarrow \ \neg\,\mbox{v\_P}(\mbox{v\_x}(\mbox{v\_U}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET621$\wedge$5.p} TPS problem BOOL-PROP-97

Trybulec's 97th Boolean property of sets




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( ( ( x@ \mbox{xx}\mbox{ and }\neg\,y@ \mbox{xx} )\mbox{ or }( y@ \mbox{xx}\mbox{ and }\neg\,x@ \mbox{xx} ) )\mbox{ and }\neg\,z@ \mbox{xx} ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz}\mbox{ or }z@ \mbox{xz} )\mbox{ or }( y@ \mbox{xz}\mbox{ and }\neg\,x@ \mbox{xz}\mbox{ or }z@ \mbox{xz} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_97\_pme},\mbox{conjecture})$

\medskip\textbf{SET646$\wedge$3.p} If x is in X and y is in Y then {$<$x,y$>$} is from X to Y.






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall x{:}\;\mbox{\$i},y{:}\;\mbox{\$i}{:}\;( \mbox{sub\_rel}@ ( \mbox{pair\_rel}@ x@ y )@ ( \mbox{cartesian\_product}@ \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$true}@ \lambda x{:}\;\mbox{\$i}{:}\;\mbox{\$true} ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET032-3.p} Range of composition






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\neg\,\mbox{range\_of}(a\circ b)\subseteq \mbox{range\_of}(a)\qquad\mbox{cnf}(\mbox{prove\_range\_of\_composition},\mbox{negated\_conjecture})$

\medskip\textbf{SET230-6.p} Corollary to cross product cancellation






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{cross\_product}(u,u) = \mbox{cross\_product}(w,w)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_cross\_product\_cancellation}_1,\mbox{negated\_conjecture})$

$u\ne w\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_cross\_product\_cancellation}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET694+4.p} Union of power sets is a subset of the power set of the union

The union of the power\_set of A and the power\_set of B is a
 subset of the power\_set of the union of A and B.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;\mbox{union}(\mbox{power\_set}(a),\mbox{power\_set}(b))\subseteq \mbox{power\_set}(\mbox{union}(a,b))\qquad\mbox{fof}(\mbox{thI}_{22},\mbox{conjecture})$

\medskip\textbf{SET940+1.p} union(unordered\_pair(A,B)) = union(A,B)






$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{union}(\mbox{unordered\_pair}(a,b)) = \mbox{set\_union}_2(a,b)\qquad\mbox{fof}(\mbox{t93\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;\mbox{union}(\mbox{unordered\_pair}(a,b)) = \mbox{set\_union}_2(a,b)\qquad\mbox{fof}(\mbox{l52\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET540-6.p} Application property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_application\_property6}_1,\mbox{negated\_conjecture})$

$\neg\,z\in \mbox{apply}(\mbox{xf},z)\qquad\mbox{cnf}(\mbox{prove\_application\_property6}_2,\mbox{negated\_conjecture})$

$\neg\,z\in \mbox{diagonalise}(\mbox{element\_relation}'\circ \mbox{xf})\qquad\mbox{cnf}(\mbox{prove\_application\_property6}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET039-4.p} Properties of apply for composition of functions, 1 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$a\in \mbox{domain\_of}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{member\_of\_domain},\mbox{hypothesis})$

$\neg\,\mbox{apply}(\mbox{another\_function},\mbox{apply}(\mbox{a\_function},a))\subseteq \mbox{apply}(\mbox{another\_function}\circ \mbox{a\_function},a)\qquad\mbox{cnf}(\mbox{prove\_apply\_for\_composition}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET936+1.p} powset(intersection(A,B)) = intersection(powset(A),powset(B))






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b) = \mbox{set\_intersection}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{powerset}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c\subseteq a ) )\qquad\mbox{fof}(\mbox{d1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{set\_intersection}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( \mbox{in}(d,a)\mbox{ and }\mbox{in}(d,b) ) ) )\qquad\mbox{fof}(\mbox{d3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k3\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_intersection}_2(a,b)\subseteq a\qquad\mbox{fof}(\mbox{t17\_xboole}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }a\subseteq c ) \ \Rightarrow \ a\subseteq \mbox{set\_intersection}_2(b,c) )\qquad\mbox{fof}(\mbox{t19\_xboole}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{axiom})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{powerset}(\mbox{set\_intersection}_2(a,b)) = \mbox{set\_intersection}_2(\mbox{powerset}(a),\mbox{powerset}(b))\qquad\mbox{fof}(\mbox{t83\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET352-6.p} The sum class of an unordered pair is the union of the pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_sum\_of\_pair}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(\mbox{unordered\_pair}(x,y))\ne \mbox{union}(x,y)\qquad\mbox{cnf}(\mbox{prove\_sum\_of\_pair}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET753$\wedge$4.p} Image of intersection is a subset of intersection of images






$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^1.ax'})$

$\forall x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o},f{:}\;\mbox{\$i}\to \mbox{\$i}{:}\;( \subseteq @ ( \mbox{fun\_image}@ f@ ( \mbox{intersection}@ x@ y ) )@ ( \mbox{intersection}@ ( \mbox{fun\_image}@ f@ x )@ ( \mbox{fun\_image}@ f@ y ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET636$\wedge$5.p} TPS problem BOOL-PROP-118

Trybulec's 118th Boolean property of sets




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o}{:}\;( \neg\,\exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} )\iff ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_118\_pme},\mbox{conjecture})$

\medskip\textbf{SET800+4.p} Greatest lower bound of sets in total order

If an order is total, the greatest lower bound of a set is greater
 than the greatest lower bound of a subset of it




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e{:}\;( \mbox{order}(r,e) \ \Rightarrow \ \forall x_1,x_2{:}\;( ( x_1\subseteq e\mbox{ and }x_2\subseteq e\mbox{ and }x_1\subseteq x_2 ) \ \Rightarrow \ \forall m_1,m_2{:}\;( ( \mbox{greatest\_lower\_bound}(m_1,x_1,r,e)\mbox{ and }\mbox{greatest\_lower\_bound}(m_2,x_2,r,e) ) \ \Rightarrow \ \mbox{apply}(r,m_2,m_1) ) ) )\qquad\mbox{fof}(\mbox{thIV}_{12},\mbox{conjecture})$

\medskip\textbf{SET743+4.p} Problem on composition of mappings 8

Consider three mappings F from A to B,G from B to C,H from
 C to A. If GoF and HoG are one-to-one, then G is one-to-one.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,c,a)\mbox{ and }\mbox{one\_to\_one}(\mbox{compose\_function}(g,f,a,b,c),a,c)\mbox{ and }\mbox{one\_to\_one}(\mbox{compose\_function}(h,g,b,c,a),b,a) ) \ \Rightarrow \ \mbox{one\_to\_one}(g,b,c) )\qquad\mbox{fof}(\mbox{thII}_{34},\mbox{conjecture})$

\medskip\textbf{SET117-6.p} Corollary 1 to every ordered pair being a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(\mbox{first}(x),\mbox{second}(x)) = x\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_ordered\_pairs\_are\_sets}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_ordered\_pairs\_are\_sets}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET145-6.p} Commutativity outside intersection






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{intersection}(x,\mbox{intersection}(y,z))\ne \mbox{intersection}(y,\mbox{intersection}(x,z))\qquad\mbox{cnf}(\mbox{prove\_commutativity\_outside\_intersection}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET806+4.p} Equality of sets defines a equivalence relation






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall x,y{:}\;( \mbox{apply}(\mbox{equal\_set\_predicate},x,y)\iff \mbox{equal\_set}(x,y) )\qquad\mbox{fof}(\mbox{rel\_equal\_set},\mbox{hypothesis})$

$\forall e{:}\;\mbox{equivalence}(\mbox{equal\_set\_predicate},\mbox{power\_set}(e))\qquad\mbox{fof}(\mbox{thIII}_{13},\mbox{conjecture})$

\medskip\textbf{SET634$\wedge$5.p} TPS problem BOOL-PROP-116

Trybulec's 116th Boolean property of sets




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }y@ \mbox{xx}\mbox{ and }\neg\,z@ \mbox{xx} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_116\_pme},\mbox{conjecture})$

\medskip\textbf{SET238-6.p} Corollary to restriction alternate definition 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$z\in \mbox{restrict}(\mbox{xr},x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_restriction\_alternate\_defn1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(\mbox{first}(z),\mbox{second}(z))\in \mbox{restrict}(\mbox{xr},x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_restriction\_alternate\_defn1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET867+1.p} union(empty\_set) = empty\_set






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{union}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff \exists d{:}\;( \mbox{in}(c,d)\mbox{ and }\mbox{in}(d,a) ) ) )\qquad\mbox{fof}(\mbox{d4\_tarski},\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\mbox{union}(\mbox{empty\_set}) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t2\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET264-6.p} Domain is monotonic 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_domain\_is\_monotonic1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{domain\_of}(x),\mbox{domain\_of}(y))\qquad\mbox{cnf}(\mbox{prove\_domain\_is\_monotonic1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET615+3.p} (X U Y) \ Z = (X \ Z) U (Y \ Z)

The difference of (the union of X and Y) and Z is the union of
 (the difference of X and Z) and (the difference of Y and Z).




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{union}(b,c)\setminus d = \mbox{union}(b\setminus d,c\setminus d)\qquad\mbox{fof}(\mbox{prove\_difference\_distributes\_over\_union},\mbox{conjecture})$

\medskip\textbf{SET837-2.p} Problem about set theory






$\mbox{v\_X} = \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_X} = \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{v\_X} = \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_x},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{v\_X} = \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a}) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_x},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_x},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_X}\ne \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ ( \mbox{v\_X} = \mbox{c\_union}(\mbox{v\_Y},\mbox{v\_Z},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnCI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnCI}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_union}(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{t\_a}) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a}) )\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnE}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

\medskip\textbf{SET175+3.p} Absorbtion for union

The union of X and  the intersection of X and Y is X.




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,\mbox{intersection}(b,c)) = b\qquad\mbox{fof}(\mbox{prove\_absorbtion\_for\_union},\mbox{conjecture})$

\medskip\textbf{SET617+3.p} X sym\ the empty set = X and the empty set sym\ X = X

The symmetric difference of X and the empty set is X and the
 symmetric difference of the empty set and X is X.




$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{union}(b\setminus c,c\setminus b)\qquad\mbox{fof}(\mbox{symmetric\_difference\_defn},\mbox{axiom})$

$\forall b{:}\;\mbox{union}(b,\mbox{empty\_set}) = b\qquad\mbox{fof}(\mbox{union\_empty\_set},\mbox{axiom})$

$\forall b{:}\;b\setminus \mbox{empty\_set} = b\qquad\mbox{fof}(\mbox{no\_difference\_with\_empty\_set}_1,\mbox{axiom})$

$\forall b{:}\;\mbox{empty\_set}\setminus b = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{no\_difference\_with\_empty\_set}_2,\mbox{axiom})$

$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{symmetric\_difference}(b,c) = \mbox{symmetric\_difference}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_symmetric\_difference},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b{:}\;( \mbox{symmetric\_difference}(b,\mbox{empty\_set}) = b\mbox{ and }\mbox{symmetric\_difference}(\mbox{empty\_set},b) = b )\qquad\mbox{fof}(\mbox{prove\_th}_{92},\mbox{conjecture})$

\medskip\textbf{SET040-4.p} Properties of apply for composition of functions, 2 of 3






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{function}(\mbox{a\_function})\qquad\mbox{cnf}(\mbox{a\_function},\mbox{hypothesis})$

$\neg\,\mbox{apply}(\mbox{another\_function}\circ \mbox{a\_function},\mbox{element})\subseteq \mbox{apply}(\mbox{another\_function},\mbox{apply}(\mbox{a\_function},\mbox{element}))\qquad\mbox{cnf}(\mbox{prove\_apply\_for\_composition}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET887+1.p} ~ ( subset(uno\_pair(A,B),uno\_pair(C,D)) \& A != C \& A != D )






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq \mbox{unordered\_pair}(b,c)\iff \neg\,a\ne \mbox{empty\_set}\mbox{ and }a\ne \mbox{singleton}(b)\mbox{ and }a\ne \mbox{singleton}(c)\mbox{ and }a\ne \mbox{unordered\_pair}(b,c) )\qquad\mbox{fof}(\mbox{l46\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c,d{:}\;\neg\,\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(c,d)\mbox{ and }a\ne c\mbox{ and }a\ne d\qquad\mbox{fof}(\mbox{t10\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;\neg\,\mbox{unordered\_pair}(a,b)\subseteq \mbox{unordered\_pair}(c,d)\mbox{ and }a\ne c\mbox{ and }a\ne d\qquad\mbox{fof}(\mbox{t28\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b,c{:}\;( \mbox{singleton}(a) = \mbox{unordered\_pair}(b,c) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t8\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET279-6.p} Domain property 5






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$u\in \mbox{domain\_of}(\mbox{restrict}(\mbox{xr},y,\mbox{singleton}(z)))\qquad\mbox{cnf}(\mbox{prove\_domain\_property5}_1,\mbox{negated\_conjecture})$

$z\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_domain\_property5}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(u,z)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{prove\_domain\_property5}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET088-6.p} A singleton set has a member, part 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{member\_of}(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_exists3}_1,\mbox{negated\_conjecture})$

$\mbox{member\_of}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_member\_exists3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET171+3.p} Union distributes over intersection

The union of X and (the intersection of Y and Z) is the
 intersection of (the union of X and Y) and (the union of X and Z).




$\forall b,c,d{:}\;( d\in \mbox{union}(b,c)\iff ( d\in b\mbox{ or }d\in c ) )\qquad\mbox{fof}(\mbox{union\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{union}(b,c) = \mbox{union}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_union},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;\mbox{union}(b,\mbox{intersection}(c,d)) = \mbox{intersection}(\mbox{union}(b,c),\mbox{union}(b,d))\qquad\mbox{fof}(\mbox{prove\_union\_distributes\_over\_intersection},\mbox{conjecture})$

\medskip\textbf{SET829-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Z},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET254-6.p} Monotonicity of restriction 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x_1,x_2)\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_restriction1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{restrict}(x_1,y,z),\mbox{restrict}(x_2,y,z))\qquad\mbox{cnf}(\mbox{prove\_monotonicity\_of\_restriction1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET975+1.p} in(o\_pair(A,B),cart\_prod(sgtn(C),D)) $\leq$$>$ ( A = C \& in(B,D) )






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(\mbox{singleton}(c),d))\iff ( a = c\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{t128\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET197-6.p} Lattice lower bound 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{intersection}(x,y),y)\qquad\mbox{cnf}(\mbox{prove\_lattice\_lower\_bound2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET627+3.p} X is disjoint from the empty set






$\forall b{:}\;\neg\,b\in \mbox{empty\_set}\qquad\mbox{fof}(\mbox{empty\_set\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{disjoint}(b,c)\iff \neg\,\mbox{intersect}(b,c) )\qquad\mbox{fof}(\mbox{disjoint\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b{:}\;( \mbox{empty}(b)\iff \forall c{:}\;\neg\,c\in b )\qquad\mbox{fof}(\mbox{empty\_defn},\mbox{axiom})$

$\forall b{:}\;\mbox{disjoint}(b,\mbox{empty\_set})\qquad\mbox{fof}(\mbox{prove\_th}_{104},\mbox{conjecture})$

\medskip\textbf{SET862-2.p} Problem about Zorn's lemma






$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_c},\mbox{v\_A},\mbox{t\_a})\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_c},\mbox{v\_B},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetD}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_A},\mbox{t\_a})\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Main\_OsubsetI\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{t\_a}),\mbox{v\_B},\mbox{t\_a}) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OsubsetI}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Ochain\_\_extend\_}_1(\mbox{v\_c},\mbox{v\_z},\mbox{t\_a}),\mbox{v\_z},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_z},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Ochain\_\_extend}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\mbox{ or }\mbox{c\_in}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_z},\mbox{v\_x},\mbox{t\_a})\mbox{ and }\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{c\_union}(\mbox{c\_insert}(\mbox{v\_x},\mbox{c\_emptyset},\mbox{tc\_set}(\mbox{t\_a})),\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Omaxchain\_\_super\_\_lemma\_}_1(\mbox{v\_c},\mbox{v\_y},\mbox{t\_a}),\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_z},\mbox{v\_y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Omaxchain\_\_super\_\_lemma}_1,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Omaxchain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_c},\mbox{c\_Zorn\_Ochain}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_y},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_c},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_in}(\mbox{v\_x}(\mbox{v\_U}),\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_x}(\mbox{v\_U}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$\mbox{v\_U} = \mbox{v\_x}(\mbox{v\_U}) \ \Rightarrow \ \neg\,\mbox{c\_in}(\mbox{v\_U},\mbox{v\_S},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_6,\mbox{negated\_conjecture})$

\medskip\textbf{SET883+1.p} subset(singleton(A),singleton(B)) =$>$ A = B






$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq \mbox{singleton}(b) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t24\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \mbox{singleton}(a)\subseteq \mbox{singleton}(b) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t6\_zfmisc}_1,\mbox{axiom})$

\medskip\textbf{SET700+4.p} Property of intersection and difference 2

A is a subset of B if and only if the intersection of A and
 of the difference of B is a subset of B.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,e{:}\;( ( a\subseteq e\mbox{ and }b\subseteq e ) \ \Rightarrow \ ( a\subseteq b\iff \mbox{intersection}(a,e\setminus b)\subseteq b ) )\qquad\mbox{fof}(\mbox{thI}_{34},\mbox{conjecture})$

\medskip\textbf{SET137-6.p} Kludge 1 to instantiate variables in unordered triples






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{union}(\mbox{singleton}(x),y) = \mbox{set\_builder}(x,y)\qquad\mbox{cnf}(\mbox{definition\_of\_set\_builder},\mbox{axiom})$

$u\in x\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple\_kludge1}_1,\mbox{negated\_conjecture})$

$v\in x\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple\_kludge1}_2,\mbox{negated\_conjecture})$

$w\in x\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple\_kludge1}_3,\mbox{negated\_conjecture})$

$\mbox{set\_builder}(u,\mbox{set\_builder}(v,\mbox{set\_builder}(w,\mbox{null\_class}))) = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_of\_triple\_kludge1}_4,\mbox{negated\_conjecture})$

\medskip\textbf{SET891+1.p} union(uno\_pair(singleton(A),singleton(B))) = uno\_pair(A,B)






$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{union}(\mbox{unordered\_pair}(a,b)) = \mbox{set\_union}_2(a,b)\qquad\mbox{fof}(\mbox{l52\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{union}(\mbox{unordered\_pair}(\mbox{singleton}(a),\mbox{singleton}(b))) = \mbox{unordered\_pair}(a,b)\qquad\mbox{fof}(\mbox{t32\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{set\_union}_2(\mbox{singleton}(a),\mbox{singleton}(b))\qquad\mbox{fof}(\mbox{t41\_enumset}_1,\mbox{axiom})$

\medskip\textbf{SET381-6.p} Relation property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_relation\_property2}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,\mbox{cross\_product}(\mbox{domain\_of}(x),\mbox{range\_of}(x)))\qquad\mbox{cnf}(\mbox{prove\_relation\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET955+1.p} Basic properties of sets, theorem 108

( in(ordered\_pair(E,F),cartesian\_product2(A,B))
 $\leq$$>$ in(ordered\_pair(E,F),cartesian\_product2(C,D)) )
 =$>$ cartesian\_product2(A,B) = cartesian\_product2(C,D)




$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \forall e,f{:}\;( \mbox{in}(\mbox{ordered\_pair}(e,f),\mbox{cartesian\_product}_2(a,b))\iff \mbox{in}(\mbox{ordered\_pair}(e,f),\mbox{cartesian\_product}_2(c,d)) ) \ \Rightarrow \ \mbox{cartesian\_product}_2(a,b) = \mbox{cartesian\_product}_2(c,d) )\qquad\mbox{fof}(\mbox{t108\_zfmisc}_1,\mbox{conjecture})$

$\forall a,b{:}\;( \forall c{:}\;( \mbox{in}(c,a)\iff \mbox{in}(c,b) ) \ \Rightarrow \ a = b )\qquad\mbox{fof}(\mbox{t2\_tarski},\mbox{axiom})$

\medskip\textbf{SET881+1.p} difference(singleton(A),unordered\_pair(A,B)) = empty\_set






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{set\_difference}(\mbox{singleton}(a),b) = \mbox{empty\_set}\iff \mbox{in}(a,b) )\qquad\mbox{fof}(\mbox{l36\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_difference}(\mbox{singleton}(a),\mbox{unordered\_pair}(a,b)) = \mbox{empty\_set}\qquad\mbox{fof}(\mbox{t22\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET884+1.p} ~ ( subset(singleton(A),unordered\_pair(B,C)) \& A != B \& A != C )






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;\neg\,\mbox{singleton}(a)\subseteq \mbox{unordered\_pair}(b,c)\mbox{ and }a\ne b\mbox{ and }a\ne c\qquad\mbox{fof}(\mbox{t25\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET519-6.p} Corollary 1 to no cycles of length 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y) = x\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_no\_cycles\_length\_2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET423-6.p} Sum compose class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{sum\_class}(\mbox{image}(\mbox{compose\_class}(x),y))\ne x\circ \mbox{sum\_class}(y)\qquad\mbox{cnf}(\mbox{prove\_sum\_compose\_class}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET761+4.p} Intersection of images

If F is injective, then the image of intersection is equal to
 the intersection of images.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,x,y{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{injective}(f,a,b)\mbox{ and }x\subseteq a\mbox{ and }y\subseteq a ) \ \Rightarrow \ \mbox{equal\_set}(\mbox{image}_3(f,\mbox{intersection}(x,y),b),\mbox{intersection}(\mbox{image}_3(f,x,b),\mbox{image}_3(f,y,b))) )\qquad\mbox{fof}(\mbox{thIIa}_{11},\mbox{conjecture})$

\medskip\textbf{SET834-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_X},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{v\_X},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Z},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_in}(\mbox{v\_x},\mbox{v\_Y},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_Z},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_X},\mbox{v\_U},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET623$\wedge$5.p} TPS problem BOOL-PROP-99

Trybulec's 99th Boolean property of sets




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( \lambda \mbox{xz}{:}\;a{:}\;( ( x@ \mbox{xz}\mbox{ and }\neg\,( y@ \mbox{xz}\mbox{ and }\neg\,z@ \mbox{xz} )\mbox{ or }( z@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} ) )\mbox{ or }( ( ( y@ \mbox{xz}\mbox{ and }\neg\,z@ \mbox{xz} )\mbox{ or }( z@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} ) )\mbox{ and }\neg\,x@ \mbox{xz} ) ) ) = ( \lambda \mbox{xz}{:}\;a{:}\;( ( ( ( x@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} )\mbox{ or }( y@ \mbox{xz}\mbox{ and }\neg\,x@ \mbox{xz} ) )\mbox{ and }\neg\,z@ \mbox{xz} )\mbox{ or }( z@ \mbox{xz}\mbox{ and }\neg\,( x@ \mbox{xz}\mbox{ and }\neg\,y@ \mbox{xz} )\mbox{ or }( y@ \mbox{xz}\mbox{ and }\neg\,x@ \mbox{xz} ) ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_99\_pme},\mbox{conjecture})$

\medskip\textbf{SET960+1.p} cart\_prod(A,B) = empty $\leq$$>$ ( A = empty | B = empty )






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a{:}\;( a = \mbox{empty\_set}\iff \forall b{:}\;\neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{d1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{cartesian\_product}_2(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff \exists e,f{:}\;( \mbox{in}(e,a)\mbox{ and }\mbox{in}(f,b)\mbox{ and }d = \mbox{ordered\_pair}(e,f) ) ) )\qquad\mbox{fof}(\mbox{d2\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{cartesian\_product}_2(a,b) = \mbox{empty\_set}\iff ( a = \mbox{empty\_set}\mbox{ or }b = \mbox{empty\_set} ) )\qquad\mbox{fof}(\mbox{t113\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET021-6.p} 2nd is unique when x is an ordered pair of sets






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_unique\_1st\_and\_2nd\_in\_pair\_of\_sets2}_1,\mbox{negated\_conjecture})$

$\mbox{second}(\mbox{ordered\_pair}(u,v))\ne v\qquad\mbox{cnf}(\mbox{prove\_unique\_1st\_and\_2nd\_in\_pair\_of\_sets2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET015-1.p} The union of sets is commutative






$\mbox{include}(\mbox{'Axioms/SET002-0.ax'})$

$\neg\,\mbox{equal\_sets}(\mbox{union}(\mbox{as},\mbox{bs}),\mbox{union}(\mbox{bs},\mbox{as}))\qquad\mbox{cnf}(\mbox{prove\_commutativity},\mbox{negated\_conjecture})$

\medskip\textbf{SET322-6.p} Range property 6






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{domain\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_range\_property6}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{range}(z,x,\mbox{universal\_class})\in \mbox{range\_of}(z)\qquad\mbox{cnf}(\mbox{prove\_range\_property6}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET246-6.p} Restriction with null class 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{xr},\mbox{null\_class},y)\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_restriction\_with\_null\_class2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET073+1.p} Corollary to unordered pair axiom






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( x\in \mbox{universal\_class} \ \Rightarrow \ \mbox{unordered\_pair}(x,y)\ne \mbox{null\_class} )\qquad\mbox{fof}(\mbox{corollary1}_1,\mbox{conjecture})$

\medskip\textbf{SET027+4.p} Transitivity of subset






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{thI}_{03},\mbox{conjecture})$

\medskip\textbf{SET419-6.p} Composition property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_composition\_property3}_1,\mbox{negated\_conjecture})$

$\mbox{apply}(\mbox{image}(\mbox{composition\_function},\mbox{singleton}(x)),y)\ne x\circ y\qquad\mbox{cnf}(\mbox{prove\_composition\_property3}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET516-6.p} Corollary to no class belongs to itself






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{singleton}(x) = x\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_no\_class\_belongs\_to\_itself}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET060-6.p} Nothing in the intersection of a set and its complement






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{prove\_special\_classes\_lemma}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET042-3.p} Ordered pairs are in cross products






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ x = u\qquad\mbox{cnf}(\mbox{first\_components\_are\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{non\_ordered\_pair}(z,x) = \mbox{non\_ordered\_pair}(z,y) ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{left\_cancellation},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{little\_set}(u)\mbox{ and }\mbox{little\_set}(v)\mbox{ and }\mbox{ordered\_pair}(x,y) = \mbox{ordered\_pair}(u,v) ) \ \Rightarrow \ y = v\qquad\mbox{cnf}(\mbox{second\_components\_are\_equal},\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{two\_sets\_equal},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{first}(\mbox{ordered\_pair}(x,y)) = x\qquad\mbox{cnf}(\mbox{property\_of\_first},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y) ) \ \Rightarrow \ \mbox{second}(\mbox{ordered\_pair}(x,y)) = y\qquad\mbox{cnf}(\mbox{property\_of\_second},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{first}(x))\qquad\mbox{cnf}(\mbox{first\_component\_is\_small},\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(\mbox{second}(x))\qquad\mbox{cnf}(\mbox{second\_component\_is\_small},\mbox{axiom})$

$\mbox{little\_set}(x) \ \Rightarrow \ x\in \mbox{singleton\_set}(x)\qquad\mbox{cnf}(\mbox{property\_of\_singleton\_sets},\mbox{axiom})$

$\mbox{little\_set}(\mbox{ordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_1,\mbox{axiom})$

$\mbox{ordered\_pair\_predicate}(x) \ \Rightarrow \ \mbox{little\_set}(x)\qquad\mbox{cnf}(\mbox{ordered\_pairs\_are\_small}_2,\mbox{axiom})$

$( x\subseteq y\mbox{ and }y\subseteq z ) \ \Rightarrow \ x\subseteq z\qquad\mbox{cnf}(\mbox{containment\_is\_transitive},\mbox{axiom})$

$\mbox{apply}(\mbox{xf},y)\subseteq \mbox{sigma}(\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf}))\qquad\mbox{cnf}(\mbox{image\_and\_apply}_1,\mbox{axiom})$

$\mbox{image}(\mbox{singleton\_set}(y),\mbox{xf})\subseteq \mbox{apply}(\mbox{xf},y)\qquad\mbox{cnf}(\mbox{image\_and\_apply}_2,\mbox{axiom})$

$\mbox{function}(y) \ \Rightarrow \ \mbox{little\_set}(\mbox{apply}(y,x))\qquad\mbox{cnf}(\mbox{function\_values\_are\_small},\mbox{axiom})$

$\mbox{relation}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_relation},\mbox{axiom})$

$\mbox{range\_of}(y\circ x)\subseteq \mbox{range\_of}(y)\qquad\mbox{cnf}(\mbox{range\_of\_composition},\mbox{axiom})$

$\mbox{range\_of}(x)\subseteq \mbox{domain\_of}(y) \ \Rightarrow \ \mbox{domain\_of}(x) = \mbox{domain\_of}(y\circ x)\qquad\mbox{cnf}(\mbox{domain\_of\_composition},\mbox{axiom})$

$( \mbox{function}(x)\mbox{ and }\mbox{function}(y) ) \ \Rightarrow \ \mbox{function}(y\circ x)\qquad\mbox{cnf}(\mbox{composition\_is\_a\_function},\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},u,v)\mbox{ and }\mbox{maps}(\mbox{xg},v,w) ) \ \Rightarrow \ \mbox{maps}(\mbox{xg}\circ \mbox{xf},u,w)\qquad\mbox{cnf}(\mbox{maps\_for\_composition},\mbox{axiom})$

$( \mbox{little\_set}(x)\mbox{ and }\mbox{little\_set}(y)\mbox{ and }\mbox{function}(\mbox{xf})\mbox{ and }\mbox{ordered\_pair}(x,y)\in \mbox{xf} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x) = y\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_1,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf})\mbox{ and }\mbox{apply}(\mbox{xf},x) = y ) \ \Rightarrow \ \mbox{ordered\_pair}(x,y)\in \mbox{xf}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_2,\mbox{axiom})$

$( \mbox{maps}(\mbox{xf},\mbox{xd},\mbox{xr})\mbox{ and }x\in \mbox{xd} ) \ \Rightarrow \ \mbox{apply}(\mbox{xf},x)\in \mbox{xr}\qquad\mbox{cnf}(\mbox{apply\_for\_functions}_3,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf}) ) \ \Rightarrow \ \mbox{apply}(\mbox{xg},\mbox{apply}(\mbox{xf},x))\subseteq \mbox{apply}(\mbox{xg}\circ \mbox{xf},x)\qquad\mbox{cnf}(\mbox{apply\_for\_composition}_1,\mbox{axiom})$

$\mbox{function}(\mbox{xf}) \ \Rightarrow \ \mbox{apply}(\mbox{xg}\circ \mbox{xf},x)\subseteq \mbox{apply}(\mbox{xg},\mbox{apply}(\mbox{xf},x))\qquad\mbox{cnf}(\mbox{apply\_for\_composition}_2,\mbox{axiom})$

$( \mbox{function}(\mbox{xf})\mbox{ and }x\in \mbox{domain\_of}(\mbox{xf}) ) \ \Rightarrow \ \mbox{apply}(\mbox{xg},\mbox{apply}(\mbox{xf},x)) = \mbox{apply}(\mbox{xg}\circ \mbox{xf},x)\qquad\mbox{cnf}(\mbox{apply\_for\_composition}_3,\mbox{axiom})$

$a\in \mbox{set\_a}\qquad\mbox{cnf}(\mbox{member\_of\_set\_a},\mbox{hypothesis})$

$b\in \mbox{set\_b}\qquad\mbox{cnf}(\mbox{member\_of\_set\_b},\mbox{hypothesis})$

$\neg\,\mbox{ordered\_pair}(a,b)\in \mbox{cross\_product}(\mbox{set\_a},\mbox{set\_b})\qquad\mbox{cnf}(\mbox{prove\_ordered\_pair\_is\_in\_cross\_product},\mbox{negated\_conjecture})$

\medskip\textbf{SET888+1.p} Basic properties of sets, theorem 29






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{symmetric\_difference}(a,b) = \mbox{symmetric\_difference}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k5\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{singleton}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c = a ) )\qquad\mbox{fof}(\mbox{d1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{symmetric\_difference}(a,b) = \mbox{set\_union}_2(\mbox{set\_difference}(a,b),\mbox{set\_difference}(b,a))\qquad\mbox{fof}(\mbox{d6\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{in}(a,\mbox{symmetric\_difference}(b,c))\iff \neg\,\mbox{in}(a,b)\iff \mbox{in}(a,c) )\qquad\mbox{fof}(\mbox{t1\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( a\ne b \ \Rightarrow \ \mbox{symmetric\_difference}(\mbox{singleton}(a),\mbox{singleton}(b)) = \mbox{unordered\_pair}(a,b) )\qquad\mbox{fof}(\mbox{t29\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET430-6.p} In a single-valued class, each image is a singleton






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{single\_valued\_class}(x)\qquad\mbox{cnf}(\mbox{prove\_image\_of\_single\_valued\_class\_domain}_1,\mbox{negated\_conjecture})$

$z\in \mbox{domain\_of}(x)\qquad\mbox{cnf}(\mbox{prove\_image\_of\_single\_valued\_class\_domain}_2,\mbox{negated\_conjecture})$

$\mbox{singleton}(\mbox{member\_of}(\mbox{image}(x,\mbox{singleton}(z))))\ne \mbox{image}(x,\mbox{singleton}(z))\qquad\mbox{cnf}(\mbox{prove\_image\_of\_single\_valued\_class\_domain}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET813+4.p} An ordinal number is a member of its successor






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+4.ax'})$

$\forall a{:}\;( a\in \mbox{on} \ \Rightarrow \ a\in \mbox{suc}(a) )\qquad\mbox{fof}(\mbox{thV}_{12},\mbox{conjecture})$

\medskip\textbf{SET838-1.p} Problem about set theory






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{v\_f}(\mbox{v\_g}(\mbox{v\_x})) = \mbox{v\_x}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{v\_f}(\mbox{v\_g}(\mbox{v\_U})) = \mbox{v\_U} \ \Rightarrow \ \mbox{v\_U} = \mbox{v\_x}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{v\_g}(\mbox{v\_f}(\mbox{v\_U})) = \mbox{v\_U} \ \Rightarrow \ \mbox{v\_g}(\mbox{v\_f}(\mbox{v\_xa}(\mbox{v\_U}))) = \mbox{v\_xa}(\mbox{v\_U})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_xa}(\mbox{v\_U}) = \mbox{v\_U} \ \Rightarrow \ \mbox{v\_g}(\mbox{v\_f}(\mbox{v\_U}))\ne \mbox{v\_U}\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET592$\wedge$5.p} TPS problem BOOL-PROP-51

Trybulec's 51st Boolean property of sets




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) )\mbox{ and }\forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( z@ \mbox{xx} ) )\mbox{ and }( \lambda \mbox{xx}{:}\;a{:}\;( y@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) ) \ \Rightarrow \ x = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_51\_pme},\mbox{conjecture})$

\medskip\textbf{SET773+4.p} Intersection of equivalence relations is an equivalence relation






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall e,r_1,r_2,r{:}\;( ( \mbox{equivalence}(r_1,e)\mbox{ and }\mbox{equivalence}(r_2,e)\mbox{ and }\forall a,b{:}\;( ( a\in e\mbox{ and }b\in e ) \ \Rightarrow \ ( \mbox{apply}(r,a,b)\iff ( \mbox{apply}(r_1,a,b)\mbox{ and }\mbox{apply}(r_2,a,b) ) ) ) ) \ \Rightarrow \ \mbox{equivalence}(r,e) )\qquad\mbox{fof}(\mbox{thIII}_{09},\mbox{conjecture})$

\medskip\textbf{SET171+4.p} Distribution of union over intersection 1

The union of X and (the intersection of Y and Z) is the
 intersection of (the union of X and Y) and (the union of X and Z).




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b,c{:}\;\mbox{equal\_set}(\mbox{union}(a,\mbox{intersection}(b,c)),\mbox{intersection}(\mbox{union}(a,b),\mbox{union}(a,c)))\qquad\mbox{fof}(\mbox{thI}_{11},\mbox{conjecture})$

\medskip\textbf{SET017-4.p} Left cancellation for non-ordered pairs






$\mbox{include}(\mbox{'Axioms/SET003-0.ax'})$

$\mbox{little\_set}(a)\qquad\mbox{cnf}(\mbox{a\_little\_set},\mbox{hypothesis})$

$\mbox{little\_set}(b)\qquad\mbox{cnf}(\mbox{b\_little\_set},\mbox{hypothesis})$

$\mbox{non\_ordered\_pair}(c,a) = \mbox{non\_ordered\_pair}(d,b)\qquad\mbox{cnf}(\mbox{equal\_non\_ordered\_pairs},\mbox{hypothesis})$

$a\ne c\qquad\mbox{cnf}(\mbox{prove\_left\_cancellation},\mbox{negated\_conjecture})$

\medskip\textbf{SET579+3.p} Trybulec's 20th Boolean property of sets






$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff ( b\subseteq c\mbox{ and }c\subseteq b ) )\qquad\mbox{fof}(\mbox{equal\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c,d{:}\;( \forall e{:}\;( e\in b\iff ( e\in c\mbox{ and }\neg\,e\in d ) ) \ \Rightarrow \ b = c\setminus d )\qquad\mbox{fof}(\mbox{prove\_th}_{20},\mbox{conjecture})$

\medskip\textbf{SET378-6.p} Corollary 2 to power class set builder






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{intersection}(\mbox{power\_class}(z),z)),\mbox{power\_class}(z))\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_power\_class\_set\_builder}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET774+4.p} The restriction of a pre-order relation is a pre-order relation






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+2.ax'})$

$\forall e,x,r{:}\;( ( \mbox{pre\_order}(r,e)\mbox{ and }x\subseteq e ) \ \Rightarrow \ \mbox{pre\_order}(r,x) )\qquad\mbox{fof}(\mbox{thIII}_{10},\mbox{conjecture})$

\medskip\textbf{SET730+4.p} Property of restriction 1

If F is a mapping from A to B,and G equal to F on A and
 C =image2(F,A), then G is a mapping from A to C.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }c\subseteq b\mbox{ and }\mbox{image}_2(f,a) = c\mbox{ and }\forall x,y{:}\;( ( x\in a\mbox{ and }y\in c ) \ \Rightarrow \ ( \mbox{apply}(g,x,y)\iff \mbox{apply}(f,x,y) ) ) ) \ \Rightarrow \ \mbox{maps}(g,a,c) )\qquad\mbox{fof}(\mbox{thII}_{21},\mbox{conjecture})$

\medskip\textbf{SET015+4.p} Commutativity of union






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall a,b{:}\;\mbox{equal\_set}(\mbox{union}(a,b),\mbox{union}(b,a))\qquad\mbox{fof}(\mbox{thI}_{07},\mbox{conjecture})$

\medskip\textbf{SET434-6.p} Function alternate definition 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,\mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class}))\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn3}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{ordered\_pair}(\mbox{single\_valued}_3(x),\mbox{single\_valued}_2(x))\in x\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn3}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{function}(x)\qquad\mbox{cnf}(\mbox{prove\_function\_alternate\_defn3}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET183-6.p} If X is a subset of  Y, then the intersection of X and Y is X






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_subclass\_property1}_1,\mbox{negated\_conjecture})$

$\mbox{intersection}(x,y)\ne x\qquad\mbox{cnf}(\mbox{prove\_subclass\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET046$\wedge$5.p} TPS problem PELL42

There is no set of non-circular sets (where a circular set is a
 set x s.t. there is a set y, s.t. x belongs to y and reversely).




$\mbox{cF}{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cF},\mbox{type})$

$\neg\,\exists \mbox{xy}{:}\;\mbox{\$i}{:}\;\forall \mbox{xx}{:}\;\mbox{\$i}{:}\;( ( ( \mbox{cF}@ \mbox{xx}@ \mbox{xy} ) \ \Rightarrow \ \neg\,\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cF}@ \mbox{xx}@ \mbox{xz}\mbox{ and }\mbox{cF}@ \mbox{xz}@ \mbox{xx} ) )\mbox{ and }( \neg\,\exists \mbox{xz}{:}\;\mbox{\$i}{:}\;( \mbox{cF}@ \mbox{xx}@ \mbox{xz}\mbox{ and }\mbox{cF}@ \mbox{xz}@ \mbox{xx} ) \ \Rightarrow \ ( \mbox{cF}@ \mbox{xx}@ \mbox{xy} ) ) )\qquad\mbox{thf}(\mbox{cPELL}_{42},\mbox{conjecture})$

\medskip\textbf{SET720+4.p} The inverse of the inverse of a mapping is equal to the mapping






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{one\_to\_one}(f,a,b) ) \ \Rightarrow \ \mbox{equal\_maps}(\mbox{inverse\_function}(\mbox{inverse\_function}(f,a,b),b,a),f,a,b) )\qquad\mbox{fof}(\mbox{thII}_{11},\mbox{conjecture})$

\medskip\textbf{SET523-6.p} Element and complement can't both be sets






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in y\qquad\mbox{cnf}(\mbox{prove\_element\_and\_complement\_not\_both\_sets}_1,\mbox{negated\_conjecture})$

$x'\in z\qquad\mbox{cnf}(\mbox{prove\_element\_and\_complement\_not\_both\_sets}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET627$\wedge$5.p} TPS problem BOOL-PROP-104

Trybulec's 104th Boolean property of sets




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall x{:}\;a\to \mbox{\$o}{:}\;\neg\,\exists \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }\mbox{\$false} )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_104\_pme},\mbox{conjecture})$

\medskip\textbf{SET718+4.p} The composition of one-to-one  mappings is one-to-one






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{one\_to\_one}(f,a,b)\mbox{ and }\mbox{one\_to\_one}(g,b,c) ) \ \Rightarrow \ \mbox{one\_to\_one}(\mbox{compose\_function}(g,f,a,b,c),a,c) )\qquad\mbox{fof}(\mbox{thII}_{09},\mbox{conjecture})$

\medskip\textbf{SET907+1.p} ( in(A,B) \& in(C,B) ) =$>$ set\_union2(unordered\_pair(A,C),B) = B






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{set\_union}_2(a,b) = b )\qquad\mbox{fof}(\mbox{t12\_xboole}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{unordered\_pair}(a,b)\subseteq c\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,c) ) )\qquad\mbox{fof}(\mbox{t38\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( ( \mbox{in}(a,b)\mbox{ and }\mbox{in}(c,b) ) \ \Rightarrow \ \mbox{set\_union}_2(\mbox{unordered\_pair}(a,c),b) = b )\qquad\mbox{fof}(\mbox{t48\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET798+4.p} If X subset Y, then a lower bound of Y is a lower bound of X






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+3.ax'})$

$\forall r,e{:}\;( \mbox{order}(r,e) \ \Rightarrow \ \forall x,y{:}\;( ( x\subseteq e\mbox{ and }y\subseteq e\mbox{ and }x\subseteq y ) \ \Rightarrow \ \forall m{:}\;( a \ \Rightarrow \ a ) ) )\qquad\mbox{fof}(\mbox{thIV}_{10},\mbox{conjecture})$

\medskip\textbf{SET931+1.p} Basic properties of sets, theorem 75

difference(A,unordered\_pair(B,C)) = empty\_set
 $\leq$$>$ ~ ( A != empty\_set \& A != singleton(B) \& A != singleton(C) \&
 A != unordered\_pair(B,C) )




$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\mbox{empty}(\mbox{empty\_set})\qquad\mbox{fof}(\mbox{fc1\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( a\subseteq \mbox{unordered\_pair}(b,c)\iff \neg\,a\ne \mbox{empty\_set}\mbox{ and }a\ne \mbox{singleton}(b)\mbox{ and }a\ne \mbox{singleton}(c)\mbox{ and }a\ne \mbox{unordered\_pair}(b,c) )\qquad\mbox{fof}(\mbox{l46\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \mbox{set\_difference}(a,b) = \mbox{empty\_set}\iff a\subseteq b )\qquad\mbox{fof}(\mbox{t37\_xboole}_1,\mbox{axiom})$

$\forall a,b,c{:}\;( \mbox{set\_difference}(a,\mbox{unordered\_pair}(b,c)) = \mbox{empty\_set}\iff \neg\,a\ne \mbox{empty\_set}\mbox{ and }a\ne \mbox{singleton}(b)\mbox{ and }a\ne \mbox{singleton}(c)\mbox{ and }a\ne \mbox{unordered\_pair}(b,c) )\qquad\mbox{fof}(\mbox{t75\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET167-6.p} Members of union 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in y\qquad\mbox{cnf}(\mbox{prove\_members\_of\_union2}_1,\mbox{negated\_conjecture})$

$\neg\,x\in \mbox{union}(y,z)\qquad\mbox{cnf}(\mbox{prove\_members\_of\_union2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET854-1.p} Problem about Zorn's lemma






$\mbox{include}(\mbox{'Axioms/MSC001-2.ax'})$

$\mbox{include}(\mbox{'Axioms/MSC001-0.ax'})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OAbrial\_\_axiom1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) = \mbox{v\_m} ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_2,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_3,\mbox{axiom})$

$\mbox{c\_in}(\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_C},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_1,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Zorn\_OUnion\_\_lemma0\_}_1(\mbox{v\_A},\mbox{v\_B},\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Union}(\mbox{v\_C},\mbox{t\_a}),\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OUnion\_\_lemma0}_2,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_y},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_trans}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_x}\ne \mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_V},\mbox{v\_U},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_V},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{v\_Y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_V},\mbox{t\_a}),\mbox{v\_U},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_V} = \mbox{v\_U} )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

\medskip\textbf{SET070-6.p} Proper class in an unordered pair, part 4

If one argument of an unordered pair is a proper class, the
 pair contains only the other.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_equals\_singleton2}_1,\mbox{negated\_conjecture})$

$\mbox{unordered\_pair}(x,y)\ne \mbox{singleton}(y)\qquad\mbox{cnf}(\mbox{prove\_unordered\_pair\_equals\_singleton2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET954+1.p} in(o\_pair(A,B),cart\_prod(C,D)) =$>$ in(o\_pair(B,A),cart\_prod(D,C))






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{ordered\_pair}(a,b) = \mbox{unordered\_pair}(\mbox{unordered\_pair}(a,b),\mbox{singleton}(a))\qquad\mbox{fof}(\mbox{d5\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\neg\,\mbox{empty}(\mbox{ordered\_pair}(a,b))\qquad\mbox{fof}(\mbox{fc1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d))\iff ( \mbox{in}(a,c)\mbox{ and }\mbox{in}(b,d) ) )\qquad\mbox{fof}(\mbox{l55\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c,d{:}\;( \mbox{in}(\mbox{ordered\_pair}(a,b),\mbox{cartesian\_product}_2(c,d)) \ \Rightarrow \ \mbox{in}(\mbox{ordered\_pair}(b,a),\mbox{cartesian\_product}_2(d,c)) )\qquad\mbox{fof}(\mbox{t107\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET518-6.p} There are no cycles of length 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in y\qquad\mbox{cnf}(\mbox{prove\_no\_cycles\_length\_2}_1,\mbox{negated\_conjecture})$

$y\in x\qquad\mbox{cnf}(\mbox{prove\_no\_cycles\_length\_2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET027-6.p} Transitivity of subset






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{subclass}(x,y)\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_subclass}_1,\mbox{negated\_conjecture})$

$\mbox{subclass}(y,z)\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_subclass}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{prove\_transitivity\_of\_subclass}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET052-6.p} Corollary to Cartesian product axiom






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_cartesian\_product}_1,\mbox{negated\_conjecture})$

$\neg\,u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_cartesian\_product}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET270-6.p} Domain property 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{domain\_of}(\mbox{cross\_product}(x,y))\ne x\qquad\mbox{cnf}(\mbox{prove\_domain\_property2}_1,\mbox{negated\_conjecture})$

$y\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_domain\_property2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET737+4.p} Problem on composition of mappings 2

Consider three mappings F from A to B,G from B to C,H from
 C to A. If HoGoF and FoHoG are injective and GoFoH surjective,
 then G is one-to-one.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,g,h,a,b,c{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{maps}(g,b,c)\mbox{ and }\mbox{maps}(h,c,a)\mbox{ and }\mbox{injective}(\mbox{compose\_function}(h,\mbox{compose\_function}(g,f,a,b,c),a,c,a),a,a)\mbox{ and }\mbox{injective}(\mbox{compose\_function}(f,\mbox{compose\_function}(h,g,b,c,a),b,a,b),b,b)\mbox{ and }\mbox{surjective}(\mbox{compose\_function}(g,\mbox{compose\_function}(f,h,c,a,b),c,b,c),c,c) ) \ \Rightarrow \ \mbox{one\_to\_one}(g,b,c) )\qquad\mbox{fof}(\mbox{thII}_{28},\mbox{conjecture})$

\medskip\textbf{SET932+1.p} subset(A,B) =$>$ subset(powerset(A),powerset(B))






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;( b = \mbox{powerset}(a)\iff \forall c{:}\;( \mbox{in}(c,b)\iff c\subseteq a ) )\qquad\mbox{fof}(\mbox{d1\_zfmisc}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b\iff \forall c{:}\;( \mbox{in}(c,a) \ \Rightarrow \ \mbox{in}(c,b) ) )\qquad\mbox{fof}(\mbox{d3\_tarski},\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;a\subseteq a\qquad\mbox{fof}(\mbox{reflexivity\_r1\_tarski},\mbox{axiom})$

$\forall a,b,c{:}\;( ( a\subseteq b\mbox{ and }b\subseteq c ) \ \Rightarrow \ a\subseteq c )\qquad\mbox{fof}(\mbox{t1\_xboole}_1,\mbox{axiom})$

$\forall a,b{:}\;( a\subseteq b \ \Rightarrow \ \mbox{powerset}(a)\subseteq \mbox{powerset}(b) )\qquad\mbox{fof}(\mbox{t79\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET315-6.p} Corollary to range property 3






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{restrict}(\mbox{xr},x,\mbox{range\_of}(\mbox{xr}))\ne \mbox{restrict}(\mbox{xr},x,\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_to\_range\_property3}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET759+4.p} Composition of images 1

If F is injective, then the inverse image of the image of X is
 equal to X.




$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET006+1.ax'})$

$\forall f,a,b,x{:}\;( ( \mbox{maps}(f,a,b)\mbox{ and }\mbox{injective}(f,a,b)\mbox{ and }x\subseteq a ) \ \Rightarrow \ \mbox{equal\_set}(\mbox{inverse\_image}_3(f,\mbox{image}_3(f,x,b),a),x) )\qquad\mbox{fof}(\mbox{thIIa}_{09},\mbox{conjecture})$

\medskip\textbf{SET068-7.p} Proper class in an unordered pair, part 2

If one argument of an unordered pair is a proper class, the
 pair contains only the other.




$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ x\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(u,v) \ \Rightarrow \ y\in \mbox{unordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_unordered\_pair},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_1\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{ordered\_pair}(u,v)\in \mbox{cross\_product}(x,y) \ \Rightarrow \ v\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{corollary\_2\_to\_cartesian\_product},\mbox{axiom})$

$\mbox{subclass}(x,x)\qquad\mbox{cnf}(\mbox{subclass\_is\_reflexive},\mbox{axiom})$

$( \mbox{subclass}(x,y)\mbox{ and }\mbox{subclass}(y,z) ) \ \Rightarrow \ \mbox{subclass}(x,z)\qquad\mbox{cnf}(\mbox{transitivity\_of\_subclass},\mbox{axiom})$

$x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y\qquad\mbox{cnf}(\mbox{equality}_1,\mbox{axiom})$

$\mbox{not\_subclass\_element}(x,y)\in y \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(y,x)\in y )\qquad\mbox{cnf}(\mbox{equality}_2,\mbox{axiom})$

$\mbox{not\_subclass\_element}(y,x)\in x \ \Rightarrow \ ( x = y\mbox{ or }\mbox{not\_subclass\_element}(x,y)\in x )\qquad\mbox{cnf}(\mbox{equality}_3,\mbox{axiom})$

$( \mbox{not\_subclass\_element}(x,y)\in y\mbox{ and }\mbox{not\_subclass\_element}(y,x)\in x ) \ \Rightarrow \ x = y\qquad\mbox{cnf}(\mbox{equality}_4,\mbox{axiom})$

$\neg\,y\in \mbox{intersection}(x',x)\qquad\mbox{cnf}(\mbox{special\_classes\_lemma},\mbox{axiom})$

$\neg\,z\in \mbox{null\_class}\qquad\mbox{cnf}(\mbox{existence\_of\_null\_class},\mbox{axiom})$

$\mbox{subclass}(\mbox{null\_class},x)\qquad\mbox{cnf}(\mbox{null\_class\_is\_subclass},\mbox{axiom})$

$\mbox{subclass}(x,\mbox{null\_class}) \ \Rightarrow \ x = \mbox{null\_class}\qquad\mbox{cnf}(\mbox{corollary\_of\_null\_class\_is\_subclass},\mbox{axiom})$

$z = \mbox{null\_class}\mbox{ or }\mbox{not\_subclass\_element}(z,\mbox{null\_class})\in z\qquad\mbox{cnf}(\mbox{null\_class\_is\_unique},\mbox{axiom})$

$\mbox{null\_class}\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{null\_class\_is\_a\_set},\mbox{axiom})$

$\mbox{unordered\_pair}(x,y) = \mbox{unordered\_pair}(y,x)\qquad\mbox{cnf}(\mbox{commutativity\_of\_unordered\_pair},\mbox{axiom})$

$\neg\,\mbox{subclass}(\mbox{singleton}(y),\mbox{unordered\_pair}(x,y))\qquad\mbox{cnf}(\mbox{prove\_singleton\_in\_unordered\_pair2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET575$\wedge$7.p} Trybulec's 15th Boolean property of sets






$\mbox{include}(\mbox{'Axioms/LCL015\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL013\string^5.ax'})$

$\mbox{include}(\mbox{'Axioms/LCL015\string^1.ax'})$

$\in {:}\;\mbox{mu}\to \mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{member\_type},\mbox{type})$

$\mbox{intersect}{:}\;\mbox{mu}\to \mbox{mu}\to \mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{intersect\_type},\mbox{type})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{intersect}@ b@ c ) )@ ( \mbox{mexists\_ind}@ \lambda d{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \in @ d@ b ) )@ ( \mbox{mbox\_s}_4@ ( \in @ d@ c ) ) ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mexists\_ind}@ \lambda d{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \in @ d@ b ) )@ ( \mbox{mbox\_s}_4@ ( \in @ d@ c ) ) ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{intersect}@ b@ c ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{intersect\_defn},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{intersect}@ b@ c ) )@ ( \mbox{mbox\_s}_4@ ( \mbox{intersect}@ c@ b ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\mbox{mvalid}@ ( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda b{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mforall\_ind}@ \lambda c{:}\;\mbox{mu}{:}\;( \mbox{mbox\_s}_4@ ( \mbox{mimplies}@ ( \mbox{mbox\_s}_4@ ( \mbox{intersect}@ b@ c ) )@ ( \mbox{mexists\_ind}@ \lambda d{:}\;\mbox{mu}{:}\;( \mbox{mand}@ ( \mbox{mbox\_s}_4@ ( \in @ d@ b ) )@ ( \mbox{mbox\_s}_4@ ( \in @ d@ c ) ) ) ) ) ) ) ) ) )\qquad\mbox{thf}(\mbox{prove\_th}_{15},\mbox{conjecture})$

\medskip\textbf{SET846-2.p} Problem about Zorn's lemma






$\mbox{c\_in}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a})) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_B},\mbox{c\_Union}(\mbox{v\_A},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_OUnion\_\_upper}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_refl}_0,\mbox{axiom})$

$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_OsuccI}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_Y},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))) \ \Rightarrow \ \mbox{c\_in}(\mbox{c\_Union}(\mbox{v\_Y},\mbox{tc\_set}(\mbox{t\_a})),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_UnionI}_0,\mbox{axiom})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{t\_a}),\mbox{c\_Union}(\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{t\_a})),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET115-6.p} 1st is unique if x is not an ordered pair of sets, part 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{ordered\_pair}(\mbox{first}_1(x),\mbox{second}_1(x))\ne x\qquad\mbox{cnf}(\mbox{prove\_unique\_1st\_and\_2nd\_in\_pair\_of\_non\_sets3}_1,\mbox{negated\_conjecture})$

$\mbox{first}(x)\ne x\qquad\mbox{cnf}(\mbox{prove\_unique\_1st\_in\_pair\_of\_non\_sets},\mbox{negated\_conjecture})$

\medskip\textbf{SET063-6.p} If X is a subset of the empty set, then X is the empty set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{subclass}(x,\mbox{null\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_of\_null\_class\_is\_subclass}_1,\mbox{negated\_conjecture})$

$x\ne \mbox{null\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_of\_null\_class\_is\_subclass}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET638$\wedge$5.p} TPS problem BOOL-PROP-120

Trybulec's 120th Boolean property of sets




$a{:}\;\mbox{\$tType}\qquad\mbox{thf}(\mbox{a\_type},\mbox{type})$

$\forall x{:}\;a\to \mbox{\$o},y{:}\;a\to \mbox{\$o},z{:}\;a\to \mbox{\$o}{:}\;( ( \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx}\mbox{ or }z@ \mbox{xx} ) )\mbox{ and }( \lambda \mbox{xx}{:}\;a{:}\;( x@ \mbox{xx}\mbox{ and }z@ \mbox{xx} ) ) = ( \lambda \mbox{xx}{:}\;a{:}\;\mbox{\$false} ) ) \ \Rightarrow \ \forall \mbox{xx}{:}\;a{:}\;( ( x@ \mbox{xx} ) \ \Rightarrow \ ( y@ \mbox{xx} ) ) )\qquad\mbox{thf}(\mbox{cBOOL\_PROP\_120\_pme},\mbox{conjecture})$

\medskip\textbf{SET629+3.p} X $\wedge$ Y is disjoint from X \ Y

The intersection of X and Y is disjoint from the difference of
 X and Y.




$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( d\in ( b\setminus c )\iff ( d\in b\mbox{ and }\neg\,d\in c ) )\qquad\mbox{fof}(\mbox{difference\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c)\iff \exists d{:}\;( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersect\_defn},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{disjoint}(b,c)\iff \neg\,\mbox{intersect}(b,c) )\qquad\mbox{fof}(\mbox{disjoint\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b,c{:}\;( \mbox{intersect}(b,c) \ \Rightarrow \ \mbox{intersect}(c,b) )\qquad\mbox{fof}(\mbox{symmetry\_of\_intersect},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{disjoint}(\mbox{intersection}(b,c),b\setminus c)\qquad\mbox{fof}(\mbox{prove\_intersection\_and\_difference\_disjoint},\mbox{conjecture})$

\medskip\textbf{SET005-1.p} Associativity of set intersection






$\mbox{include}(\mbox{'Axioms/SET001-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET001-2.ax'})$

$\mbox{intersection}(a,b,\mbox{aIb})\qquad\mbox{cnf}(\mbox{a\_intersection\_b},\mbox{axiom})$

$\mbox{intersection}(b,c,\mbox{bIc})\qquad\mbox{cnf}(\mbox{b\_intersection\_c},\mbox{axiom})$

$\mbox{intersection}(a,\mbox{bIc},\mbox{aIbIc})\qquad\mbox{cnf}(\mbox{a\_intersection\_bIc},\mbox{axiom})$

$\neg\,\mbox{intersection}(\mbox{aIb},c,\mbox{aIbIc})\qquad\mbox{cnf}(\mbox{prove\_aIb\_intersection\_c\_is\_aIbIc},\mbox{negated\_conjecture})$

\medskip\textbf{SET374-6.p} Power class is closed under union






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{power\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_under\_union}_1,\mbox{negated\_conjecture})$

$y\in \mbox{power\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_under\_union}_2,\mbox{negated\_conjecture})$

$\neg\,\mbox{union}(x,y)\in \mbox{power\_class}(z)\qquad\mbox{cnf}(\mbox{prove\_power\_class\_under\_union}_3,\mbox{negated\_conjecture})$

\medskip\textbf{SET199+3.p} If Z (= X and Z (= Y, then Z (= X $\wedge$ Y

If Z is a subset of X and Z is a subset of Y, then Z is a
 subset of the intersection of X and Y.




$\forall b,c,d{:}\;( d\in \mbox{intersection}(b,c)\iff ( d\in b\mbox{ and }d\in c ) )\qquad\mbox{fof}(\mbox{intersection\_defn},\mbox{axiom})$

$\forall b,c{:}\;( b\subseteq c\iff \forall d{:}\;( d\in b \ \Rightarrow \ d\in c ) )\qquad\mbox{fof}(\mbox{subset\_defn},\mbox{axiom})$

$\forall b,c{:}\;\mbox{intersection}(b,c) = \mbox{intersection}(c,b)\qquad\mbox{fof}(\mbox{commutativity\_of\_intersection},\mbox{axiom})$

$\forall b{:}\;b\subseteq b\qquad\mbox{fof}(\mbox{reflexivity\_of\_subset},\mbox{axiom})$

$\forall b,c{:}\;( b = c\iff \forall d{:}\;( d\in b\iff d\in c ) )\qquad\mbox{fof}(\mbox{equal\_member\_defn},\mbox{axiom})$

$\forall b,c,d{:}\;( ( b\subseteq c\mbox{ and }b\subseteq d ) \ \Rightarrow \ b\subseteq \mbox{intersection}(c,d) )\qquad\mbox{fof}(\mbox{prove\_intersection\_of\_subsets},\mbox{conjecture})$

\medskip\textbf{SET154-6.p} Union with complement is universal class






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{union}(x',x)\ne \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_union\_with\_complement}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET464-6.p} Class image under identity






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{image}(\mbox{restrict}(\mbox{identity\_relation},x,x),y)\ne \mbox{intersection}(x,y)\qquad\mbox{cnf}(\mbox{prove\_class\_image\_under\_identity}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET354-6.p} Sum of ordered pair






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_sum\_of\_ordered\_pair}_1,\mbox{negated\_conjecture})$

$\mbox{sum\_class}(\mbox{ordered\_pair}(x,y))\ne \mbox{unordered\_pair}(x,\mbox{singleton}(y))\qquad\mbox{cnf}(\mbox{prove\_sum\_of\_ordered\_pair}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET421-6.p} Compose class is a function






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\neg\,\mbox{function}(\mbox{compose\_class}(x))\qquad\mbox{cnf}(\mbox{prove\_compose\_class\_is\_a\_function}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET105+1.p} Special member 3 of an ordered pair






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( \mbox{unordered\_pair}(\mbox{null\_class},\mbox{singleton}(\mbox{null\_class})) = \mbox{ordered\_pair}(x,y)\mbox{ or }x\in \mbox{universal\_class}\mbox{ or }y\in \mbox{universal\_class} )\qquad\mbox{fof}(\mbox{property\_3\_of\_ordered\_pair},\mbox{conjecture})$

\medskip\textbf{SET379-6.p} Corollary 3 to power class set builder






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},z),z)\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_power\_class\_set\_builder}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{subclass}(\mbox{image}(\mbox{successor\_relation},\mbox{intersection}(\mbox{power\_class}(z),z)),\mbox{intersection}(\mbox{power\_class}(z),z))\qquad\mbox{cnf}(\mbox{prove\_corollary\_1\_to\_power\_class\_set\_builder}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET520-6.p} Corollary 2 to no cycles of length 2






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(x,y) = y\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_no\_cycles\_length\_2}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET853-2.p} Problem about Zorn's lemma






$\mbox{c\_in}(\mbox{v\_x},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_0,\mbox{negated\_conjecture})$

$\mbox{c\_in}(\mbox{v\_xa},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_1,\mbox{negated\_conjecture})$

$\mbox{c\_lessequals}(\mbox{v\_xa},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_2,\mbox{negated\_conjecture})$

$\mbox{v\_xa}\ne \mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a})\qquad\mbox{cnf}(\mbox{cls\_conjecture}_3,\mbox{negated\_conjecture})$

$\neg\,\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_xa},\mbox{t\_a}),\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_x},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_conjecture}_4,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_U},\mbox{v\_x},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_U},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_U},\mbox{t\_a}),\mbox{v\_x},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{v\_U} = \mbox{v\_x} )\qquad\mbox{cnf}(\mbox{cls\_conjecture}_5,\mbox{negated\_conjecture})$

$( \mbox{c\_lessequals}(\mbox{v\_B},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\mbox{ and }\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_B},\mbox{tc\_set}(\mbox{t\_a})) ) \ \Rightarrow \ \mbox{v\_A} = \mbox{v\_B}\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_antisym}_0,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_A},\mbox{v\_A},\mbox{tc\_set}(\mbox{t\_a}))\qquad\mbox{cnf}(\mbox{cls\_Set\_Osubset\_\_refl}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_in}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_0,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_1,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}) = \mbox{v\_m} ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_2,\mbox{axiom})$

$( \mbox{c\_in}(\mbox{v\_m},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_in}(\mbox{v\_n},\mbox{c\_Zorn\_OTFin}(\mbox{v\_S},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ and }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{c\_Zorn\_OTFin\_\_linear\_\_lemma1\_}_1(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{t\_a}),\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) ) \ \Rightarrow \ ( \mbox{c\_lessequals}(\mbox{v\_n},\mbox{v\_m},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\mbox{ or }\mbox{c\_lessequals}(\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_m},\mbox{t\_a}),\mbox{v\_n},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) )\qquad\mbox{cnf}(\mbox{cls\_Zorn\_OTFin\_\_linear\_\_lemma1}_3,\mbox{axiom})$

$\mbox{c\_lessequals}(\mbox{v\_x},\mbox{v\_y},\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a}))) \ \Rightarrow \ \mbox{c\_lessequals}(\mbox{v\_x},\mbox{c\_Zorn\_Osucc}(\mbox{v\_S},\mbox{v\_y},\mbox{t\_a}),\mbox{tc\_set}(\mbox{tc\_set}(\mbox{t\_a})))\qquad\mbox{cnf}(\mbox{cls\_Zorn\_Osucc\_\_trans}_0,\mbox{axiom})$

\medskip\textbf{SET077-6.p} Every singleton is a set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,\mbox{singleton}(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_singletons\_are\_sets}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET122-6.p} Corollary 4 to components of equal ordered pairs being equal






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\neg\,y\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_OP\_determines\_components2}_1,\mbox{negated\_conjecture})$

$\mbox{second}(\mbox{ordered\_pair}(x,y))\ne \mbox{ordered\_pair}(x,y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_OP\_determines\_components2}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET067+1.p} If one argument is a proper class, pair contains only the other






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;\mbox{subclass}(\mbox{unordered\_pair}(x,x),\mbox{unordered\_pair}(x,y))\qquad\mbox{fof}(\mbox{pair\_contains\_other},\mbox{conjecture})$

\medskip\textbf{SET874+1.p} union(singleton(A),unordered\_pair(A,B)) = unordered\_pair(A,B)






$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \neg\,\mbox{in}(b,a) )\qquad\mbox{fof}(\mbox{antisymmetry\_r2\_hidden},\mbox{axiom})$

$\forall a,b{:}\;\mbox{unordered\_pair}(a,b) = \mbox{unordered\_pair}(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,b) = \mbox{set\_union}_2(b,a)\qquad\mbox{fof}(\mbox{commutativity\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b,c{:}\;( c = \mbox{unordered\_pair}(a,b)\iff \forall d{:}\;( \mbox{in}(d,c)\iff ( d = a\mbox{ or }d = b ) ) )\qquad\mbox{fof}(\mbox{d2\_tarski},\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(a,b)) )\qquad\mbox{fof}(\mbox{fc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \neg\,\mbox{empty}(a) \ \Rightarrow \ \neg\,\mbox{empty}(\mbox{set\_union}_2(b,a)) )\qquad\mbox{fof}(\mbox{fc3\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(a,a) = a\qquad\mbox{fof}(\mbox{idempotence\_k2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;( \mbox{in}(a,b) \ \Rightarrow \ \mbox{set\_union}_2(\mbox{singleton}(a),b) = b )\qquad\mbox{fof}(\mbox{l23\_zfmisc}_1,\mbox{axiom})$

$\exists a{:}\;\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc1\_xboole}_0,\mbox{axiom})$

$\exists a{:}\;\neg\,\mbox{empty}(a)\qquad\mbox{fof}(\mbox{rc2\_xboole}_0,\mbox{axiom})$

$\forall a,b{:}\;\mbox{set\_union}_2(\mbox{singleton}(a),\mbox{unordered\_pair}(a,b)) = \mbox{unordered\_pair}(a,b)\qquad\mbox{fof}(\mbox{t14\_zfmisc}_1,\mbox{conjecture})$

\medskip\textbf{SET392-6.p} Right identity for composition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\circ \mbox{identity\_relation}\ne \mbox{restrict}(x,\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_right\_identity\_for\_composition}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET071+1.p} If both arguments are proper classes, pair is null






$\mbox{include}(\mbox{'Axioms/SET005+0.ax'})$

$\forall x,y{:}\;( ( \neg\,x\in \mbox{universal\_class}\mbox{ and }\neg\,y\in \mbox{universal\_class} ) \ \Rightarrow \ \mbox{unordered\_pair}(x,y) = \mbox{null\_class} )\qquad\mbox{fof}(\mbox{null\_unordered\_pair},\mbox{conjecture})$

\medskip\textbf{SET393-6.p} Left identity for composition






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{identity\_relation}\circ x\ne \mbox{restrict}(x,\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_left\_identity\_for\_composition}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET090-6.p} The member of a singleton set is unique






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$u\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_member\_of\_singleton\_is\_unique}_1,\mbox{negated\_conjecture})$

$\mbox{member\_of}(\mbox{singleton}(u))\ne u\qquad\mbox{cnf}(\mbox{prove\_member\_of\_singleton\_is\_unique}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET512-6.p} Corollary 2 to special members of ordered pairs






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{unordered\_pair}(\mbox{null\_class},\mbox{singleton}(\mbox{singleton}(y)))\ne \mbox{ordered\_pair}(\mbox{universal\_class},y)\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_property\_1\_of\_ordered\_pair}_1,\mbox{negated\_conjecture})$

\medskip\textbf{SET479-6.p} Replacement property 1






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$x\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property1}_1,\mbox{negated\_conjecture})$

$\neg\,\mbox{domain\_of}(x)\in \mbox{universal\_class}\qquad\mbox{cnf}(\mbox{prove\_replacement\_property1}_2,\mbox{negated\_conjecture})$

\medskip\textbf{SET680$\wedge$3.p} !x in D, x the domain of R (X to Y) iff ?y in E : $<$x,y$>$ in R

For every element x in D, x is in the domain of a relation R from
 X to Y iff there exists an element y in E such that $<$x,y$>$ is in R.




$\mbox{include}(\mbox{'Axioms/SET008\string^0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET008\string^2.ax'})$

$\forall r{:}\;\mbox{\$i}\to \mbox{\$i}\to \mbox{\$o},x{:}\;\mbox{\$i}\to \mbox{\$o},y{:}\;\mbox{\$i}\to \mbox{\$o}{:}\;( ( \mbox{is\_rel\_on}@ r@ x@ y ) \ \Rightarrow \ \forall u{:}\;\mbox{\$i}{:}\;( ( x@ u ) \ \Rightarrow \ ( ( \mbox{rel\_domain}@ r@ u )\iff \exists v{:}\;\mbox{\$i}{:}\;( y@ v\mbox{ and }r@ u@ v ) ) ) )\qquad\mbox{thf}(\mbox{thm},\mbox{conjecture})$

\medskip\textbf{SET062$\wedge$6.p} TPS problem from BASIC-FO-THMS

Trybulec's 27th Boolean property of sets.




$\mbox{cA}{:}\;\mbox{\$i}\to \mbox{\$o}\qquad\mbox{thf}(\mbox{cA},\mbox{type})$

$\forall z_3{:}\;\mbox{\$i}{:}\;( \mbox{\$false} \ \Rightarrow \ ( \mbox{cA}@ z_3 ) )\qquad\mbox{thf}(\mbox{cSET76\_pme},\mbox{conjecture})$

\medskip\textbf{SET603+4.p} The difference of X and the empty set is X






$\mbox{include}(\mbox{'Axioms/SET006+0.ax'})$

$\forall e{:}\;\mbox{equal\_set}(e\setminus \mbox{empty\_set},e)\qquad\mbox{fof}(\mbox{thI}_{30},\mbox{conjecture})$

\medskip\textbf{SET505-6.p} Corollary 2 to universal class not set






$\mbox{include}(\mbox{'Axioms/SET004-0.ax'})$

$\mbox{include}(\mbox{'Axioms/SET004-1.ax'})$

$\mbox{ordered\_pair}(\mbox{universal\_class},y)\in \mbox{cross\_product}(\mbox{universal\_class},\mbox{universal\_class})\qquad\mbox{cnf}(\mbox{prove\_corollary\_2\_to\_universal\_class\_not\_set}_1,\mbox{negated\_conjecture})$

\medskip
\end{document}