:: ZF_FUND1 semantic presentation begin registrationlet "V" be ($#m1_hidden :::"Universe":::); cluster ($#v1_relat_1 :::"Relation-like"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" "V"; end; definitioncanceled; let "V" be ($#m1_hidden :::"Universe":::); let "x", "y" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "V")); :: original: :::"(#)"::: redefine func "x" :::"(#)"::: "y" -> ($#v1_relat_1 :::"Relation-like"::: ) ($#m1_subset_1 :::"Element"::: ) "of" "V"; end; :: deftheorem ZF_FUND1:def 1 : canceled; definitionfunc :::"decode"::: -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "," (Set ($#k1_zf_lang :::"VAR"::: ) ) means :: ZF_FUND1:def 2 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zf_lang :::"x."::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "p")) ")" )))); end; :: deftheorem defines :::"decode"::: ZF_FUND1:def 2 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "," (Set ($#k1_zf_lang :::"VAR"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) )) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "holds" (Bool (Set (Set (Var "b1")) ($#k3_funct_2 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k2_zf_lang :::"x."::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "p")) ")" )))) ")" )); definitionlet "v1" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ); func :::"x"."::: "v1" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) means :: ZF_FUND1:def 3 (Bool (Set ($#k2_zf_lang :::"x."::: ) it) ($#r1_hidden :::"="::: ) "v1"); end; :: deftheorem defines :::"x"."::: ZF_FUND1:def 3 : (Bool "for" (Set (Var "v1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k3_zf_fund1 :::"x"."::: ) (Set (Var "v1")))) "iff" (Bool (Set ($#k2_zf_lang :::"x."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Var "v1"))) ")" ))); definitionlet "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ); func :::"code"::: "A" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) equals :: ZF_FUND1:def 4 (Set (Set "(" (Set ($#k2_zf_fund1 :::"decode"::: ) ) ($#k2_funct_1 :::"""::: ) ")" ) ($#k7_relat_1 :::".:"::: ) "A"); end; :: deftheorem defines :::"code"::: ZF_FUND1:def 4 : (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "holds" (Bool (Set ($#k4_zf_fund1 :::"code"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k2_zf_fund1 :::"decode"::: ) ) ($#k2_funct_1 :::"""::: ) ")" ) ($#k7_relat_1 :::".:"::: ) (Set (Var "A"))))); registrationlet "A" be ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ); cluster (Set ($#k4_zf_fund1 :::"code"::: ) "A") -> ($#v1_finset_1 :::"finite"::: ) ; end; definitionlet "H" be ($#m2_finseq_1 :::"ZF-formula":::); let "E" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; func :::"Diagram"::: "(" "H" "," "E" ")" -> ($#m1_hidden :::"set"::: ) means :: ZF_FUND1:def 5 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "," "E" "st" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k4_zf_fund1 :::"code"::: ) (Set "(" ($#k2_zf_model :::"Free"::: ) "H" ")" ) ")" ))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k5_zf_model :::"St"::: ) "(" "H" "," "E" ")" )) ")" )) ")" )); end; :: deftheorem defines :::"Diagram"::: ZF_FUND1:def 5 : (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"ZF-formula":::) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b3")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set (Var "H")) "," (Set (Var "E")) ")" )) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "," (Set (Var "E")) "st" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k4_zf_fund1 :::"code"::: ) (Set "(" ($#k2_zf_model :::"Free"::: ) (Set (Var "H")) ")" ) ")" ))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k5_zf_model :::"St"::: ) "(" (Set (Var "H")) "," (Set (Var "E")) ")" )) ")" )) ")" )) ")" )))); definitionlet "V" be ($#m1_hidden :::"Universe":::); let "X" be ($#m1_subset_1 :::"Subclass":::) "of" (Set (Const "V")); attr "X" is :::"closed_wrt_A1"::: means :: ZF_FUND1:def 6 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "V" "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "{" (Set ($#k5_classes2 :::"{"::: ) (Set ($#k6_classes2 :::"["::: ) (Set "(" ($#k2_ordinal4 :::"0-element_of"::: ) "V" ")" ) "," (Set (Var "x")) ($#k6_classes2 :::"]"::: ) ) "," (Set ($#k6_classes2 :::"["::: ) (Set "(" ($#k3_ordinal4 :::"1-element_of"::: ) "V" ")" ) "," (Set (Var "y")) ($#k6_classes2 :::"]"::: ) ) ($#k5_classes2 :::"}"::: ) ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" "V" : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "y"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) ")" ) "}" ($#r2_hidden :::"in"::: ) "X")); attr "X" is :::"closed_wrt_A2"::: means :: ZF_FUND1:def 7 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "V" "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set ($#k5_classes2 :::"{"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k5_classes2 :::"}"::: ) ) ($#r2_hidden :::"in"::: ) "X")); attr "X" is :::"closed_wrt_A3"::: means :: ZF_FUND1:def 8 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "V" "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set ($#k3_classes2 :::"union"::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) "X")); attr "X" is :::"closed_wrt_A4"::: means :: ZF_FUND1:def 9 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "V" "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "{" (Set ($#k1_classes2 :::"{"::: ) (Set ($#k6_classes2 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k6_classes2 :::"]"::: ) ) ($#k1_classes2 :::"}"::: ) ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" "V" : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) ")" ) "}" ($#r2_hidden :::"in"::: ) "X")); attr "X" is :::"closed_wrt_A5"::: means :: ZF_FUND1:def 10 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "V" "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "{" (Set "(" (Set (Var "x")) ($#k7_classes2 :::"\/"::: ) (Set (Var "y")) ")" ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" "V" : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) ")" ) "}" ($#r2_hidden :::"in"::: ) "X")); attr "X" is :::"closed_wrt_A6"::: means :: ZF_FUND1:def 11 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "V" "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "{" (Set "(" (Set (Var "x")) ($#k9_classes2 :::"\"::: ) (Set (Var "y")) ")" ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" "V" : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) ")" ) "}" ($#r2_hidden :::"in"::: ) "X")); attr "X" is :::"closed_wrt_A7"::: means :: ZF_FUND1:def 12 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "V" "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "X") & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool "{" (Set "(" (Set (Var "x")) ($#k1_zf_fund1 :::"(#)"::: ) (Set (Var "y")) ")" ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" "V" : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) ")" ) "}" ($#r2_hidden :::"in"::: ) "X")); end; :: deftheorem defines :::"closed_wrt_A1"::: ZF_FUND1:def 6 : (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_zf_fund1 :::"closed_wrt_A1"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "{" (Set ($#k5_classes2 :::"{"::: ) (Set ($#k6_classes2 :::"["::: ) (Set "(" ($#k2_ordinal4 :::"0-element_of"::: ) (Set (Var "V")) ")" ) "," (Set (Var "x")) ($#k6_classes2 :::"]"::: ) ) "," (Set ($#k6_classes2 :::"["::: ) (Set "(" ($#k3_ordinal4 :::"1-element_of"::: ) (Set (Var "V")) ")" ) "," (Set (Var "y")) ($#k6_classes2 :::"]"::: ) ) ($#k5_classes2 :::"}"::: ) ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "y"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) ")" ) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" ))); :: deftheorem defines :::"closed_wrt_A2"::: ZF_FUND1:def 7 : (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v2_zf_fund1 :::"closed_wrt_A2"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k5_classes2 :::"{"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k5_classes2 :::"}"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" ))); :: deftheorem defines :::"closed_wrt_A3"::: ZF_FUND1:def 8 : (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v3_zf_fund1 :::"closed_wrt_A3"::: ) ) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k3_classes2 :::"union"::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" ))); :: deftheorem defines :::"closed_wrt_A4"::: ZF_FUND1:def 9 : (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v4_zf_fund1 :::"closed_wrt_A4"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "{" (Set ($#k1_classes2 :::"{"::: ) (Set ($#k6_classes2 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k6_classes2 :::"]"::: ) ) ($#k1_classes2 :::"}"::: ) ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) ")" ) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" ))); :: deftheorem defines :::"closed_wrt_A5"::: ZF_FUND1:def 10 : (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v5_zf_fund1 :::"closed_wrt_A5"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "{" (Set "(" (Set (Var "x")) ($#k7_classes2 :::"\/"::: ) (Set (Var "y")) ")" ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) ")" ) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" ))); :: deftheorem defines :::"closed_wrt_A6"::: ZF_FUND1:def 11 : (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v6_zf_fund1 :::"closed_wrt_A6"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "{" (Set "(" (Set (Var "x")) ($#k9_classes2 :::"\"::: ) (Set (Var "y")) ")" ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) ")" ) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" ))); :: deftheorem defines :::"closed_wrt_A7"::: ZF_FUND1:def 12 : (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v7_zf_fund1 :::"closed_wrt_A7"::: ) ) "iff" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "{" (Set "(" (Set (Var "x")) ($#k1_zf_fund1 :::"(#)"::: ) (Set (Var "y")) ")" ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) ")" ) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" ))); definitionlet "V" be ($#m1_hidden :::"Universe":::); let "X" be ($#m1_subset_1 :::"Subclass":::) "of" (Set (Const "V")); attr "X" is :::"closed_wrt_A1-A7"::: means :: ZF_FUND1:def 13 (Bool "(" (Bool "X" "is" ($#v1_zf_fund1 :::"closed_wrt_A1"::: ) ) & (Bool "X" "is" ($#v2_zf_fund1 :::"closed_wrt_A2"::: ) ) & (Bool "X" "is" ($#v3_zf_fund1 :::"closed_wrt_A3"::: ) ) & (Bool "X" "is" ($#v4_zf_fund1 :::"closed_wrt_A4"::: ) ) & (Bool "X" "is" ($#v5_zf_fund1 :::"closed_wrt_A5"::: ) ) & (Bool "X" "is" ($#v6_zf_fund1 :::"closed_wrt_A6"::: ) ) & (Bool "X" "is" ($#v7_zf_fund1 :::"closed_wrt_A7"::: ) ) ")" ); end; :: deftheorem defines :::"closed_wrt_A1-A7"::: ZF_FUND1:def 13 : (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) "iff" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_zf_fund1 :::"closed_wrt_A1"::: ) ) & (Bool (Set (Var "X")) "is" ($#v2_zf_fund1 :::"closed_wrt_A2"::: ) ) & (Bool (Set (Var "X")) "is" ($#v3_zf_fund1 :::"closed_wrt_A3"::: ) ) & (Bool (Set (Var "X")) "is" ($#v4_zf_fund1 :::"closed_wrt_A4"::: ) ) & (Bool (Set (Var "X")) "is" ($#v5_zf_fund1 :::"closed_wrt_A5"::: ) ) & (Bool (Set (Var "X")) "is" ($#v6_zf_fund1 :::"closed_wrt_A6"::: ) ) & (Bool (Set (Var "X")) "is" ($#v7_zf_fund1 :::"closed_wrt_A7"::: ) ) ")" ) ")" ))); theorem :: ZF_FUND1:1 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "o")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "V"))) & "(" (Bool (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "implies" (Bool (Set (Var "o")) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V"))) ")" & "(" (Bool (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "implies" (Bool (Set (Var "o")) "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V"))) ")" ")" )))) ; theorem :: ZF_FUND1:2 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "o")) "," (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) )) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "implies" (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "o")) ($#k1_tarski :::"}"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" & "(" (Bool (Bool (Set ($#k1_tarski :::"{"::: ) (Set (Var "o")) ($#k1_tarski :::"}"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "implies" (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" & "(" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "implies" (Bool (Set ($#k3_tarski :::"union"::: ) (Set (Var "A"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" ")" )))) ; theorem :: ZF_FUND1:3 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) )) "holds" (Bool (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))) ; theorem :: ZF_FUND1:4 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set (Set (Var "A")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "B"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "A")) ($#k6_subset_1 :::"\"::: ) (Set (Var "B"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "A")) ($#k3_relat_1 :::"(#)"::: ) (Set (Var "B"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" )))) ; theorem :: ZF_FUND1:5 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "A")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "B"))) ($#r2_hidden :::"in"::: ) (Set (Var "X")))))) ; theorem :: ZF_FUND1:6 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "o")) "," (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set ($#k2_tarski :::"{"::: ) (Set (Var "o")) "," (Set (Var "p")) ($#k2_tarski :::"}"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" )))) ; theorem :: ZF_FUND1:7 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) )) "holds" (Bool (Set ($#k4_ordinal1 :::"omega"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "X"))))) ; theorem :: ZF_FUND1:8 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "fs")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) )) "holds" (Bool (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "fs")) "," (Set ($#k4_ordinal1 :::"omega"::: ) ) ")" ) ($#r1_tarski :::"c="::: ) (Set (Var "X")))))) ; theorem :: ZF_FUND1:9 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "fs")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "fs")) "," (Set (Var "a")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))))) ; theorem :: ZF_FUND1:10 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "fs")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "fs")) "," (Set ($#k4_ordinal1 :::"omega"::: ) ) ")" )) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "{" (Set "(" (Set (Var "a")) ($#k1_zf_fund1 :::"(#)"::: ) (Set (Var "x")) ")" ) where x "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X"))))))) ; theorem :: ZF_FUND1:11 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) (Bool "for" (Set (Var "fs")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set (Var "fs"))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r1_tarski :::"c="::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "fs")) "," (Set (Var "a")) ")" ))) "holds" (Bool "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set "(" (Set (Var "fs")) ($#k7_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "n")) ($#k6_domain_1 :::"}"::: ) ) ")" ) "," (Set (Var "a")) ")" )) & (Bool "ex" (Set (Var "u")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "n")) "," (Set (Var "u")) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "b")))) ")" ) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X")))))))) ; theorem :: ZF_FUND1:12 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "{" (Set "(" (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_domain_1 :::"["::: ) (Set (Var "n")) "," (Set (Var "x")) ($#k1_domain_1 :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "y")) ")" ) where x, y "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) ")" ) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X"))))))) ; theorem :: ZF_FUND1:13 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "B")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "B")) "is" ($#v1_finset_1 :::"finite"::: ) ) & (Bool "(" "for" (Set (Var "o")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "o")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" )) "holds" (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))))) ; theorem :: ZF_FUND1:14 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "fs")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "fs")) "," (Set (Var "A")) ")" ))) "holds" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))))))) ; theorem :: ZF_FUND1:15 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "a")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "fs")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "a")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "fs")) "," (Set (Var "a")) ")" ))) "holds" (Bool "{" (Set "(" (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_domain_1 :::"["::: ) (Set (Var "n")) "," (Set (Var "x")) ($#k1_domain_1 :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "y")) ")" ) where x "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X")))))))) ; theorem :: ZF_FUND1:16 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "a")) "," (Set (Var "y")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) (Bool "for" (Set (Var "fs")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Bool "not" (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set (Var "fs")))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "a")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set (Var "fs")) "," (Set (Var "a")) ")" )) & (Bool (Set (Var "b")) ($#r1_tarski :::"c="::: ) (Set ($#k1_funct_2 :::"Funcs"::: ) "(" (Set "(" (Set (Var "fs")) ($#k4_subset_1 :::"\/"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "n")) ($#k6_domain_1 :::"}"::: ) ) ")" ) "," (Set (Var "a")) ")" )) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) & (Bool (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_domain_1 :::"["::: ) (Set (Var "n")) "," (Set (Var "x")) ($#k1_domain_1 :::"]"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Var "b"))) ")" ) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X")))))))) ; theorem :: ZF_FUND1:17 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "{" (Set ($#k5_classes2 :::"{"::: ) (Set ($#k6_classes2 :::"["::: ) (Set "(" ($#k2_ordinal4 :::"0-element_of"::: ) (Set (Var "V")) ")" ) "," (Set (Var "x")) ($#k6_classes2 :::"]"::: ) ) "," (Set ($#k6_classes2 :::"["::: ) (Set "(" ($#k3_ordinal4 :::"1-element_of"::: ) (Set (Var "V")) ")" ) "," (Set (Var "x")) ($#k6_classes2 :::"]"::: ) ) ($#k5_classes2 :::"}"::: ) ) where x "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "V")) : (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "a"))) "}" ($#r2_hidden :::"in"::: ) (Set (Var "X")))))) ; theorem :: ZF_FUND1:18 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "E")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "holds" (Bool "(" (Bool (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set "(" (Set (Var "v1")) ($#k4_zf_lang :::"'='"::: ) (Set (Var "v2")) ")" ) "," (Set (Var "E")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set "(" (Set (Var "v1")) ($#k5_zf_lang :::"'in'"::: ) (Set (Var "v2")) ")" ) "," (Set (Var "E")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" ))))) ; theorem :: ZF_FUND1:19 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "E")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"ZF-formula":::) "st" (Bool (Bool (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set (Var "H")) "," (Set (Var "E")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set "(" ($#k6_zf_lang :::"'not'"::: ) (Set (Var "H")) ")" ) "," (Set (Var "E")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))))) ; theorem :: ZF_FUND1:20 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "E")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "for" (Set (Var "H")) "," (Set (Var "H9")) "being" ($#m2_finseq_1 :::"ZF-formula":::) "st" (Bool (Bool (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set (Var "H")) "," (Set (Var "E")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set (Var "H9")) "," (Set (Var "E")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set "(" (Set (Var "H")) ($#k7_zf_lang :::"'&'"::: ) (Set (Var "H9")) ")" ) "," (Set (Var "E")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))))) ; theorem :: ZF_FUND1:21 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "E")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"ZF-formula":::) (Bool "for" (Set (Var "v1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "st" (Bool (Bool (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set (Var "H")) "," (Set (Var "E")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set "(" ($#k8_zf_lang :::"All"::: ) "(" (Set (Var "v1")) "," (Set (Var "H")) ")" ")" ) "," (Set (Var "E")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X")))))))) ; theorem :: ZF_FUND1:22 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"ZF-formula":::) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) ) & (Bool (Set (Var "E")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set ($#k5_zf_fund1 :::"Diagram"::: ) "(" (Set (Var "H")) "," (Set (Var "E")) ")" ) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))))) ; theorem :: ZF_FUND1:23 (Bool "for" (Set (Var "V")) "being" ($#m1_hidden :::"Universe":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subclass":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v8_zf_fund1 :::"closed_wrt_A1-A7"::: ) )) "holds" (Bool "(" (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k2_ordinal4 :::"0-element_of"::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k3_ordinal4 :::"1-element_of"::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" )))) ; theorem :: ZF_FUND1:24 (Bool "for" (Set (Var "o")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k2_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set (Var "p")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) ($#k2_tarski :::"}"::: ) ) ($#k3_relat_1 :::"(#)"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_tarski :::"]"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set (Var "q")) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_tarski :::"]"::: ) ) ($#k2_tarski :::"}"::: ) ))) ; theorem :: ZF_FUND1:25 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "o")) "," (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "r")))) "holds" (Bool (Set (Set ($#k2_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set (Var "p")) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set (Var "q")) "," (Set (Var "r")) ($#k4_tarski :::"]"::: ) ) ($#k2_tarski :::"}"::: ) ) ($#k3_relat_1 :::"(#)"::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_tarski :::"]"::: ) ) ($#k2_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set (Var "s")) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set (Var "q")) "," (Set (Var "t")) ($#k4_tarski :::"]"::: ) ) ($#k2_tarski :::"}"::: ) ))) ; theorem :: ZF_FUND1:26 (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "holds" (Bool "(" (Bool (Set ($#k4_zf_fund1 :::"code"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v1")) ($#k6_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k3_zf_fund1 :::"x"."::: ) (Set (Var "v1")) ")" ) ($#k6_domain_1 :::"}"::: ) )) & (Bool (Set ($#k4_zf_fund1 :::"code"::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "v1")) "," (Set (Var "v2")) ($#k7_domain_1 :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k7_domain_1 :::"{"::: ) (Set "(" ($#k3_zf_fund1 :::"x"."::: ) (Set (Var "v1")) ")" ) "," (Set "(" ($#k3_zf_fund1 :::"x"."::: ) (Set (Var "v2")) ")" ) ($#k7_domain_1 :::"}"::: ) )) ")" )) ; theorem :: ZF_FUND1:27 (Bool "for" (Set (Var "o")) "," (Set (Var "q")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "o")) "," (Set (Var "q")) ($#k2_tarski :::"}"::: ) )) "iff" (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k4_tarski :::"["::: ) (Set (Var "o")) "," (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "o")) ")" ) ($#k4_tarski :::"]"::: ) ) "," (Set ($#k4_tarski :::"["::: ) (Set (Var "q")) "," (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "q")) ")" ) ($#k4_tarski :::"]"::: ) ) ($#k2_tarski :::"}"::: ) )) ")" ))) ; theorem :: ZF_FUND1:28 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_zf_lang :::"VAR"::: ) )) & (Bool (Set ($#k2_zf_fund1 :::"decode"::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set (Set ($#k2_zf_fund1 :::"decode"::: ) ) ($#k2_funct_1 :::"""::: ) ) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) & (Bool (Set ($#k9_xtuple_0 :::"dom"::: ) (Set "(" (Set ($#k2_zf_fund1 :::"decode"::: ) ) ($#k2_funct_1 :::"""::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_zf_lang :::"VAR"::: ) )) & (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set "(" (Set ($#k2_zf_fund1 :::"decode"::: ) ) ($#k2_funct_1 :::"""::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )) ")" ) ; theorem :: ZF_FUND1:29 (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "holds" (Bool (Set (Var "A")) "," (Set ($#k4_zf_fund1 :::"code"::: ) (Set (Var "A"))) ($#r2_tarski :::"are_equipotent"::: ) )) ; theorem :: ZF_FUND1:30 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k3_zf_fund1 :::"x"."::: ) (Set "(" ($#k2_zf_lang :::"x."::: ) (Set "(" ($#k5_card_1 :::"card"::: ) (Set (Var "A")) ")" ) ")" )))) ; theorem :: ZF_FUND1:31 (Bool "for" (Set (Var "fs")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "," (Set (Var "E")) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "fs")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "fs"))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "fs")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "E"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "fs"))) ($#r2_hidden :::"in"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "fs")) "," (Set (Var "E")) ")" )) & (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_ordinal1 :::"omega"::: ) )) ")" )))) ; theorem :: ZF_FUND1:32 (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "," (Set (Var "E")) (Bool "for" (Set (Var "v1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k2_zf_fund1 :::"decode"::: ) ) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k3_zf_fund1 :::"x"."::: ) (Set (Var "v1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "v1"))) & (Bool (Set (Set "(" (Set ($#k2_zf_fund1 :::"decode"::: ) ) ($#k2_funct_1 :::"""::: ) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set ($#k3_zf_fund1 :::"x"."::: ) (Set (Var "v1")))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k3_zf_fund1 :::"x"."::: ) (Set (Var "v1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")))) ")" )))) ; theorem :: ZF_FUND1:33 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k4_zf_fund1 :::"code"::: ) (Set (Var "A")))) "iff" (Bool "ex" (Set (Var "v1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set ($#k3_zf_fund1 :::"x"."::: ) (Set (Var "v1")))) ")" )) ")" ))) ; theorem :: ZF_FUND1:34 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "holds" (Bool "(" (Bool (Set ($#k4_zf_fund1 :::"code"::: ) (Set "(" (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_zf_fund1 :::"code"::: ) (Set (Var "A")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k4_zf_fund1 :::"code"::: ) (Set (Var "B")) ")" ))) & (Bool (Set ($#k4_zf_fund1 :::"code"::: ) (Set "(" (Set (Var "A")) ($#k7_subset_1 :::"\"::: ) (Set (Var "B")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_zf_fund1 :::"code"::: ) (Set (Var "A")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" ($#k4_zf_fund1 :::"code"::: ) (Set (Var "B")) ")" ))) ")" )) ; theorem :: ZF_FUND1:35 (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "," (Set (Var "E")) (Bool "for" (Set (Var "v1")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"ZF-formula":::) "st" (Bool (Bool (Set (Var "v1")) ($#r2_hidden :::"in"::: ) (Set ($#k2_zf_model :::"Free"::: ) (Set (Var "H"))))) "holds" (Bool (Set (Set "(" (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k4_zf_fund1 :::"code"::: ) (Set "(" ($#k2_zf_model :::"Free"::: ) (Set (Var "H")) ")" ) ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_zf_fund1 :::"x"."::: ) (Set (Var "v1")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "v1")))))))) ; theorem :: ZF_FUND1:36 (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "H")) "being" ($#m2_finseq_1 :::"ZF-formula":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "," (Set (Var "E")) "st" (Bool (Bool (Set (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k4_zf_fund1 :::"code"::: ) (Set "(" ($#k2_zf_model :::"Free"::: ) (Set (Var "H")) ")" ) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k4_zf_fund1 :::"code"::: ) (Set "(" ($#k2_zf_model :::"Free"::: ) (Set (Var "H")) ")" ) ")" ))) & (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k5_zf_model :::"St"::: ) "(" (Set (Var "H")) "," (Set (Var "E")) ")" ))) "holds" (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k5_zf_model :::"St"::: ) "(" (Set (Var "H")) "," (Set (Var "E")) ")" ))))) ; theorem :: ZF_FUND1:37 (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "fs")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k4_ordinal1 :::"omega"::: ) ) (Bool "for" (Set (Var "E")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k9_funct_2 :::"Funcs"::: ) "(" (Set (Var "fs")) "," (Set (Var "E")) ")" ))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k1_zf_lang :::"VAR"::: ) ) "," (Set (Var "E")) "st" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set ($#k2_zf_fund1 :::"decode"::: ) ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "fs")))))))) ;