SET007 Axioms: SET007+121.ax
%------------------------------------------------------------------------------
% File : SET007+121 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Mostowski's Fundamental Operations - Part I
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : zf_fund1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 78 ( 4 unt; 0 def)
% Number of atoms : 593 ( 49 equ)
% Maximal formula atoms : 18 ( 7 avg)
% Number of connectives : 604 ( 89 ~; 3 |; 274 &)
% ( 27 <=>; 211 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-3 aty)
% Number of functors : 62 ( 62 usr; 6 con; 0-5 aty)
% Number of variables : 269 ( 246 !; 23 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(d1_zf_fund1,axiom,
! [A,B,C] :
( C = k1_zf_fund1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E,F,G] :
( D = k4_tarski(E,G)
& r2_hidden(k4_tarski(E,F),A)
& r2_hidden(k4_tarski(F,G),B) ) ) ) ).
fof(d2_zf_fund1,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_ordinal2,k1_zf_lang)
& m2_relset_1(A,k5_ordinal2,k1_zf_lang) )
=> ( A = k3_zf_fund1
<=> ! [B] :
( m1_subset_1(B,k5_ordinal2)
=> k8_funct_2(k5_ordinal2,k1_zf_lang,A,B) = k2_zf_lang(k4_card_1(B)) ) ) ) ).
fof(d3_zf_fund1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( B = k4_zf_fund1(A)
<=> k2_zf_lang(B) = A ) ) ) ).
fof(d4_zf_fund1,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_zf_lang)) )
=> k5_zf_fund1(A) = k9_relat_1(k2_funct_1(k3_zf_fund1),A) ) ).
fof(d5_zf_fund1,axiom,
! [A] :
( ( v1_zf_lang(A)
& m2_finseq_1(A,k5_numbers) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( C = k10_zf_fund1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k1_zf_lang,B)
& m2_relset_1(E,k1_zf_lang,B)
& D = k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,E),k5_zf_fund1(k6_zf_fund1(A)))
& r2_hidden(E,k5_zf_model(A,B)) ) ) ) ) ) ).
fof(d7_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v2_zf_fund1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k5_classes2(A,C,D),B) ) ) ) ) ) ) ).
fof(d8_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v3_zf_fund1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,B)
=> r2_hidden(k3_classes2(A,C),B) ) ) ) ) ) ).
fof(d13_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v8_zf_fund1(B,A)
<=> ( v1_zf_fund1(B,A)
& v2_zf_fund1(B,A)
& v3_zf_fund1(B,A)
& v4_zf_fund1(B,A)
& v5_zf_fund1(B,A)
& v6_zf_fund1(B,A)
& v7_zf_fund1(B,A) ) ) ) ) ).
fof(t1_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C,D] :
( r1_tarski(B,A)
& ( r2_hidden(C,B)
=> m1_subset_1(C,A) )
& ( ( r2_hidden(C,D)
& r2_hidden(D,B) )
=> m1_subset_1(C,A) ) ) ) ) ).
fof(t2_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C,D] :
( v8_zf_fund1(B,A)
=> ( ( r2_hidden(C,B)
=> r2_hidden(k1_tarski(C),B) )
& ( r2_hidden(k1_tarski(C),B)
=> r2_hidden(C,B) )
& ( r2_hidden(D,B)
=> r2_hidden(k3_tarski(D),B) ) ) ) ) ) ).
fof(t3_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v8_zf_fund1(B,A)
=> r2_hidden(k1_xboole_0,B) ) ) ) ).
fof(t4_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C,D] :
( ( v8_zf_fund1(B,A)
& r2_hidden(C,B)
& r2_hidden(D,B) )
=> ( r2_hidden(k2_xboole_0(C,D),B)
& r2_hidden(k4_xboole_0(C,D),B)
& r2_hidden(k1_zf_fund1(C,D),B) ) ) ) ) ).
fof(t5_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C,D] :
( ( v8_zf_fund1(B,A)
& r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(k3_xboole_0(C,D),B) ) ) ) ).
fof(t6_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C,D] :
( ( v8_zf_fund1(B,A)
& r2_hidden(C,B)
& r2_hidden(D,B) )
=> ( r2_hidden(k2_tarski(C,D),B)
& r2_hidden(k4_tarski(C,D),B) ) ) ) ) ).
fof(t7_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v8_zf_fund1(B,A)
=> r1_tarski(k5_ordinal2,B) ) ) ) ).
fof(t8_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( v1_finset_1(C)
& m1_subset_1(C,k1_zfmisc_1(k5_ordinal2)) )
=> ( v8_zf_fund1(B,A)
=> r1_tarski(k1_funct_2(C,k5_ordinal2),B) ) ) ) ) ).
fof(t9_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ! [D] :
( ( v1_finset_1(D)
& m1_subset_1(D,k1_zfmisc_1(k5_ordinal2)) )
=> ( ( v8_zf_fund1(C,A)
& r2_hidden(B,C) )
=> r2_hidden(k1_funct_2(D,B),C) ) ) ) ) ) ).
fof(t13_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ( v8_zf_fund1(B,A)
& v1_finset_1(C)
& ! [D] :
( r2_hidden(D,C)
=> r2_hidden(D,B) ) )
=> r2_hidden(C,B) ) ) ) ).
fof(t14_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ! [D,E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(k5_ordinal2)) )
=> ( ( v8_zf_fund1(C,A)
& r1_tarski(D,C)
& r2_hidden(B,k1_funct_2(E,D)) )
=> r2_hidden(B,C) ) ) ) ) ) ).
fof(t18_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( ( v8_zf_fund1(B,A)
& r2_hidden(C,B) )
=> ! [D] :
( m2_subset_1(D,k5_numbers,k1_zf_lang)
=> ! [E] :
( m2_subset_1(E,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k10_zf_fund1(k4_zf_lang(D,E),C),B)
& r2_hidden(k10_zf_fund1(k5_zf_lang(D,E),C),B) ) ) ) ) ) ) ) ).
fof(t19_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( ( v8_zf_fund1(B,A)
& r2_hidden(C,B) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( r2_hidden(k10_zf_fund1(D,C),B)
=> r2_hidden(k10_zf_fund1(k6_zf_lang(D),C),B) ) ) ) ) ) ) ).
fof(t20_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( ( v8_zf_fund1(B,A)
& r2_hidden(C,B) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ! [E] :
( ( v1_zf_lang(E)
& m2_finseq_1(E,k5_numbers) )
=> ( ( r2_hidden(k10_zf_fund1(D,C),B)
& r2_hidden(k10_zf_fund1(E,C),B) )
=> r2_hidden(k10_zf_fund1(k7_zf_lang(D,E),C),B) ) ) ) ) ) ) ) ).
fof(t21_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( ( v8_zf_fund1(B,A)
& r2_hidden(C,B) )
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ! [E] :
( m2_subset_1(E,k5_numbers,k1_zf_lang)
=> ( r2_hidden(k10_zf_fund1(D,C),B)
=> r2_hidden(k10_zf_fund1(k8_zf_lang(E,D),C),B) ) ) ) ) ) ) ) ).
fof(t22_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( ( v8_zf_fund1(B,A)
& r2_hidden(C,B) )
=> r2_hidden(k10_zf_fund1(D,C),B) ) ) ) ) ) ).
fof(t23_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ! [C] :
( m1_subset_1(C,k5_ordinal2)
=> ( v8_zf_fund1(B,A)
=> ( r2_hidden(C,B)
& r2_hidden(k2_ordinal4(A),B)
& r2_hidden(k3_ordinal4(A),B) ) ) ) ) ) ).
fof(t24_zf_fund1,axiom,
! [A,B,C] : k1_zf_fund1(k2_tarski(k4_tarski(A,B),k4_tarski(B,B)),k1_tarski(k4_tarski(B,C))) = k2_tarski(k4_tarski(A,C),k4_tarski(B,C)) ).
fof(t25_zf_fund1,axiom,
! [A,B,C,D,E,F] :
( A != B
=> k1_zf_fund1(k2_tarski(k4_tarski(C,A),k4_tarski(D,B)),k2_tarski(k4_tarski(A,E),k4_tarski(B,F))) = k2_tarski(k4_tarski(C,E),k4_tarski(D,F)) ) ).
fof(t26_zf_fund1,axiom,
$true ).
fof(t27_zf_fund1,axiom,
! [A] :
( m2_subset_1(A,k5_numbers,k1_zf_lang)
=> ! [B] :
( m2_subset_1(B,k5_numbers,k1_zf_lang)
=> ( k5_zf_fund1(k8_zf_fund1(A)) = k7_zf_fund1(k4_zf_fund1(A))
& k5_zf_fund1(k9_zf_fund1(A,B)) = k2_tarski(k4_zf_fund1(A),k4_zf_fund1(B)) ) ) ) ).
fof(t28_zf_fund1,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ( k1_relat_1(C) = k2_tarski(A,B)
<=> C = k2_tarski(k4_tarski(A,k1_funct_1(C,A)),k4_tarski(B,k1_funct_1(C,B))) ) ) ).
fof(t29_zf_fund1,axiom,
( k1_relat_1(k3_zf_fund1) = k5_ordinal2
& k2_relat_1(k3_zf_fund1) = k1_zf_lang
& v2_funct_1(k3_zf_fund1)
& v2_funct_1(k2_funct_1(k3_zf_fund1))
& k1_relat_1(k2_funct_1(k3_zf_fund1)) = k1_zf_lang
& k2_relat_1(k2_funct_1(k3_zf_fund1)) = k5_ordinal2 ) ).
fof(t30_zf_fund1,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_zf_lang)) )
=> r2_tarski(A,k5_zf_fund1(A)) ) ).
fof(t31_zf_fund1,axiom,
! [A] :
( m1_subset_1(A,k5_ordinal2)
=> A = k4_zf_fund1(k2_zf_lang(k4_card_1(A))) ) ).
fof(t32_zf_fund1,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k5_ordinal2)) )
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,B)
& m2_relset_1(C,k1_zf_lang,B) )
=> ( k1_relat_1(k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,C),A)) = A
& r1_tarski(k2_relat_1(k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,C),A)),B)
& r2_hidden(k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,C),A),k1_funct_2(A,B))
& k1_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,C)) = k5_ordinal2
& r1_tarski(k2_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,B,k3_zf_fund1,C)),B) ) ) ) ) ).
fof(t33_zf_fund1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ( k8_funct_2(k5_numbers,k1_zf_lang,k3_zf_fund1,k4_zf_fund1(C)) = C
& k1_funct_1(k2_funct_1(k3_zf_fund1),C) = k4_zf_fund1(C)
& k8_funct_2(k5_numbers,A,k7_funct_2(k5_ordinal2,k1_zf_lang,A,k3_zf_fund1,B),k4_zf_fund1(C)) = k8_funct_2(k1_zf_lang,A,B,C) ) ) ) ) ).
fof(t34_zf_fund1,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zf_lang)) )
=> ( r2_hidden(A,k5_zf_fund1(B))
<=> ? [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
& r2_hidden(C,B)
& A = k4_zf_fund1(C) ) ) ) ).
fof(t35_zf_fund1,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_zf_lang)) )
=> ! [B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zf_lang)) )
=> ( k5_zf_fund1(k4_subset_1(k1_zf_lang,A,B)) = k4_subset_1(k5_ordinal2,k5_zf_fund1(A),k5_zf_fund1(B))
& k5_zf_fund1(k6_subset_1(k1_zf_lang,A,B)) = k6_subset_1(k5_ordinal2,k5_zf_fund1(A),k5_zf_fund1(B)) ) ) ) ).
fof(t36_zf_fund1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zf_lang,A)
& m2_relset_1(B,k1_zf_lang,A) )
=> ! [C] :
( m2_subset_1(C,k5_numbers,k1_zf_lang)
=> ! [D] :
( ( v1_zf_lang(D)
& m2_finseq_1(D,k5_numbers) )
=> ( r2_hidden(C,k6_zf_fund1(D))
=> k1_funct_1(k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,A,k3_zf_fund1,B),k5_zf_fund1(k6_zf_fund1(D))),k4_zf_fund1(C)) = k8_funct_2(k1_zf_lang,A,B,C) ) ) ) ) ) ).
fof(t37_zf_fund1,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_zf_lang(B)
& m2_finseq_1(B,k5_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k1_zf_lang,A)
& m2_relset_1(C,k1_zf_lang,A) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,A)
& m2_relset_1(D,k1_zf_lang,A) )
=> ( ( k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,A,k3_zf_fund1,C),k5_zf_fund1(k6_zf_fund1(B))) = k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,A,k3_zf_fund1,D),k5_zf_fund1(k6_zf_fund1(B)))
& r2_hidden(C,k5_zf_model(B,A)) )
=> r2_hidden(D,k5_zf_model(B,A)) ) ) ) ) ) ).
fof(t38_zf_fund1,axiom,
! [A,B] :
( ( v1_finset_1(B)
& m1_subset_1(B,k1_zfmisc_1(k5_ordinal2)) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ~ ( r2_hidden(A,k1_funct_2(B,C))
& ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k1_zf_lang,C)
& m2_relset_1(D,k1_zf_lang,C) )
=> A != k7_relat_1(k7_funct_2(k5_ordinal2,k1_zf_lang,C,k3_zf_fund1,D),B) ) ) ) ) ).
fof(dt_k1_zf_fund1,axiom,
$true ).
fof(dt_k2_zf_fund1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> m1_subset_1(k2_zf_fund1(A,B,C),A) ) ).
fof(redefinition_k2_zf_fund1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A)
& m1_subset_1(B,A)
& m1_subset_1(C,A) )
=> k2_zf_fund1(A,B,C) = k1_zf_fund1(B,C) ) ).
fof(dt_k3_zf_fund1,axiom,
( v1_funct_1(k3_zf_fund1)
& v1_funct_2(k3_zf_fund1,k5_ordinal2,k1_zf_lang)
& m2_relset_1(k3_zf_fund1,k5_ordinal2,k1_zf_lang) ) ).
fof(dt_k4_zf_fund1,axiom,
! [A] :
( m1_subset_1(A,k1_zf_lang)
=> m2_subset_1(k4_zf_fund1(A),k1_numbers,k5_numbers) ) ).
fof(dt_k5_zf_fund1,axiom,
! [A] :
( ( v1_finset_1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_zf_lang)) )
=> ( v1_finset_1(k5_zf_fund1(A))
& m1_subset_1(k5_zf_fund1(A),k1_zfmisc_1(k5_ordinal2)) ) ) ).
fof(dt_k6_zf_fund1,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> ( v1_finset_1(k6_zf_fund1(A))
& m1_subset_1(k6_zf_fund1(A),k1_zfmisc_1(k1_zf_lang)) ) ) ).
fof(redefinition_k6_zf_fund1,axiom,
! [A] :
( ( v1_zf_lang(A)
& m1_finseq_1(A,k5_numbers) )
=> k6_zf_fund1(A) = k1_zf_model(A) ) ).
fof(dt_k7_zf_fund1,axiom,
! [A] :
( m1_subset_1(A,k5_ordinal2)
=> ( v1_finset_1(k7_zf_fund1(A))
& m1_subset_1(k7_zf_fund1(A),k1_zfmisc_1(k5_ordinal2)) ) ) ).
fof(redefinition_k7_zf_fund1,axiom,
! [A] :
( m1_subset_1(A,k5_ordinal2)
=> k7_zf_fund1(A) = k1_tarski(A) ) ).
fof(dt_k8_zf_fund1,axiom,
! [A] :
( m1_subset_1(A,k1_zf_lang)
=> ( v1_finset_1(k8_zf_fund1(A))
& m1_subset_1(k8_zf_fund1(A),k1_zfmisc_1(k1_zf_lang)) ) ) ).
fof(redefinition_k8_zf_fund1,axiom,
! [A] :
( m1_subset_1(A,k1_zf_lang)
=> k8_zf_fund1(A) = k1_tarski(A) ) ).
fof(dt_k9_zf_fund1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang) )
=> ( v1_finset_1(k9_zf_fund1(A,B))
& m1_subset_1(k9_zf_fund1(A,B),k1_zfmisc_1(k1_zf_lang)) ) ) ).
fof(commutativity_k9_zf_fund1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang) )
=> k9_zf_fund1(A,B) = k9_zf_fund1(B,A) ) ).
fof(redefinition_k9_zf_fund1,axiom,
! [A,B] :
( ( m1_subset_1(A,k1_zf_lang)
& m1_subset_1(B,k1_zf_lang) )
=> k9_zf_fund1(A,B) = k2_tarski(A,B) ) ).
fof(dt_k10_zf_fund1,axiom,
$true ).
fof(d6_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v1_zf_fund1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ( r2_hidden(C,B)
=> r2_hidden(a_2_0_zf_fund1(A,C),B) ) ) ) ) ) ).
fof(d9_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v4_zf_fund1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(a_3_0_zf_fund1(A,C,D),B) ) ) ) ) ) ) ).
fof(d10_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v5_zf_fund1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(a_3_1_zf_fund1(A,C,D),B) ) ) ) ) ) ) ).
fof(d11_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v6_zf_fund1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(a_3_2_zf_fund1(A,C,D),B) ) ) ) ) ) ) ).
fof(d12_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(A)) )
=> ( v7_zf_fund1(B,A)
<=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ( ( r2_hidden(C,B)
& r2_hidden(D,B) )
=> r2_hidden(a_3_3_zf_fund1(A,C,D),B) ) ) ) ) ) ) ).
fof(t10_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(A)) )
=> ! [E] :
( ( v1_finset_1(E)
& m1_subset_1(E,k1_zfmisc_1(k5_ordinal2)) )
=> ( ( v8_zf_fund1(D,A)
& r2_hidden(B,k1_funct_2(E,k5_ordinal2))
& r2_hidden(C,D) )
=> r2_hidden(a_3_4_zf_fund1(A,B,C),D) ) ) ) ) ) ) ).
fof(t11_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(A)) )
=> ! [E] :
( m1_subset_1(E,k5_ordinal2)
=> ! [F] :
( ( v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(k5_ordinal2)) )
=> ( ( v8_zf_fund1(D,A)
& r2_hidden(E,F)
& r2_hidden(B,D)
& r2_hidden(C,D)
& r1_tarski(C,k1_funct_2(F,B)) )
=> r2_hidden(a_5_0_zf_fund1(A,B,C,E,F),D) ) ) ) ) ) ) ) ).
fof(t12_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(A)) )
=> ! [E] :
( m1_subset_1(E,k5_ordinal2)
=> ! [F] :
( ( v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(k5_ordinal2)) )
=> ( ( v8_zf_fund1(D,A)
& r2_hidden(B,D)
& r2_hidden(C,D)
& r1_tarski(C,k1_funct_2(F,B)) )
=> ( r2_hidden(E,F)
| r2_hidden(a_4_0_zf_fund1(A,B,C,E),D) ) ) ) ) ) ) ) ) ).
fof(t15_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( ( ~ v1_xboole_0(D)
& m1_subset_1(D,k1_zfmisc_1(A)) )
=> ! [E] :
( m1_subset_1(E,k5_ordinal2)
=> ! [F] :
( ( v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(k5_ordinal2)) )
=> ( ( v8_zf_fund1(D,A)
& r2_hidden(B,D)
& r1_tarski(B,D)
& r2_hidden(C,k1_funct_2(F,B)) )
=> ( r2_hidden(E,F)
| r2_hidden(a_4_1_zf_fund1(A,B,C,E),D) ) ) ) ) ) ) ) ) ).
fof(t16_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( m1_subset_1(C,A)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( ( ~ v1_xboole_0(E)
& m1_subset_1(E,k1_zfmisc_1(A)) )
=> ! [F] :
( m1_subset_1(F,k5_ordinal2)
=> ! [G] :
( ( v1_finset_1(G)
& m1_subset_1(G,k1_zfmisc_1(k5_ordinal2)) )
=> ( ( v8_zf_fund1(E,A)
& r2_hidden(B,E)
& r1_tarski(B,E)
& r2_hidden(C,k1_funct_2(G,B))
& r1_tarski(D,k1_funct_2(k4_subset_1(k5_ordinal2,G,k7_zf_fund1(F)),B))
& r2_hidden(D,E) )
=> ( r2_hidden(F,G)
| r2_hidden(a_5_1_zf_fund1(A,B,C,D,F),E) ) ) ) ) ) ) ) ) ) ).
fof(t17_zf_fund1,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B] :
( m1_subset_1(B,A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(A)) )
=> ( ( v8_zf_fund1(C,A)
& r2_hidden(B,C) )
=> r2_hidden(a_2_1_zf_fund1(A,B),C) ) ) ) ) ).
fof(fraenkel_a_2_0_zf_fund1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B) )
=> ( r2_hidden(A,a_2_0_zf_fund1(B,C))
<=> ? [D,E] :
( m1_subset_1(D,B)
& m1_subset_1(E,B)
& A = k2_tarski(k4_tarski(k2_ordinal4(B),D),k4_tarski(k3_ordinal4(B),E))
& r2_hidden(D,E)
& r2_hidden(D,C)
& r2_hidden(E,C) ) ) ) ).
fof(fraenkel_a_3_0_zf_fund1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B)
& m1_subset_1(D,B) )
=> ( r2_hidden(A,a_3_0_zf_fund1(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,B)
& m1_subset_1(F,B)
& A = k1_classes2(B,k6_classes2(B,E,F))
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_3_1_zf_fund1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B)
& m1_subset_1(D,B) )
=> ( r2_hidden(A,a_3_1_zf_fund1(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,B)
& m1_subset_1(F,B)
& A = k7_classes2(B,E,F)
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_3_2_zf_fund1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B)
& m1_subset_1(D,B) )
=> ( r2_hidden(A,a_3_2_zf_fund1(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,B)
& m1_subset_1(F,B)
& A = k9_classes2(B,E,F)
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_3_3_zf_fund1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B)
& m1_subset_1(D,B) )
=> ( r2_hidden(A,a_3_3_zf_fund1(B,C,D))
<=> ? [E,F] :
( m1_subset_1(E,B)
& m1_subset_1(F,B)
& A = k2_zf_fund1(B,E,F)
& r2_hidden(E,C)
& r2_hidden(F,D) ) ) ) ).
fof(fraenkel_a_3_4_zf_fund1,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B)
& m1_subset_1(D,B) )
=> ( r2_hidden(A,a_3_4_zf_fund1(B,C,D))
<=> ? [E] :
( m1_subset_1(E,B)
& A = k2_zf_fund1(B,C,E)
& r2_hidden(E,D) ) ) ) ).
fof(fraenkel_a_5_0_zf_fund1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B)
& m1_subset_1(D,B)
& m1_subset_1(E,k5_ordinal2)
& v1_finset_1(F)
& m1_subset_1(F,k1_zfmisc_1(k5_ordinal2)) )
=> ( r2_hidden(A,a_5_0_zf_fund1(B,C,D,E,F))
<=> ? [G] :
( m1_subset_1(G,B)
& A = G
& r2_hidden(G,k1_funct_2(k6_subset_1(k5_ordinal2,F,k7_zf_fund1(E)),C))
& ? [H] : r2_hidden(k2_xboole_0(k1_tarski(k4_tarski(E,H)),G),D) ) ) ) ).
fof(fraenkel_a_4_0_zf_fund1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B)
& m1_subset_1(D,B)
& m1_subset_1(E,k5_ordinal2) )
=> ( r2_hidden(A,a_4_0_zf_fund1(B,C,D,E))
<=> ? [F,G] :
( m1_subset_1(F,B)
& m1_subset_1(G,B)
& A = k2_xboole_0(k1_tarski(k4_tarski(E,F)),G)
& r2_hidden(F,C)
& r2_hidden(G,D) ) ) ) ).
fof(fraenkel_a_4_1_zf_fund1,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B)
& m1_subset_1(D,B)
& m1_subset_1(E,k5_ordinal2) )
=> ( r2_hidden(A,a_4_1_zf_fund1(B,C,D,E))
<=> ? [F] :
( m1_subset_1(F,B)
& A = k2_xboole_0(k1_tarski(k4_tarski(E,F)),D)
& r2_hidden(F,C) ) ) ) ).
fof(fraenkel_a_5_1_zf_fund1,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B)
& m1_subset_1(D,B)
& m1_subset_1(E,B)
& m1_subset_1(F,k5_ordinal2) )
=> ( r2_hidden(A,a_5_1_zf_fund1(B,C,D,E,F))
<=> ? [G] :
( m1_subset_1(G,B)
& A = G
& r2_hidden(G,C)
& r2_hidden(k2_xboole_0(k1_tarski(k4_tarski(F,G)),D),E) ) ) ) ).
fof(fraenkel_a_2_1_zf_fund1,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_classes2(B)
& m1_subset_1(C,B) )
=> ( r2_hidden(A,a_2_1_zf_fund1(B,C))
<=> ? [D] :
( m1_subset_1(D,B)
& A = k2_tarski(k4_tarski(k2_ordinal4(B),D),k4_tarski(k3_ordinal4(B),D))
& r2_hidden(D,C) ) ) ) ).
%------------------------------------------------------------------------------