SET007 Axioms: SET007+448.ax
%------------------------------------------------------------------------------
% File : SET007+448 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Examples of Category Structures
% 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 : altcat_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 92 ( 2 unt; 0 def)
% Number of atoms : 546 ( 73 equ)
% Maximal formula atoms : 24 ( 5 avg)
% Number of connectives : 533 ( 79 ~; 1 |; 230 &)
% ( 19 <=>; 204 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 8 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 39 ( 37 usr; 1 prp; 0-4 aty)
% Number of functors : 60 ( 60 usr; 12 con; 0-6 aty)
% Number of variables : 244 ( 221 !; 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(fc1_altcat_2,axiom,
! [A] :
( v1_relat_1(k1_pboole(A))
& v3_relat_1(k1_pboole(A))
& v1_funct_1(k1_pboole(A))
& v1_funcop_1(k1_pboole(A)) ) ).
fof(fc2_altcat_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_funcop_1(A) )
=> ( v1_relat_1(k2_funct_4(A))
& v1_funct_1(k2_funct_4(A))
& v1_funcop_1(k2_funct_4(A)) ) ) ).
fof(fc3_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( ~ v3_struct_0(k6_altcat_2(A))
& v2_altcat_1(k6_altcat_2(A))
& v6_altcat_1(k6_altcat_2(A))
& v11_altcat_1(k6_altcat_2(A))
& v12_altcat_1(k6_altcat_2(A)) ) ) ).
fof(rc1_altcat_2,axiom,
? [A] :
( l1_altcat_1(A)
& ~ v3_struct_0(A)
& v1_altcat_1(A) ) ).
fof(cc1_altcat_2,axiom,
! [A] :
( l2_altcat_1(A)
=> ( ( ~ v3_struct_0(A)
& v12_altcat_1(A) )
=> ( ~ v3_struct_0(A)
& v1_altcat_2(A) ) ) ) ).
fof(rc2_altcat_2,axiom,
? [A] :
( l1_altcat_1(A)
& ~ v3_struct_0(A)
& v1_altcat_2(A) ) ).
fof(rc3_altcat_2,axiom,
? [A] :
( l2_altcat_1(A)
& ~ v3_struct_0(A)
& v1_altcat_2(A) ) ).
fof(fc4_altcat_2,axiom,
( v3_struct_0(k7_altcat_2)
& v6_altcat_1(k7_altcat_2) ) ).
fof(rc4_altcat_2,axiom,
? [A] :
( l2_altcat_1(A)
& v3_struct_0(A)
& v6_altcat_1(A) ) ).
fof(rc5_altcat_2,axiom,
! [A] :
( l2_altcat_1(A)
=> ? [B] :
( m1_altcat_2(B,A)
& v6_altcat_1(B) ) ) ).
fof(fc5_altcat_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k8_altcat_2(A,B))
& v2_altcat_1(k8_altcat_2(A,B))
& v6_altcat_1(k8_altcat_2(A,B)) ) ) ).
fof(rc6_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ? [B] :
( m1_altcat_2(B,A)
& ~ v3_struct_0(B)
& v2_altcat_1(B)
& v6_altcat_1(B) ) ) ).
fof(rc7_altcat_2,axiom,
! [A] :
( l2_altcat_1(A)
=> ? [B] :
( m1_altcat_2(B,A)
& v6_altcat_1(B)
& v2_altcat_2(B,A) ) ) ).
fof(rc8_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ? [B] :
( m1_altcat_2(B,A)
& ~ v3_struct_0(B)
& v6_altcat_1(B)
& v2_altcat_2(B,A) ) ) ).
fof(fc6_altcat_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( ~ v3_struct_0(k8_altcat_2(A,B))
& v2_altcat_1(k8_altcat_2(A,B))
& v6_altcat_1(k8_altcat_2(A,B))
& v2_altcat_2(k8_altcat_2(A,B),A)
& v3_altcat_2(k8_altcat_2(A,B),A) ) ) ).
fof(rc9_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ? [B] :
( m1_altcat_2(B,A)
& ~ v3_struct_0(B)
& v6_altcat_1(B)
& v2_altcat_2(B,A)
& v3_altcat_2(B,A) ) ) ).
fof(cc2_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_altcat_2(B,A)
=> ( ( ~ v3_struct_0(B)
& v2_altcat_2(B,A) )
=> v2_altcat_1(B) ) ) ) ).
fof(cc3_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_altcat_2(B,A)
=> ( ( ~ v3_struct_0(B)
& v2_altcat_1(B) )
=> ( ~ v3_struct_0(B)
& v11_altcat_1(B) ) ) ) ) ).
fof(cc4_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_altcat_2(B,A)
=> ( ( ~ v3_struct_0(B)
& v2_altcat_1(B)
& v3_altcat_2(B,A) )
=> ( ~ v3_struct_0(B)
& v12_altcat_1(B)
& v1_altcat_2(B) ) ) ) ) ).
fof(rc10_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ? [B] :
( m1_altcat_2(B,A)
& ~ v3_struct_0(B)
& v2_altcat_1(B)
& v11_altcat_1(B)
& v12_altcat_1(B)
& v1_altcat_2(B)
& v3_altcat_2(B,A) ) ) ).
fof(t1_altcat_2,axiom,
! [A,B,C,D] : k15_funct_3(k2_funcop_1(A,C),k2_funcop_1(B,D)) = k2_funcop_1(k2_zfmisc_1(A,B),k4_tarski(C,D)) ).
fof(t2_altcat_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> k2_funct_4(k5_relat_1(A,B)) = k5_relat_1(k2_funct_4(A),B) ) ) ).
fof(t3_altcat_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ! [C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k2_funct_4(k5_relat_1(k15_funct_3(B,C),A)) = k5_relat_1(k15_funct_3(C,B),k2_funct_4(A)) ) ) ) ).
fof(t4_altcat_2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,A)
=> ( r1_pzfmisc1(A,B,C)
=> ! [E] :
( m3_pboole(E,A,B,C)
=> ! [F] :
( m3_pboole(F,A,C,D)
=> m3_pboole(k13_pboole(E,F),A,B,D) ) ) ) ) ) ) ).
fof(t5_altcat_2,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( m1_pboole(D,B)
=> ! [E] :
( m1_pboole(E,B)
=> ! [F] :
( m3_pboole(F,B,D,E)
=> m3_pboole(k7_pboole(A,B,C,F),A,k7_pboole(A,B,C,D),k7_pboole(A,B,C,E)) ) ) ) ) ) ).
fof(t6_altcat_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_pboole(C,k2_zfmisc_1(A,B))
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,B)
=> k1_binop_1(k18_pralg_1(A,B,C),E,D) = k1_binop_1(C,D,E) ) ) ) ) ) ).
fof(d1_altcat_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> ! [B] :
( ( v1_relat_1(B)
& v1_funct_1(B) )
=> ( r1_altcat_2(A,B)
<=> ( r1_tarski(k1_relat_1(A),k1_relat_1(B))
& ! [C] :
( r2_hidden(C,k1_relat_1(A))
=> r1_tarski(k1_funct_1(A,C),k1_funct_1(B,C)) ) ) ) ) ) ).
fof(d2_altcat_2,axiom,
! [A,B,C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,B)
=> ( r2_altcat_2(A,B,C,D)
<=> ( r1_tarski(A,B)
& ! [E] :
( r2_hidden(E,A)
=> r1_tarski(k1_funct_1(C,E),k1_funct_1(D,E)) ) ) ) ) ) ).
fof(t7_altcat_2,axiom,
$true ).
fof(t8_altcat_2,axiom,
! [A,B,C] :
( m1_pboole(C,A)
=> ! [D] :
( m1_pboole(D,B)
=> ( ( r2_altcat_2(A,B,C,D)
& r2_altcat_2(B,A,D,C) )
=> C = D ) ) ) ).
fof(t9_altcat_2,axiom,
! [A,B,C,D] :
( m1_pboole(D,A)
=> ! [E] :
( m1_pboole(E,B)
=> ! [F] :
( m1_pboole(F,C)
=> ( ( r2_altcat_2(A,B,D,E)
& r2_altcat_2(B,C,E,F) )
=> r2_altcat_2(A,C,D,F) ) ) ) ) ).
fof(t10_altcat_2,axiom,
! [A,B] :
( m1_pboole(B,A)
=> ! [C] :
( m1_pboole(C,A)
=> ( r2_altcat_2(A,A,B,C)
<=> r2_pboole(A,B,C) ) ) ) ).
fof(t11_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> r1_tarski(k12_mcart_1(u2_cat_1(A),u2_cat_1(A),k6_cat_1(A,C,D),k6_cat_1(A,B,C)),k1_relat_1(u5_cat_1(A))) ) ) ) ) ).
fof(t12_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> r1_tarski(k9_relat_1(u5_cat_1(A),k12_mcart_1(u2_cat_1(A),u2_cat_1(A),k6_cat_1(A,C,D),k6_cat_1(A,B,C))),k6_cat_1(A,B,D)) ) ) ) ) ).
fof(d3_altcat_2,axiom,
! [A] :
( l1_cat_1(A)
=> ! [B] :
( m1_pboole(B,k2_zfmisc_1(u1_cat_1(A),u1_cat_1(A)))
=> ( B = k4_altcat_2(A)
<=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> k1_binop_1(B,C,D) = k6_cat_1(A,C,D) ) ) ) ) ) ).
fof(t13_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> r2_hidden(k10_cat_1(A,B),k1_binop_1(k4_altcat_2(A),B,B)) ) ) ).
fof(d4_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_pboole(B,k3_zfmisc_1(u1_cat_1(A),u1_cat_1(A),u1_cat_1(A)),k3_altcat_1(u1_cat_1(A),k4_altcat_2(A),k4_altcat_2(A)),k2_altcat_1(u1_cat_1(A),k4_altcat_2(A)))
=> ( B = k5_altcat_2(A)
<=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> k4_altcat_1(u1_cat_1(A),k4_altcat_2(A),B,C,D,E) = k7_relat_1(u5_cat_1(A),k2_zfmisc_1(k1_binop_1(k4_altcat_2(A),D,E),k1_binop_1(k4_altcat_2(A),C,D))) ) ) ) ) ) ) ).
fof(t14_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ~ ( k6_cat_1(A,B,C) != k1_xboole_0
& k6_cat_1(A,C,D) != k1_xboole_0
& ~ ! [E] :
( m1_cat_1(E,A,B,C)
=> ! [F] :
( m1_cat_1(F,A,C,D)
=> k1_binop_1(k4_altcat_1(u1_cat_1(A),k4_altcat_2(A),k5_altcat_2(A),B,C,D),F,E) = k9_cat_1(A,B,C,D,E,F) ) ) ) ) ) ) ) ).
fof(t15_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> v3_altcat_1(k5_altcat_2(A),u1_cat_1(A),k4_altcat_2(A)) ) ).
fof(t16_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( v5_altcat_1(k5_altcat_2(A),u1_cat_1(A),k4_altcat_2(A))
& v4_altcat_1(k5_altcat_2(A),u1_cat_1(A),k4_altcat_2(A)) ) ) ).
fof(d5_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> k6_altcat_2(A) = g2_altcat_1(u1_cat_1(A),k4_altcat_2(A),k5_altcat_2(A)) ) ).
fof(t17_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> v11_altcat_1(k6_altcat_2(A)) ) ).
fof(t18_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> v12_altcat_1(k6_altcat_2(A)) ) ).
fof(t19_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> v2_altcat_1(k6_altcat_2(A)) ) ).
fof(d6_altcat_2,axiom,
! [A] :
( l1_altcat_1(A)
=> ( v1_altcat_2(A)
<=> ! [B] :
~ ( r2_hidden(B,u1_struct_0(A))
& k1_binop_1(u1_altcat_1(A),B,B) = k1_xboole_0 ) ) ) ).
fof(d7_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_altcat_1(A) )
=> ( v1_altcat_2(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> k1_altcat_1(A,B,B) != k1_xboole_0 ) ) ) ).
fof(d8_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& l2_altcat_1(A) )
=> ( v11_altcat_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(A))
=> ! [F] :
( m1_subset_1(F,k1_altcat_1(A,B,C))
=> ! [G] :
( m1_subset_1(G,k1_altcat_1(A,C,D))
=> ! [H] :
( m1_subset_1(H,k1_altcat_1(A,D,E))
=> ~ ( k1_altcat_1(A,B,C) != k1_xboole_0
& k1_altcat_1(A,C,D) != k1_xboole_0
& k1_altcat_1(A,D,E) != k1_xboole_0
& k5_altcat_1(A,B,C,E,F,k5_altcat_1(A,C,D,E,G,H)) != k5_altcat_1(A,B,D,E,k5_altcat_1(A,B,C,D,F,G),H) ) ) ) ) ) ) ) ) ) ) ).
fof(d9_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ( v12_altcat_1(A)
<=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ( k1_altcat_1(A,B,B) != k1_xboole_0
& ? [C] :
( m1_subset_1(C,k1_altcat_1(A,B,B))
& ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,k1_altcat_1(A,D,B))
=> ! [F] :
( m1_subset_1(F,k1_altcat_1(A,B,D))
=> ( ( k1_altcat_1(A,D,B) != k1_xboole_0
=> k5_altcat_1(A,D,B,B,E,C) = E )
& ( k1_altcat_1(A,B,D) != k1_xboole_0
=> k5_altcat_1(A,B,B,D,C,F) = F ) ) ) ) ) ) ) ) ) ) ).
fof(d10_altcat_2,axiom,
! [A] :
( ( v6_altcat_1(A)
& l2_altcat_1(A) )
=> ( A = k7_altcat_2
<=> v1_xboole_0(u1_struct_0(A)) ) ) ).
fof(t20_altcat_2,axiom,
! [A] :
( ( v3_struct_0(A)
& v6_altcat_1(A)
& l2_altcat_1(A) )
=> A = k7_altcat_2 ) ).
fof(d11_altcat_2,axiom,
! [A] :
( l2_altcat_1(A)
=> ! [B] :
( l2_altcat_1(B)
=> ( m1_altcat_2(B,A)
<=> ( r1_tarski(u1_struct_0(B),u1_struct_0(A))
& r2_altcat_2(k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(A)),u1_altcat_1(B),u1_altcat_1(A))
& r2_altcat_2(k3_zfmisc_1(u1_struct_0(B),u1_struct_0(B),u1_struct_0(B)),k3_zfmisc_1(u1_struct_0(A),u1_struct_0(A),u1_struct_0(A)),u2_altcat_1(B),u2_altcat_1(A)) ) ) ) ) ).
fof(t21_altcat_2,axiom,
! [A] :
( l2_altcat_1(A)
=> m1_altcat_2(A,A) ) ).
fof(t22_altcat_2,axiom,
! [A] :
( l2_altcat_1(A)
=> ! [B] :
( l2_altcat_1(B)
=> ! [C] :
( l2_altcat_1(C)
=> ( ( m1_altcat_2(A,B)
& m1_altcat_2(B,C) )
=> m1_altcat_2(A,C) ) ) ) ) ).
fof(t23_altcat_2,axiom,
! [A] :
( l2_altcat_1(A)
=> ! [B] :
( l2_altcat_1(B)
=> ( ( m1_altcat_2(A,B)
& m1_altcat_2(B,A) )
=> g2_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A)) = g2_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B)) ) ) ) ).
fof(d12_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( ( v6_altcat_1(C)
& m1_altcat_2(C,A) )
=> ( C = k8_altcat_2(A,B)
<=> ( u1_struct_0(C) = k1_struct_0(A,B)
& u1_altcat_1(C) = k9_funct_2(B,B,k1_altcat_1(A,B,B))
& u2_altcat_1(C) = k7_cat_1(k3_mcart_1(B,B,B),k4_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A),B,B,B)) ) ) ) ) ) ).
fof(t24_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_struct_0(A))
=> ! [C] :
( m1_subset_1(C,u1_struct_0(k8_altcat_2(A,B)))
=> C = B ) ) ) ).
fof(t25_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_1(B)
& m1_altcat_2(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_altcat_1(C)
& m1_altcat_2(C,A) )
=> ( ( r1_tarski(u1_struct_0(B),u1_struct_0(C))
& r2_altcat_2(k2_zfmisc_1(u1_struct_0(B),u1_struct_0(B)),k2_zfmisc_1(u1_struct_0(C),u1_struct_0(C)),u1_altcat_1(B),u1_altcat_1(C)) )
=> m1_altcat_2(B,C) ) ) ) ) ).
fof(d13_altcat_2,axiom,
! [A] :
( l2_altcat_1(A)
=> ! [B] :
( m1_altcat_2(B,A)
=> ( v2_altcat_2(B,A)
<=> u1_altcat_1(B) = k1_realset1(u1_altcat_1(A),u1_struct_0(B)) ) ) ) ).
fof(d14_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_altcat_2(B,A)
=> ( ( ~ v3_struct_0(B)
=> ( v3_altcat_2(B,A)
<=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( C = D
=> r2_hidden(k8_altcat_1(A,D),k1_altcat_1(B,C,C)) ) ) ) ) )
& ( v3_struct_0(B)
=> v3_altcat_2(B,A) ) ) ) ) ).
fof(t26_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( m1_altcat_2(B,A)
=> ( ( u1_struct_0(B) = u1_struct_0(A)
& u1_altcat_1(B) = u1_altcat_1(A) )
=> g2_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B)) = g2_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A)) ) ) ) ).
fof(t27_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_1(B)
& m1_altcat_2(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_altcat_1(C)
& m1_altcat_2(C,A) )
=> ( ( u1_struct_0(B) = u1_struct_0(C)
& u1_altcat_1(B) = u1_altcat_1(C) )
=> g2_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B)) = g2_altcat_1(u1_struct_0(C),u1_altcat_1(C),u2_altcat_1(C)) ) ) ) ) ).
fof(t28_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( v2_altcat_2(B,A)
& m1_altcat_2(B,A) )
=> ( u1_struct_0(B) = u1_struct_0(A)
=> g2_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B)) = g2_altcat_1(u1_struct_0(A),u1_altcat_1(A),u2_altcat_1(A)) ) ) ) ).
fof(t29_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_2(B,A)
& m1_altcat_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( C = E
& D = F )
=> k1_altcat_1(A,C,D) = k1_altcat_1(B,E,F) ) ) ) ) ) ) ) ).
fof(t30_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_altcat_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> m1_subset_1(C,u1_struct_0(A)) ) ) ) ).
fof(t31_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_2(B,A)
& m1_altcat_2(B,A) )
=> ! [C] :
( ( ~ v3_struct_0(C)
& v2_altcat_2(C,A)
& m1_altcat_2(C,A) )
=> ( u1_struct_0(B) = u1_struct_0(C)
=> g2_altcat_1(u1_struct_0(B),u1_altcat_1(B),u2_altcat_1(B)) = g2_altcat_1(u1_struct_0(C),u1_altcat_1(C),u2_altcat_1(C)) ) ) ) ) ).
fof(t32_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_altcat_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( C = E
& D = F )
=> r1_tarski(k1_altcat_1(B,E,F),k1_altcat_1(A,C,D)) ) ) ) ) ) ) ) ).
fof(t33_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_1(B)
& m1_altcat_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(B))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ~ ( k1_altcat_1(B,C,D) != k1_xboole_0
& k1_altcat_1(B,D,E) != k1_xboole_0
& ? [F] :
( m1_subset_1(F,u1_struct_0(A))
& ? [G] :
( m1_subset_1(G,u1_struct_0(A))
& ? [H] :
( m1_subset_1(H,u1_struct_0(A))
& F = C
& G = D
& H = E
& ? [I] :
( m1_subset_1(I,k1_altcat_1(A,F,G))
& ? [J] :
( m1_subset_1(J,k1_altcat_1(A,G,H))
& ? [K] :
( m1_subset_1(K,k1_altcat_1(B,C,D))
& ? [L] :
( m1_subset_1(L,k1_altcat_1(B,D,E))
& I = K
& J = L
& k5_altcat_1(A,F,G,H,I,J) != k5_altcat_1(B,C,D,E,K,L) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t34_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& m1_altcat_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(A))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ! [E] :
( m1_subset_1(E,u1_struct_0(B))
=> ! [F] :
( m1_subset_1(F,u1_struct_0(B))
=> ( ( C = E
& D = F )
=> ( k1_altcat_1(B,E,F) = k1_xboole_0
| ! [G] :
( m1_subset_1(G,k1_altcat_1(B,E,F))
=> m1_subset_1(G,k1_altcat_1(A,C,D)) ) ) ) ) ) ) ) ) ) ).
fof(t35_altcat_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v2_altcat_1(A)
& v11_altcat_1(A)
& v12_altcat_1(A)
& l2_altcat_1(A) )
=> ! [B] :
( ( ~ v3_struct_0(B)
& v2_altcat_1(B)
& v3_altcat_2(B,A)
& m1_altcat_2(B,A) )
=> ! [C] :
( m1_subset_1(C,u1_struct_0(B))
=> ! [D] :
( m1_subset_1(D,u1_struct_0(A))
=> ( C = D
=> k8_altcat_1(B,C) = k8_altcat_1(A,D) ) ) ) ) ) ).
fof(dt_m1_altcat_2,axiom,
! [A] :
( l2_altcat_1(A)
=> ! [B] :
( m1_altcat_2(B,A)
=> l2_altcat_1(B) ) ) ).
fof(existence_m1_altcat_2,axiom,
! [A] :
( l2_altcat_1(A)
=> ? [B] : m1_altcat_2(B,A) ) ).
fof(reflexivity_r1_altcat_2,axiom,
! [A,B] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_relat_1(B)
& v1_funct_1(B) )
=> r1_altcat_2(A,A) ) ).
fof(reflexivity_r2_altcat_2,axiom,
! [A,B,C,D] :
( ( m1_pboole(C,A)
& m1_pboole(D,B) )
=> r2_altcat_2(A,B,C,C) ) ).
fof(redefinition_r2_altcat_2,axiom,
! [A,B,C,D] :
( ( m1_pboole(C,A)
& m1_pboole(D,B) )
=> ( r2_altcat_2(A,B,C,D)
<=> r1_altcat_2(C,D) ) ) ).
fof(dt_k1_altcat_2,axiom,
! [A,B] :
( m1_pboole(B,k2_zfmisc_1(A,A))
=> m1_pboole(k1_altcat_2(A,B),k2_zfmisc_1(A,A)) ) ).
fof(redefinition_k1_altcat_2,axiom,
! [A,B] :
( m1_pboole(B,k2_zfmisc_1(A,A))
=> k1_altcat_2(A,B) = k2_funct_4(B) ) ).
fof(dt_k2_altcat_2,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,k2_zfmisc_1(A,A))
& m1_pboole(C,k2_zfmisc_1(A,A))
& m3_pboole(D,k2_zfmisc_1(A,A),B,C) )
=> m3_pboole(k2_altcat_2(A,B,C,D),k2_zfmisc_1(A,A),k1_altcat_2(A,B),k1_altcat_2(A,C)) ) ).
fof(redefinition_k2_altcat_2,axiom,
! [A,B,C,D] :
( ( m1_pboole(B,k2_zfmisc_1(A,A))
& m1_pboole(C,k2_zfmisc_1(A,A))
& m3_pboole(D,k2_zfmisc_1(A,A),B,C) )
=> k2_altcat_2(A,B,C,D) = k2_funct_4(D) ) ).
fof(dt_k3_altcat_2,axiom,
! [A,B,C] :
( ( v1_funcop_1(B)
& m1_pboole(B,A)
& v1_funcop_1(C)
& m1_pboole(C,A) )
=> ( v1_funcop_1(k3_altcat_2(A,B,C))
& m1_pboole(k3_altcat_2(A,B,C),A) ) ) ).
fof(redefinition_k3_altcat_2,axiom,
! [A,B,C] :
( ( v1_funcop_1(B)
& m1_pboole(B,A)
& v1_funcop_1(C)
& m1_pboole(C,A) )
=> k3_altcat_2(A,B,C) = k13_pboole(B,C) ) ).
fof(dt_k4_altcat_2,axiom,
! [A] :
( l1_cat_1(A)
=> m1_pboole(k4_altcat_2(A),k2_zfmisc_1(u1_cat_1(A),u1_cat_1(A))) ) ).
fof(dt_k5_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> m3_pboole(k5_altcat_2(A),k3_zfmisc_1(u1_cat_1(A),u1_cat_1(A),u1_cat_1(A)),k3_altcat_1(u1_cat_1(A),k4_altcat_2(A),k4_altcat_2(A)),k2_altcat_1(u1_cat_1(A),k4_altcat_2(A))) ) ).
fof(dt_k6_altcat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ( ~ v3_struct_0(k6_altcat_2(A))
& v6_altcat_1(k6_altcat_2(A))
& l2_altcat_1(k6_altcat_2(A)) ) ) ).
fof(dt_k7_altcat_2,axiom,
( v6_altcat_1(k7_altcat_2)
& l2_altcat_1(k7_altcat_2) ) ).
fof(dt_k8_altcat_2,axiom,
! [A,B] :
( ( ~ v3_struct_0(A)
& l2_altcat_1(A)
& m1_subset_1(B,u1_struct_0(A)) )
=> ( v6_altcat_1(k8_altcat_2(A,B))
& m1_altcat_2(k8_altcat_2(A,B),A) ) ) ).
fof(s1_altcat_2,axiom,
( v1_relat_1(a_0_0_altcat_2)
& v1_funct_1(a_0_0_altcat_2) ) ).
fof(s2_altcat_2,axiom,
( f2_s2_altcat_2 = a_0_1_altcat_2
=> k1_relat_1(f2_s2_altcat_2) = a_0_2_altcat_2 ) ).
fof(s3_altcat_2,axiom,
( ( f2_s3_altcat_2 = a_0_3_altcat_2
& p1_s3_altcat_2(f3_s3_altcat_2) )
=> k1_funct_1(f2_s3_altcat_2,f3_s3_altcat_2) = f4_s3_altcat_2(f3_s3_altcat_2) ) ).
fof(fraenkel_a_0_0_altcat_2,axiom,
! [A] :
( r2_hidden(A,a_0_0_altcat_2)
<=> ? [B] :
( m1_subset_1(B,f1_s1_altcat_2)
& A = k4_tarski(B,f2_s1_altcat_2(B))
& p1_s1_altcat_2(B) ) ) ).
fof(fraenkel_a_0_1_altcat_2,axiom,
! [A] :
( r2_hidden(A,a_0_1_altcat_2)
<=> ? [B] :
( m1_subset_1(B,f1_s2_altcat_2)
& A = k4_tarski(B,f3_s2_altcat_2(B))
& p1_s2_altcat_2(B) ) ) ).
fof(fraenkel_a_0_2_altcat_2,axiom,
! [A] :
( r2_hidden(A,a_0_2_altcat_2)
<=> ? [B] :
( m1_subset_1(B,f1_s2_altcat_2)
& A = B
& p1_s2_altcat_2(B) ) ) ).
fof(fraenkel_a_0_3_altcat_2,axiom,
! [A] :
( r2_hidden(A,a_0_3_altcat_2)
<=> ? [B] :
( m1_subset_1(B,f1_s3_altcat_2)
& A = k4_tarski(B,f4_s3_altcat_2(B))
& p1_s3_altcat_2(B) ) ) ).
%------------------------------------------------------------------------------