SET007 Axioms: SET007+237.ax
%------------------------------------------------------------------------------
% File : SET007+237 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Subcategories and Products of Categories
% 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 : cat_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 118 ( 13 unt; 0 def)
% Number of atoms : 891 ( 94 equ)
% Maximal formula atoms : 22 ( 7 avg)
% Number of connectives : 833 ( 60 ~; 1 |; 416 &)
% ( 9 <=>; 347 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 10 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-4 aty)
% Number of functors : 56 ( 56 usr; 1 con; 0-6 aty)
% Number of variables : 454 ( 443 !; 11 ?)
% 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_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ~ v1_xboole_0(k5_cat_2(A,B)) ) ).
fof(rc1_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ? [B] :
( m3_cat_2(B,A)
& v1_cat_1(B)
& v2_cat_1(B) ) ) ).
fof(fc2_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ( v1_cat_1(k11_cat_2(A,B))
& v2_cat_1(k11_cat_2(A,B)) ) ) ).
fof(t1_cat_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m2_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ( v1_funct_1(k3_funct_5(D))
& v1_funct_2(k3_funct_5(D),A,k1_fraenkel(B,C))
& m2_relset_1(k3_funct_5(D),A,k1_fraenkel(B,C)) ) ) ) ) ) ).
fof(t2_cat_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(B,A),C)
& m2_relset_1(D,k2_zfmisc_1(B,A),C) )
=> ( v1_funct_1(k5_funct_5(D))
& v1_funct_2(k5_funct_5(D),A,k1_fraenkel(B,C))
& m2_relset_1(k5_funct_5(D),A,k1_fraenkel(B,C)) ) ) ) ) ) ).
fof(t3_cat_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(A,B),C)
& m2_relset_1(F,k2_zfmisc_1(A,B),C) )
=> k8_funct_2(k2_zfmisc_1(A,B),C,F,k1_domain_1(A,B,D,E)) = k8_funct_2(B,C,k1_cat_2(A,B,C,k1_fraenkel(B,C),k2_cat_2(A,B,C,F),D),E) ) ) ) ) ) ) ).
fof(t4_cat_2,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( m1_subset_1(D,A)
=> ! [E] :
( m1_subset_1(E,B)
=> ! [F] :
( ( v1_funct_1(F)
& v1_funct_2(F,k2_zfmisc_1(A,B),C)
& m2_relset_1(F,k2_zfmisc_1(A,B),C) )
=> k8_funct_2(k2_zfmisc_1(A,B),C,F,k1_domain_1(A,B,D,E)) = k8_funct_2(A,C,k1_cat_2(B,A,C,k1_fraenkel(A,C),k3_cat_2(A,B,C,F),E),D) ) ) ) ) ) ) ).
fof(d1_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> k4_cat_2(A,B,C) = k10_pboole(u2_cat_1(A),k10_cat_1(B,C)) ) ) ) ).
fof(t5_cat_2,axiom,
$true ).
fof(t6_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> k8_funct_2(u2_cat_1(B),u2_cat_1(A),k4_cat_2(B,A,C),D) = k10_cat_1(A,C) ) ) ) ) ).
fof(t7_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> k8_funct_2(u1_cat_1(B),u1_cat_1(A),k12_cat_1(B,A,k4_cat_2(B,A,C)),D) = C ) ) ) ) ).
fof(d2_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( C = k5_cat_2(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> m2_cat_1(D,A,B) ) ) ) ) ).
fof(d3_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( m1_cat_2(C,A,B)
<=> ! [D] :
( m1_subset_1(D,C)
=> m2_cat_1(D,A,B) ) ) ) ) ) ).
fof(d4_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ( m3_cat_2(B,A)
<=> ( r1_tarski(u1_cat_1(B),u1_cat_1(A))
& ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> ( ( C = E
& D = F )
=> r1_tarski(k6_cat_1(B,C,D),k6_cat_1(A,E,F)) ) ) ) ) )
& r1_tarski(u5_cat_1(B),u5_cat_1(A))
& ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ( C = D
=> k10_cat_1(B,C) = k10_cat_1(A,D) ) ) ) ) ) ) ) ).
fof(t8_cat_2,axiom,
$true ).
fof(t9_cat_2,axiom,
$true ).
fof(t10_cat_2,axiom,
$true ).
fof(t11_cat_2,axiom,
$true ).
fof(t12_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> m1_subset_1(C,u1_cat_1(A)) ) ) ) ).
fof(t13_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> r1_tarski(u2_cat_1(B),u2_cat_1(A)) ) ) ).
fof(t14_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m1_subset_1(C,u2_cat_1(B))
=> m1_subset_1(C,u2_cat_1(A)) ) ) ) ).
fof(t15_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m1_subset_1(C,u2_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> ( C = D
=> ( k2_cat_1(B,C) = k2_cat_1(A,D)
& k3_cat_1(B,C) = k3_cat_1(A,D) ) ) ) ) ) ) ).
fof(t16_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> ! [G] :
( m1_cat_1(G,B,C,D)
=> ( ( C = E
& D = F )
=> ( k6_cat_1(B,C,D) = k1_xboole_0
| m1_cat_1(G,A,E,F) ) ) ) ) ) ) ) ) ) ).
fof(t17_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m1_subset_1(C,u2_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(A))
=> ! [F] :
( m1_subset_1(F,u2_cat_1(A))
=> ( ( C = E
& D = F
& k2_cat_1(B,D) = k3_cat_1(B,C) )
=> k4_cat_1(B,C,D) = k4_cat_1(A,E,F) ) ) ) ) ) ) ) ).
fof(t18_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> m3_cat_2(A,A) ) ).
fof(t19_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> m2_cat_1(k15_cat_1(B),B,A) ) ) ).
fof(d5_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> k8_cat_2(A,B) = k15_cat_1(B) ) ) ).
fof(t20_cat_2,axiom,
$true ).
fof(t21_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m1_subset_1(C,u2_cat_1(B))
=> k8_funct_2(u2_cat_1(B),u2_cat_1(A),k8_cat_2(A,B),C) = C ) ) ) ).
fof(t22_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> k8_funct_2(u1_cat_1(B),u1_cat_1(A),k12_cat_1(B,A,k8_cat_2(A,B)),C) = C ) ) ) ).
fof(t23_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> k13_cat_1(B,A,k8_cat_2(A,B),C) = C ) ) ) ).
fof(t24_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> v10_cat_1(k8_cat_2(A,B),B,A) ) ) ).
fof(t25_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ( v9_cat_1(k8_cat_2(A,B),B,A)
<=> ! [C] :
( m1_subset_1(C,u1_cat_1(B))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> ( ( C = E
& D = F )
=> k6_cat_1(B,C,D) = k6_cat_1(A,E,F) ) ) ) ) ) ) ) ) ).
fof(d6_cat_2,axiom,
! [A] :
( l1_cat_1(A)
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ( r1_cat_2(A,B)
<=> ( m3_cat_2(A,B)
& ! [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(B))
=> ! [F] :
( m1_subset_1(F,u1_cat_1(B))
=> ( ( C = E
& D = F )
=> k6_cat_1(A,C,D) = k6_cat_1(B,E,F) ) ) ) ) ) ) ) ) ) ).
fof(t26_cat_2,axiom,
$true ).
fof(t27_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ( r1_cat_2(B,A)
<=> v9_cat_1(k8_cat_2(A,B),B,A) ) ) ) ).
fof(d7_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> k11_cat_2(A,B) = g1_cat_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B)),k9_cat_2(u2_cat_1(A),u2_cat_1(B),u1_cat_1(A),u1_cat_1(B),u3_cat_1(A),u3_cat_1(B)),k9_cat_2(u2_cat_1(A),u2_cat_1(B),u1_cat_1(A),u1_cat_1(B),u4_cat_1(A),u4_cat_1(B)),k10_cat_2(u2_cat_1(A),u2_cat_1(B),u5_cat_1(A),u5_cat_1(B)),k9_cat_2(u1_cat_1(A),u1_cat_1(B),u2_cat_1(A),u2_cat_1(B),u6_cat_1(A),u6_cat_1(B))) ) ) ).
fof(t32_cat_2,axiom,
$true ).
fof(t33_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ( u1_cat_1(k11_cat_2(A,B)) = k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B))
& u2_cat_1(k11_cat_2(A,B)) = k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))
& u3_cat_1(k11_cat_2(A,B)) = k9_cat_2(u2_cat_1(A),u2_cat_1(B),u1_cat_1(A),u1_cat_1(B),u3_cat_1(A),u3_cat_1(B))
& u4_cat_1(k11_cat_2(A,B)) = k9_cat_2(u2_cat_1(A),u2_cat_1(B),u1_cat_1(A),u1_cat_1(B),u4_cat_1(A),u4_cat_1(B))
& u5_cat_1(k11_cat_2(A,B)) = k10_cat_2(u2_cat_1(A),u2_cat_1(B),u5_cat_1(A),u5_cat_1(B))
& u6_cat_1(k11_cat_2(A,B)) = k9_cat_2(u1_cat_1(A),u1_cat_1(B),u2_cat_1(A),u2_cat_1(B),u6_cat_1(A),u6_cat_1(B)) ) ) ) ).
fof(t34_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> m1_subset_1(k1_domain_1(u1_cat_1(A),u1_cat_1(B),C,D),u1_cat_1(k11_cat_2(A,B))) ) ) ) ) ).
fof(t35_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(k11_cat_2(A,B)))
=> ? [D] :
( m1_subset_1(D,u1_cat_1(A))
& ? [E] :
( m1_subset_1(E,u1_cat_1(B))
& C = k12_cat_2(A,B,D,E) ) ) ) ) ) ).
fof(t36_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> m1_subset_1(k1_domain_1(u2_cat_1(A),u2_cat_1(B),C,D),u2_cat_1(k11_cat_2(A,B))) ) ) ) ) ).
fof(t37_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u2_cat_1(k11_cat_2(A,B)))
=> ? [D] :
( m1_subset_1(D,u2_cat_1(A))
& ? [E] :
( m1_subset_1(E,u2_cat_1(B))
& C = k13_cat_2(A,B,D,E) ) ) ) ) ) ).
fof(t38_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> ( k2_cat_1(k11_cat_2(A,B),k13_cat_2(A,B,C,D)) = k12_cat_2(A,B,k2_cat_1(A,C),k2_cat_1(B,D))
& k3_cat_1(k11_cat_2(A,B),k13_cat_2(A,B,C,D)) = k12_cat_2(A,B,k3_cat_1(A,C),k3_cat_1(B,D)) ) ) ) ) ) ).
fof(t39_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(B))
=> ! [F] :
( m1_subset_1(F,u2_cat_1(B))
=> ( ( k2_cat_1(A,D) = k3_cat_1(A,C)
& k2_cat_1(B,F) = k3_cat_1(B,E) )
=> k4_cat_1(k11_cat_2(A,B),k13_cat_2(A,B,C,E),k13_cat_2(A,B,D,F)) = k13_cat_2(A,B,k4_cat_1(A,C,D),k4_cat_1(B,E,F)) ) ) ) ) ) ) ) ).
fof(t40_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(A))
=> ! [E] :
( m1_subset_1(E,u2_cat_1(B))
=> ! [F] :
( m1_subset_1(F,u2_cat_1(B))
=> ( k2_cat_1(k11_cat_2(A,B),k13_cat_2(A,B,D,F)) = k3_cat_1(k11_cat_2(A,B),k13_cat_2(A,B,C,E))
=> k4_cat_1(k11_cat_2(A,B),k13_cat_2(A,B,C,E),k13_cat_2(A,B,D,F)) = k13_cat_2(A,B,k4_cat_1(A,C,D),k4_cat_1(B,E,F)) ) ) ) ) ) ) ) ).
fof(t41_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> k10_cat_1(k11_cat_2(A,B),k12_cat_2(A,B,C,D)) = k13_cat_2(A,B,k10_cat_1(A,C),k10_cat_1(B,D)) ) ) ) ) ).
fof(t42_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [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(B))
=> ! [F] :
( m1_subset_1(F,u1_cat_1(B))
=> k6_cat_1(k11_cat_2(A,B),k12_cat_2(A,B,C,E),k12_cat_2(A,B,D,F)) = k2_zfmisc_1(k6_cat_1(A,C,D),k6_cat_1(B,E,F)) ) ) ) ) ) ) ).
fof(t43_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(A))
=> ! [E] :
( m1_cat_1(E,A,C,D)
=> ! [F] :
( m1_subset_1(F,u1_cat_1(B))
=> ! [G] :
( m1_subset_1(G,u1_cat_1(B))
=> ! [H] :
( m1_cat_1(H,B,F,G)
=> ~ ( k6_cat_1(A,C,D) != k1_xboole_0
& k6_cat_1(B,F,G) != k1_xboole_0
& ~ m1_cat_1(k13_cat_2(A,B,E,H),k11_cat_2(A,B),k12_cat_2(A,B,C,F),k12_cat_2(A,B,D,G)) ) ) ) ) ) ) ) ) ) ).
fof(t44_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,k11_cat_2(A,B),C)
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> m2_cat_1(k1_funct_1(k3_funct_5(D),k10_cat_1(A,E)),B,C) ) ) ) ) ) ).
fof(t45_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,k11_cat_2(A,B),C)
=> ! [E] :
( m1_subset_1(E,u1_cat_1(B))
=> m2_cat_1(k1_funct_1(k5_funct_5(D),k10_cat_1(B,E)),A,C) ) ) ) ) ) ).
fof(d8_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,k11_cat_2(A,B),C)
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> k14_cat_2(A,B,C,D,E) = k1_funct_1(k3_funct_5(D),k10_cat_1(A,E)) ) ) ) ) ) ).
fof(t46_cat_2,axiom,
$true ).
fof(t47_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,k11_cat_2(A,B),C)
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ! [F] :
( m1_subset_1(F,u2_cat_1(B))
=> k8_funct_2(u2_cat_1(B),u2_cat_1(C),k14_cat_2(A,B,C,D,E),F) = k8_funct_2(u2_cat_1(k11_cat_2(A,B)),u2_cat_1(C),D,k13_cat_2(A,B,k10_cat_1(A,E),F)) ) ) ) ) ) ) ).
fof(t48_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,k11_cat_2(A,B),C)
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ! [F] :
( m1_subset_1(F,u1_cat_1(B))
=> k8_funct_2(u1_cat_1(B),u1_cat_1(C),k12_cat_1(B,C,k14_cat_2(A,B,C,D,E)),F) = k8_funct_2(u1_cat_1(k11_cat_2(A,B)),u1_cat_1(C),k12_cat_1(k11_cat_2(A,B),C,D),k12_cat_2(A,B,E,F)) ) ) ) ) ) ) ).
fof(d9_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,k11_cat_2(A,B),C)
=> ! [E] :
( m1_subset_1(E,u1_cat_1(B))
=> k15_cat_2(A,B,C,D,E) = k1_funct_1(k5_funct_5(D),k10_cat_1(B,E)) ) ) ) ) ) ).
fof(t49_cat_2,axiom,
$true ).
fof(t50_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,k11_cat_2(A,B),C)
=> ! [E] :
( m1_subset_1(E,u1_cat_1(B))
=> ! [F] :
( m1_subset_1(F,u2_cat_1(A))
=> k8_funct_2(u2_cat_1(A),u2_cat_1(C),k15_cat_2(A,B,C,D,E),F) = k8_funct_2(u2_cat_1(k11_cat_2(A,B)),u2_cat_1(C),D,k13_cat_2(A,B,F,k10_cat_1(B,E))) ) ) ) ) ) ) ).
fof(t51_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,k11_cat_2(A,B),C)
=> ! [E] :
( m1_subset_1(E,u1_cat_1(A))
=> ! [F] :
( m1_subset_1(F,u1_cat_1(B))
=> k8_funct_2(u1_cat_1(A),u1_cat_1(C),k12_cat_1(A,C,k15_cat_2(A,B,C,D,F)),E) = k8_funct_2(u1_cat_1(k11_cat_2(A,B)),u1_cat_1(C),k12_cat_1(k11_cat_2(A,B),C,D),k12_cat_2(A,B,E,F)) ) ) ) ) ) ) ).
fof(t52_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_cat_1(A),k7_cat_2(B,C))
& m2_relset_1(D,u1_cat_1(A),k7_cat_2(B,C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_cat_1(B),k7_cat_2(A,C))
& m2_relset_1(E,u1_cat_1(B),k7_cat_2(A,C)) )
=> ~ ( ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> ! [G] :
( m1_subset_1(G,u1_cat_1(B))
=> k8_funct_2(u2_cat_1(A),u2_cat_1(C),k6_cat_2(u1_cat_1(B),A,C,k7_cat_2(A,C),E,G),k10_cat_1(A,F)) = k8_funct_2(u2_cat_1(B),u2_cat_1(C),k6_cat_2(u1_cat_1(A),B,C,k7_cat_2(B,C),D,F),k10_cat_1(B,G)) ) )
& ! [F] :
( m1_subset_1(F,u2_cat_1(B))
=> ! [G] :
( m1_subset_1(G,u2_cat_1(A))
=> k4_cat_1(C,k8_funct_2(u2_cat_1(B),u2_cat_1(C),k6_cat_2(u1_cat_1(A),B,C,k7_cat_2(B,C),D,k2_cat_1(A,G)),F),k8_funct_2(u2_cat_1(A),u2_cat_1(C),k6_cat_2(u1_cat_1(B),A,C,k7_cat_2(A,C),E,k3_cat_1(B,F)),G)) = k4_cat_1(C,k8_funct_2(u2_cat_1(A),u2_cat_1(C),k6_cat_2(u1_cat_1(B),A,C,k7_cat_2(A,C),E,k2_cat_1(B,F)),G),k8_funct_2(u2_cat_1(B),u2_cat_1(C),k6_cat_2(u1_cat_1(A),B,C,k7_cat_2(B,C),D,k3_cat_1(A,G)),F)) ) )
& ! [F] :
( m2_cat_1(F,k11_cat_2(B,A),C)
=> ~ ! [G] :
( m1_subset_1(G,u2_cat_1(B))
=> ! [H] :
( m1_subset_1(H,u2_cat_1(A))
=> k8_funct_2(u2_cat_1(k11_cat_2(B,A)),u2_cat_1(C),F,k13_cat_2(B,A,G,H)) = k4_cat_1(C,k8_funct_2(u2_cat_1(A),u2_cat_1(C),k6_cat_2(u1_cat_1(B),A,C,k7_cat_2(A,C),E,k2_cat_1(B,G)),H),k8_funct_2(u2_cat_1(B),u2_cat_1(C),k6_cat_2(u1_cat_1(A),B,C,k7_cat_2(B,C),D,k3_cat_1(A,H)),G)) ) ) ) ) ) ) ) ) ) ).
fof(t53_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,u1_cat_1(A),k7_cat_2(B,C))
& m2_relset_1(D,u1_cat_1(A),k7_cat_2(B,C)) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,u1_cat_1(B),k7_cat_2(A,C))
& m2_relset_1(E,u1_cat_1(B),k7_cat_2(A,C)) )
=> ( ? [F] :
( m2_cat_1(F,k11_cat_2(B,A),C)
& ! [G] :
( m1_subset_1(G,u1_cat_1(A))
=> ! [H] :
( m1_subset_1(H,u1_cat_1(B))
=> ( k15_cat_2(B,A,C,F,G) = k6_cat_2(u1_cat_1(A),B,C,k7_cat_2(B,C),D,G)
& k14_cat_2(B,A,C,F,H) = k6_cat_2(u1_cat_1(B),A,C,k7_cat_2(A,C),E,H) ) ) ) )
=> ! [F] :
( m1_subset_1(F,u2_cat_1(B))
=> ! [G] :
( m1_subset_1(G,u2_cat_1(A))
=> k4_cat_1(C,k8_funct_2(u2_cat_1(B),u2_cat_1(C),k6_cat_2(u1_cat_1(A),B,C,k7_cat_2(B,C),D,k2_cat_1(A,G)),F),k8_funct_2(u2_cat_1(A),u2_cat_1(C),k6_cat_2(u1_cat_1(B),A,C,k7_cat_2(A,C),E,k3_cat_1(B,F)),G)) = k4_cat_1(C,k8_funct_2(u2_cat_1(A),u2_cat_1(C),k6_cat_2(u1_cat_1(B),A,C,k7_cat_2(A,C),E,k2_cat_1(B,F)),G),k8_funct_2(u2_cat_1(B),u2_cat_1(C),k6_cat_2(u1_cat_1(A),B,C,k7_cat_2(B,C),D,k3_cat_1(A,G)),F)) ) ) ) ) ) ) ) ) ).
fof(t54_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> m2_cat_1(k9_funct_3(u2_cat_1(A),u2_cat_1(B)),k11_cat_2(A,B),A) ) ) ).
fof(t55_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> m2_cat_1(k10_funct_3(u2_cat_1(A),u2_cat_1(B)),k11_cat_2(A,B),B) ) ) ).
fof(d10_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> k16_cat_2(A,B) = k9_funct_3(u2_cat_1(A),u2_cat_1(B)) ) ) ).
fof(d11_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> k17_cat_2(A,B) = k10_funct_3(u2_cat_1(A),u2_cat_1(B)) ) ) ).
fof(t56_cat_2,axiom,
$true ).
fof(t57_cat_2,axiom,
$true ).
fof(t58_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> k8_funct_2(u2_cat_1(k11_cat_2(A,B)),u2_cat_1(A),k16_cat_2(A,B),k13_cat_2(A,B,C,D)) = C ) ) ) ) ).
fof(t59_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> k8_funct_2(u1_cat_1(k11_cat_2(A,B)),u1_cat_1(A),k12_cat_1(k11_cat_2(A,B),A,k16_cat_2(A,B)),k12_cat_2(A,B,C,D)) = C ) ) ) ) ).
fof(t60_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u2_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u2_cat_1(B))
=> k8_funct_2(u2_cat_1(k11_cat_2(A,B)),u2_cat_1(B),k17_cat_2(A,B),k13_cat_2(A,B,C,D)) = D ) ) ) ) ).
fof(t61_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_subset_1(C,u1_cat_1(A))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(B))
=> k8_funct_2(u1_cat_1(k11_cat_2(A,B)),u1_cat_1(B),k12_cat_1(k11_cat_2(A,B),B,k17_cat_2(A,B)),k12_cat_2(A,B,C,D)) = D ) ) ) ) ).
fof(t62_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,C)
=> m2_cat_1(k14_funct_3(u2_cat_1(A),u2_cat_1(B),u2_cat_1(C),D,E),A,k11_cat_2(B,C)) ) ) ) ) ) ).
fof(t63_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( m2_cat_1(D,A,B)
=> ! [E] :
( m2_cat_1(E,A,C)
=> ! [F] :
( m1_subset_1(F,u1_cat_1(A))
=> k8_funct_2(u1_cat_1(A),u1_cat_1(k11_cat_2(B,C)),k12_cat_1(A,k11_cat_2(B,C),k18_cat_2(A,B,C,D,E)),F) = k12_cat_2(B,C,k8_funct_2(u1_cat_1(A),u1_cat_1(B),k12_cat_1(A,B,D),F),k8_funct_2(u1_cat_1(A),u1_cat_1(C),k12_cat_1(A,C,E),F)) ) ) ) ) ) ) ).
fof(t64_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( ( v2_cat_1(D)
& l1_cat_1(D) )
=> ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m2_cat_1(F,C,D)
=> k9_cat_2(u2_cat_1(A),u2_cat_1(C),u2_cat_1(B),u2_cat_1(D),E,F) = k18_cat_2(k11_cat_2(A,C),B,D,k14_cat_1(k11_cat_2(A,C),A,B,k16_cat_2(A,C),E),k14_cat_1(k11_cat_2(A,C),C,D,k17_cat_2(A,C),F)) ) ) ) ) ) ) ).
fof(t65_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( ( v2_cat_1(D)
& l1_cat_1(D) )
=> ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m2_cat_1(F,C,D)
=> m2_cat_1(k9_cat_2(u2_cat_1(A),u2_cat_1(C),u2_cat_1(B),u2_cat_1(D),E,F),k11_cat_2(A,C),k11_cat_2(B,D)) ) ) ) ) ) ) ).
fof(t66_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( ( v2_cat_1(C)
& l1_cat_1(C) )
=> ! [D] :
( ( v2_cat_1(D)
& l1_cat_1(D) )
=> ! [E] :
( m2_cat_1(E,A,B)
=> ! [F] :
( m2_cat_1(F,C,D)
=> ! [G] :
( m1_subset_1(G,u1_cat_1(A))
=> ! [H] :
( m1_subset_1(H,u1_cat_1(C))
=> k8_funct_2(u1_cat_1(k11_cat_2(A,C)),u1_cat_1(k11_cat_2(B,D)),k12_cat_1(k11_cat_2(A,C),k11_cat_2(B,D),k19_cat_2(A,C,B,D,E,F)),k12_cat_2(A,C,G,H)) = k12_cat_2(B,D,k8_funct_2(u1_cat_1(A),u1_cat_1(B),k12_cat_1(A,B,E),G),k8_funct_2(u1_cat_1(C),u1_cat_1(D),k12_cat_1(C,D,F),H)) ) ) ) ) ) ) ) ) ).
fof(dt_m1_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ! [C] :
( m1_cat_2(C,A,B)
=> ~ v1_xboole_0(C) ) ) ).
fof(existence_m1_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ? [C] : m1_cat_2(C,A,B) ) ).
fof(dt_m2_cat_2,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_cat_2(C,A,B) )
=> ! [D] :
( m2_cat_2(D,A,B,C)
=> m2_cat_1(D,A,B) ) ) ).
fof(existence_m2_cat_2,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_cat_2(C,A,B) )
=> ? [D] : m2_cat_2(D,A,B,C) ) ).
fof(redefinition_m2_cat_2,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_cat_2(C,A,B) )
=> ! [D] :
( m2_cat_2(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(dt_m3_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m3_cat_2(B,A)
=> ( v2_cat_1(B)
& l1_cat_1(B) ) ) ) ).
fof(existence_m3_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ? [B] : m3_cat_2(B,A) ) ).
fof(dt_k1_cat_2,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(C)
& m1_fraenkel(D,B,C)
& v1_funct_1(E)
& v1_funct_2(E,A,D)
& m1_relset_1(E,A,D)
& m1_subset_1(F,A) )
=> m2_fraenkel(k1_cat_2(A,B,C,D,E,F),B,C,D) ) ).
fof(redefinition_k1_cat_2,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(C)
& m1_fraenkel(D,B,C)
& v1_funct_1(E)
& v1_funct_2(E,A,D)
& m1_relset_1(E,A,D)
& m1_subset_1(F,A) )
=> k1_cat_2(A,B,C,D,E,F) = k1_funct_1(E,F) ) ).
fof(dt_k2_cat_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m1_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ( v1_funct_1(k2_cat_2(A,B,C,D))
& v1_funct_2(k2_cat_2(A,B,C,D),A,k1_fraenkel(B,C))
& m2_relset_1(k2_cat_2(A,B,C,D),A,k1_fraenkel(B,C)) ) ) ).
fof(redefinition_k2_cat_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m1_relset_1(D,k2_zfmisc_1(A,B),C) )
=> k2_cat_2(A,B,C,D) = k3_funct_5(D) ) ).
fof(dt_k3_cat_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m1_relset_1(D,k2_zfmisc_1(A,B),C) )
=> ( v1_funct_1(k3_cat_2(A,B,C,D))
& v1_funct_2(k3_cat_2(A,B,C,D),B,k1_fraenkel(A,C))
& m2_relset_1(k3_cat_2(A,B,C,D),B,k1_fraenkel(A,C)) ) ) ).
fof(redefinition_k3_cat_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,k2_zfmisc_1(A,B),C)
& m1_relset_1(D,k2_zfmisc_1(A,B),C) )
=> k3_cat_2(A,B,C,D) = k5_funct_5(D) ) ).
fof(dt_k4_cat_2,axiom,
! [A,B,C] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_subset_1(C,u1_cat_1(B)) )
=> m2_cat_1(k4_cat_2(A,B,C),A,B) ) ).
fof(dt_k5_cat_2,axiom,
$true ).
fof(dt_k6_cat_2,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& m1_cat_2(D,B,C)
& v1_funct_1(E)
& v1_funct_2(E,A,D)
& m1_relset_1(E,A,D)
& m1_subset_1(F,A) )
=> m2_cat_2(k6_cat_2(A,B,C,D,E,F),B,C,D) ) ).
fof(redefinition_k6_cat_2,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& m1_cat_2(D,B,C)
& v1_funct_1(E)
& v1_funct_2(E,A,D)
& m1_relset_1(E,A,D)
& m1_subset_1(F,A) )
=> k6_cat_2(A,B,C,D,E,F) = k1_funct_1(E,F) ) ).
fof(dt_k7_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m1_cat_2(k7_cat_2(A,B),A,B) ) ).
fof(redefinition_k7_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> k7_cat_2(A,B) = k5_cat_2(A,B) ) ).
fof(dt_k8_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m3_cat_2(B,A) )
=> m2_cat_1(k8_cat_2(A,B),B,A) ) ).
fof(dt_k9_cat_2,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,A,C)
& m1_relset_1(E,A,C)
& v1_funct_1(F)
& v1_funct_2(F,B,D)
& m1_relset_1(F,B,D) )
=> ( v1_funct_1(k9_cat_2(A,B,C,D,E,F))
& v1_funct_2(k9_cat_2(A,B,C,D,E,F),k2_zfmisc_1(A,B),k2_zfmisc_1(C,D))
& m2_relset_1(k9_cat_2(A,B,C,D,E,F),k2_zfmisc_1(A,B),k2_zfmisc_1(C,D)) ) ) ).
fof(redefinition_k9_cat_2,axiom,
! [A,B,C,D,E,F] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& ~ v1_xboole_0(D)
& v1_funct_1(E)
& v1_funct_2(E,A,C)
& m1_relset_1(E,A,C)
& v1_funct_1(F)
& v1_funct_2(F,B,D)
& m1_relset_1(F,B,D) )
=> k9_cat_2(A,B,C,D,E,F) = k15_funct_3(E,F) ) ).
fof(dt_k10_cat_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& m1_relset_1(D,k2_zfmisc_1(B,B),B) )
=> ( v1_funct_1(k10_cat_2(A,B,C,D))
& m2_relset_1(k10_cat_2(A,B,C,D),k2_zfmisc_1(k2_zfmisc_1(A,B),k2_zfmisc_1(A,B)),k2_zfmisc_1(A,B)) ) ) ).
fof(redefinition_k10_cat_2,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_funct_1(C)
& m1_relset_1(C,k2_zfmisc_1(A,A),A)
& v1_funct_1(D)
& m1_relset_1(D,k2_zfmisc_1(B,B),B) )
=> k10_cat_2(A,B,C,D) = k3_funct_4(C,D) ) ).
fof(dt_k11_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> ( v2_cat_1(k11_cat_2(A,B))
& l1_cat_1(k11_cat_2(A,B)) ) ) ).
fof(dt_k12_cat_2,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_subset_1(C,u1_cat_1(A))
& m1_subset_1(D,u1_cat_1(B)) )
=> m1_subset_1(k12_cat_2(A,B,C,D),u1_cat_1(k11_cat_2(A,B))) ) ).
fof(redefinition_k12_cat_2,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_subset_1(C,u1_cat_1(A))
& m1_subset_1(D,u1_cat_1(B)) )
=> k12_cat_2(A,B,C,D) = k4_tarski(C,D) ) ).
fof(dt_k13_cat_2,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_subset_1(C,u2_cat_1(A))
& m1_subset_1(D,u2_cat_1(B)) )
=> m1_subset_1(k13_cat_2(A,B,C,D),u2_cat_1(k11_cat_2(A,B))) ) ).
fof(redefinition_k13_cat_2,axiom,
! [A,B,C,D] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& m1_subset_1(C,u2_cat_1(A))
& m1_subset_1(D,u2_cat_1(B)) )
=> k13_cat_2(A,B,C,D) = k4_tarski(C,D) ) ).
fof(dt_k14_cat_2,axiom,
! [A,B,C,D,E] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& m2_cat_1(D,k11_cat_2(A,B),C)
& m1_subset_1(E,u1_cat_1(A)) )
=> m2_cat_1(k14_cat_2(A,B,C,D,E),B,C) ) ).
fof(dt_k15_cat_2,axiom,
! [A,B,C,D,E] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& m2_cat_1(D,k11_cat_2(A,B),C)
& m1_subset_1(E,u1_cat_1(B)) )
=> m2_cat_1(k15_cat_2(A,B,C,D,E),A,C) ) ).
fof(dt_k16_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m2_cat_1(k16_cat_2(A,B),k11_cat_2(A,B),A) ) ).
fof(dt_k17_cat_2,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B) )
=> m2_cat_1(k17_cat_2(A,B),k11_cat_2(A,B),B) ) ).
fof(dt_k18_cat_2,axiom,
! [A,B,C,D,E] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& m2_cat_1(D,A,B)
& m2_cat_1(E,A,C) )
=> m2_cat_1(k18_cat_2(A,B,C,D,E),A,k11_cat_2(B,C)) ) ).
fof(redefinition_k18_cat_2,axiom,
! [A,B,C,D,E] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& m2_cat_1(D,A,B)
& m2_cat_1(E,A,C) )
=> k18_cat_2(A,B,C,D,E) = k13_funct_3(D,E) ) ).
fof(dt_k19_cat_2,axiom,
! [A,B,C,D,E,F] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& v2_cat_1(D)
& l1_cat_1(D)
& m2_cat_1(E,A,C)
& m2_cat_1(F,B,D) )
=> m2_cat_1(k19_cat_2(A,B,C,D,E,F),k11_cat_2(A,B),k11_cat_2(C,D)) ) ).
fof(redefinition_k19_cat_2,axiom,
! [A,B,C,D,E,F] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& v2_cat_1(B)
& l1_cat_1(B)
& v2_cat_1(C)
& l1_cat_1(C)
& v2_cat_1(D)
& l1_cat_1(D)
& m2_cat_1(E,A,C)
& m2_cat_1(F,B,D) )
=> k19_cat_2(A,B,C,D,E,F) = k15_funct_3(E,F) ) ).
fof(t28_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_cat_1(A))) )
=> ( ~ v1_xboole_0(k3_tarski(a_2_0_cat_2(A,B)))
& m1_subset_1(k3_tarski(a_2_0_cat_2(A,B)),k1_zfmisc_1(u2_cat_1(A))) ) ) ) ).
fof(t29_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_cat_1(A))) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ( C = k3_tarski(a_2_0_cat_2(A,B))
=> ( v1_funct_1(k2_partfun1(u2_cat_1(A),u1_cat_1(A),u3_cat_1(A),C))
& v1_funct_2(k2_partfun1(u2_cat_1(A),u1_cat_1(A),u3_cat_1(A),C),C,B)
& m2_relset_1(k2_partfun1(u2_cat_1(A),u1_cat_1(A),u3_cat_1(A),C),C,B)
& v1_funct_1(k2_partfun1(u2_cat_1(A),u1_cat_1(A),u4_cat_1(A),C))
& v1_funct_2(k2_partfun1(u2_cat_1(A),u1_cat_1(A),u4_cat_1(A),C),C,B)
& m2_relset_1(k2_partfun1(u2_cat_1(A),u1_cat_1(A),u4_cat_1(A),C),C,B)
& v1_funct_1(k1_realset1(u5_cat_1(A),C))
& m2_relset_1(k1_realset1(u5_cat_1(A),C),k2_zfmisc_1(C,C),C)
& v1_funct_1(k2_partfun1(u1_cat_1(A),u2_cat_1(A),u6_cat_1(A),B))
& v1_funct_2(k2_partfun1(u1_cat_1(A),u2_cat_1(A),u6_cat_1(A),B),B,C)
& m2_relset_1(k2_partfun1(u1_cat_1(A),u2_cat_1(A),u6_cat_1(A),B),B,C) ) ) ) ) ) ).
fof(t30_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_cat_1(A))) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,C,B)
& m2_relset_1(D,C,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,C,B)
& m2_relset_1(E,C,B) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,k2_zfmisc_1(C,C),C) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,B,C)
& m2_relset_1(G,B,C) )
=> ( ( C = k3_tarski(a_2_0_cat_2(A,B))
& D = k2_partfun1(u2_cat_1(A),u1_cat_1(A),u3_cat_1(A),C)
& E = k2_partfun1(u2_cat_1(A),u1_cat_1(A),u4_cat_1(A),C)
& F = k1_realset1(u5_cat_1(A),C)
& G = k2_partfun1(u1_cat_1(A),u2_cat_1(A),u6_cat_1(A),B) )
=> r1_cat_2(g1_cat_1(B,C,D,E,F,G),A) ) ) ) ) ) ) ) ) ).
fof(t31_cat_2,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_cat_1(A))) )
=> ! [C] :
( ~ v1_xboole_0(C)
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,C,B)
& m2_relset_1(D,C,B) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,C,B)
& m2_relset_1(E,C,B) )
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,k2_zfmisc_1(C,C),C) )
=> ! [G] :
( ( v1_funct_1(G)
& v1_funct_2(G,B,C)
& m2_relset_1(G,B,C) )
=> ( r1_cat_2(g1_cat_1(B,C,D,E,F,G),A)
=> ( C = k3_tarski(a_2_0_cat_2(A,B))
& D = k2_partfun1(u2_cat_1(A),u1_cat_1(A),u3_cat_1(A),C)
& E = k2_partfun1(u2_cat_1(A),u1_cat_1(A),u4_cat_1(A),C)
& F = k1_realset1(u5_cat_1(A),C)
& G = k2_partfun1(u1_cat_1(A),u2_cat_1(A),u6_cat_1(A),B) ) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_2_0_cat_2,axiom,
! [A,B,C] :
( ( v2_cat_1(B)
& l1_cat_1(B)
& ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(u1_cat_1(B))) )
=> ( r2_hidden(A,a_2_0_cat_2(B,C))
<=> ? [D,E] :
( m1_subset_1(D,u1_cat_1(B))
& m1_subset_1(E,u1_cat_1(B))
& A = k6_cat_1(B,D,E)
& r2_hidden(D,C)
& r2_hidden(E,C) ) ) ) ).
%------------------------------------------------------------------------------