SET007 Axioms: SET007+330.ax
%------------------------------------------------------------------------------
% File : SET007+330 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Comma Category
% 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 : commacat [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 32 ( 5 unt; 0 def)
% Number of atoms : 296 ( 40 equ)
% Maximal formula atoms : 25 ( 9 avg)
% Number of connectives : 267 ( 3 ~; 1 |; 179 &)
% ( 5 <=>; 79 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 12 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of predicates : 13 ( 11 usr; 1 prp; 0-3 aty)
% Number of functors : 43 ( 43 usr; 0 con; 1-7 aty)
% Number of variables : 160 ( 136 !; 24 ?)
% 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_commacat,axiom,
$true ).
fof(d2_commacat,axiom,
$true ).
fof(d3_commacat,axiom,
$true ).
fof(d4_commacat,axiom,
$true ).
fof(t1_commacat,axiom,
$true ).
fof(t2_commacat,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,C)
=> ! [E] :
( m2_cat_1(E,B,C)
=> ! [F] :
( m2_subset_1(F,k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E))
=> ( ? [G] :
( m1_subset_1(G,u1_cat_1(A))
& ? [H] :
( m1_subset_1(H,u1_cat_1(B))
& ? [I] :
( m1_subset_1(I,u2_cat_1(C))
& r2_hidden(I,k6_cat_1(C,k13_cat_1(A,C,D,G),k13_cat_1(B,C,E,H))) ) ) )
=> ( F = k1_domain_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C),k1_domain_1(u1_cat_1(A),u1_cat_1(B),k14_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),F),k15_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),F)),k3_domain_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C),F))
& r2_hidden(k3_domain_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C),F),k6_cat_1(C,k13_cat_1(A,C,D,k14_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),F)),k13_cat_1(B,C,E,k15_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),F))))
& k2_cat_1(C,k3_domain_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C),F)) = k13_cat_1(A,C,D,k14_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),F))
& k3_cat_1(C,k3_domain_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C),F)) = k13_cat_1(B,C,E,k15_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),F)) ) ) ) ) ) ) ) ) ).
fof(t3_commacat,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,C)
=> ! [E] :
( m2_cat_1(E,B,C)
=> ! [F] :
( m2_subset_1(F,k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_commacat(A,B,C,D,E))
=> ( ? [G] :
( m1_subset_1(G,u1_cat_1(A))
& ? [H] :
( m1_subset_1(H,u1_cat_1(B))
& ? [I] :
( m1_subset_1(I,u2_cat_1(C))
& r2_hidden(I,k6_cat_1(C,k13_cat_1(A,C,D,G),k13_cat_1(B,C,E,H))) ) ) )
=> ( F = k1_domain_1(k2_zfmisc_1(k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B)),k1_domain_1(k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E),k3_commacat(A,B,C,D,E,F),k4_commacat(A,B,C,D,E,F)),k1_domain_1(u2_cat_1(A),u2_cat_1(B),k16_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),F),k17_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),F)))
& k2_cat_1(A,k16_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),F)) = k14_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),k3_commacat(A,B,C,D,E,F))
& k3_cat_1(A,k16_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),F)) = k14_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),k4_commacat(A,B,C,D,E,F))
& k2_cat_1(B,k17_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),F)) = k15_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),k3_commacat(A,B,C,D,E,F))
& k3_cat_1(B,k17_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),F)) = k15_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),k4_commacat(A,B,C,D,E,F))
& k4_cat_1(C,k8_funct_2(u2_cat_1(A),u2_cat_1(C),D,k16_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),F)),k3_domain_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C),k4_commacat(A,B,C,D,E,F))) = k4_cat_1(C,k3_domain_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C),k3_commacat(A,B,C,D,E,F)),k8_funct_2(u2_cat_1(B),u2_cat_1(C),E,k17_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),F))) ) ) ) ) ) ) ) ) ).
fof(d7_commacat,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,C)
=> ! [E] :
( m2_cat_1(E,B,C)
=> ! [F] :
( m2_subset_1(F,k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_commacat(A,B,C,D,E))
=> ! [G] :
( m2_subset_1(G,k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_commacat(A,B,C,D,E))
=> ( k4_commacat(A,B,C,D,E,F) = k3_commacat(A,B,C,D,E,G)
=> ( ! [H] :
( m1_subset_1(H,u1_cat_1(A))
=> ! [I] :
( m1_subset_1(I,u1_cat_1(B))
=> ! [J] :
( m1_subset_1(J,u2_cat_1(C))
=> ~ r2_hidden(J,k6_cat_1(C,k13_cat_1(A,C,D,H),k13_cat_1(B,C,E,I))) ) ) )
| k5_commacat(A,B,C,D,E,F,G) = k1_domain_1(k2_zfmisc_1(k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B)),k1_domain_1(k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E),k3_commacat(A,B,C,D,E,F),k4_commacat(A,B,C,D,E,G)),k1_domain_1(u2_cat_1(A),u2_cat_1(B),k4_cat_1(A,k16_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),F),k16_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),G)),k4_cat_1(B,k17_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),F),k17_domain_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),u2_cat_1(A),u2_cat_1(B),G)))) ) ) ) ) ) ) ) ) ) ).
fof(d9_commacat,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,C)
=> ! [E] :
( m2_cat_1(E,B,C)
=> ( ? [F] :
( m1_subset_1(F,u1_cat_1(A))
& ? [G] :
( m1_subset_1(G,u1_cat_1(B))
& ? [H] :
( m1_subset_1(H,u2_cat_1(C))
& r2_hidden(H,k6_cat_1(C,k13_cat_1(A,C,D,F),k13_cat_1(B,C,E,G))) ) ) )
=> ! [F] :
( ( v1_cat_1(F)
& v2_cat_1(F)
& l1_cat_1(F) )
=> ( F = k7_commacat(A,B,C,D,E)
<=> ( u1_cat_1(F) = k1_commacat(A,B,C,D,E)
& u2_cat_1(F) = k2_commacat(A,B,C,D,E)
& ! [G] :
( m2_subset_1(G,k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_commacat(A,B,C,D,E))
=> k1_funct_1(u3_cat_1(F),G) = k3_commacat(A,B,C,D,E,G) )
& ! [G] :
( m2_subset_1(G,k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_commacat(A,B,C,D,E))
=> k1_funct_1(u4_cat_1(F),G) = k4_commacat(A,B,C,D,E,G) )
& ! [G] :
( m2_subset_1(G,k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E))
=> k1_funct_1(u6_cat_1(F),G) = k1_domain_1(k2_zfmisc_1(k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B)),k1_domain_1(k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E),G,G),k1_domain_1(u2_cat_1(A),u2_cat_1(B),k10_cat_1(A,k14_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),G)),k10_cat_1(B,k15_domain_1(u1_cat_1(A),u1_cat_1(B),u2_cat_1(C),G)))) )
& u5_cat_1(F) = k6_commacat(A,B,C,D,E) ) ) ) ) ) ) ) ) ) ).
fof(t4_commacat,axiom,
! [A,B] :
( u1_cat_1(k8_cat_1(B,A)) = k1_tarski(B)
& u2_cat_1(k8_cat_1(B,A)) = k1_tarski(A) ) ).
fof(t5_commacat,axiom,
! [A,B,C] :
( m1_subset_1(C,u1_cat_1(k8_cat_1(B,A)))
=> ! [D] :
( m1_subset_1(D,u1_cat_1(k8_cat_1(B,A)))
=> k6_cat_1(k8_cat_1(B,A),C,D) = k1_tarski(A) ) ) ).
fof(d10_commacat,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> k8_commacat(A,B) = k8_cat_1(B,k10_cat_1(A,B)) ) ) ).
fof(d11_commacat,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> k9_commacat(A,B) = k7_commacat(k8_commacat(A,B),A,A,k8_cat_2(A,k8_commacat(A,B)),k15_cat_1(A)) ) ) ).
fof(d12_commacat,axiom,
! [A] :
( ( v2_cat_1(A)
& l1_cat_1(A) )
=> ! [B] :
( m1_subset_1(B,u1_cat_1(A))
=> k10_commacat(A,B) = k7_commacat(A,k8_commacat(A,B),A,k15_cat_1(A),k8_cat_2(A,k8_commacat(A,B))) ) ) ).
fof(dt_k1_commacat,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,C)
& m2_cat_1(E,B,C) )
=> ( ~ v1_xboole_0(k1_commacat(A,B,C,D,E))
& m1_subset_1(k1_commacat(A,B,C,D,E),k1_zfmisc_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)))) ) ) ).
fof(dt_k2_commacat,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,C)
& m2_cat_1(E,B,C) )
=> ( ~ v1_xboole_0(k2_commacat(A,B,C,D,E))
& m1_subset_1(k2_commacat(A,B,C,D,E),k1_zfmisc_1(k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))))) ) ) ).
fof(dt_k3_commacat,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)
& m2_cat_1(D,A,C)
& m2_cat_1(E,B,C)
& m1_subset_1(F,k2_commacat(A,B,C,D,E)) )
=> m2_subset_1(k3_commacat(A,B,C,D,E,F),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E)) ) ).
fof(redefinition_k3_commacat,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)
& m2_cat_1(D,A,C)
& m2_cat_1(E,B,C)
& m1_subset_1(F,k2_commacat(A,B,C,D,E)) )
=> k3_commacat(A,B,C,D,E,F) = k17_mcart_1(F) ) ).
fof(dt_k4_commacat,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)
& m2_cat_1(D,A,C)
& m2_cat_1(E,B,C)
& m1_subset_1(F,k2_commacat(A,B,C,D,E)) )
=> m2_subset_1(k4_commacat(A,B,C,D,E,F),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E)) ) ).
fof(redefinition_k4_commacat,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)
& m2_cat_1(D,A,C)
& m2_cat_1(E,B,C)
& m1_subset_1(F,k2_commacat(A,B,C,D,E)) )
=> k4_commacat(A,B,C,D,E,F) = k18_mcart_1(F) ) ).
fof(dt_k5_commacat,axiom,
! [A,B,C,D,E,F,G] :
( ( 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,C)
& m2_cat_1(E,B,C)
& m1_subset_1(F,k2_commacat(A,B,C,D,E))
& m1_subset_1(G,k2_commacat(A,B,C,D,E)) )
=> m2_subset_1(k5_commacat(A,B,C,D,E,F,G),k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_commacat(A,B,C,D,E)) ) ).
fof(dt_k6_commacat,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,C)
& m2_cat_1(E,B,C) )
=> ( v1_funct_1(k6_commacat(A,B,C,D,E))
& m2_relset_1(k6_commacat(A,B,C,D,E),k12_mcart_1(k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_commacat(A,B,C,D,E),k2_commacat(A,B,C,D,E)),k2_commacat(A,B,C,D,E)) ) ) ).
fof(dt_k7_commacat,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,C)
& m2_cat_1(E,B,C) )
=> ( v1_cat_1(k7_commacat(A,B,C,D,E))
& v2_cat_1(k7_commacat(A,B,C,D,E))
& l1_cat_1(k7_commacat(A,B,C,D,E)) ) ) ).
fof(dt_k8_commacat,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ( v1_cat_1(k8_commacat(A,B))
& m3_cat_2(k8_commacat(A,B),A) ) ) ).
fof(dt_k9_commacat,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ( v1_cat_1(k9_commacat(A,B))
& v2_cat_1(k9_commacat(A,B))
& l1_cat_1(k9_commacat(A,B)) ) ) ).
fof(dt_k10_commacat,axiom,
! [A,B] :
( ( v2_cat_1(A)
& l1_cat_1(A)
& m1_subset_1(B,u1_cat_1(A)) )
=> ( v1_cat_1(k10_commacat(A,B))
& v2_cat_1(k10_commacat(A,B))
& l1_cat_1(k10_commacat(A,B)) ) ) ).
fof(d5_commacat,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,C)
=> ! [E] :
( m2_cat_1(E,B,C)
=> ( ? [F] :
( m1_subset_1(F,u1_cat_1(A))
& ? [G] :
( m1_subset_1(G,u1_cat_1(B))
& ? [H] :
( m1_subset_1(H,u2_cat_1(C))
& r2_hidden(H,k6_cat_1(C,k13_cat_1(A,C,D,F),k13_cat_1(B,C,E,G))) ) ) )
=> k1_commacat(A,B,C,D,E) = a_5_0_commacat(A,B,C,D,E) ) ) ) ) ) ) ).
fof(d6_commacat,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,C)
=> ! [E] :
( m2_cat_1(E,B,C)
=> ( ? [F] :
( m1_subset_1(F,u1_cat_1(A))
& ? [G] :
( m1_subset_1(G,u1_cat_1(B))
& ? [H] :
( m1_subset_1(H,u2_cat_1(C))
& r2_hidden(H,k6_cat_1(C,k13_cat_1(A,C,D,F),k13_cat_1(B,C,E,G))) ) ) )
=> k2_commacat(A,B,C,D,E) = a_5_1_commacat(A,B,C,D,E) ) ) ) ) ) ) ).
fof(d8_commacat,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,C)
=> ! [E] :
( m2_cat_1(E,B,C)
=> ! [F] :
( ( v1_funct_1(F)
& m2_relset_1(F,k12_mcart_1(k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_commacat(A,B,C,D,E),k2_commacat(A,B,C,D,E)),k2_commacat(A,B,C,D,E)) )
=> ( F = k6_commacat(A,B,C,D,E)
<=> ( k1_relat_1(F) = a_5_2_commacat(A,B,C,D,E)
& ! [G] :
( m2_subset_1(G,k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_commacat(A,B,C,D,E))
=> ! [H] :
( m2_subset_1(H,k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(A),u1_cat_1(B)),u2_cat_1(C)),k1_commacat(A,B,C,D,E),k1_commacat(A,B,C,D,E)),k2_zfmisc_1(u2_cat_1(A),u2_cat_1(B))),k2_commacat(A,B,C,D,E))
=> ( r2_hidden(k1_domain_1(k2_commacat(A,B,C,D,E),k2_commacat(A,B,C,D,E),G,H),k1_relat_1(F))
=> k1_funct_1(F,k1_domain_1(k2_commacat(A,B,C,D,E),k2_commacat(A,B,C,D,E),G,H)) = k5_commacat(A,B,C,D,E,H,G) ) ) ) ) ) ) ) ) ) ) ) ).
fof(fraenkel_a_5_0_commacat,axiom,
! [A,B,C,D,E,F] :
( ( 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,B,D)
& m2_cat_1(F,C,D) )
=> ( r2_hidden(A,a_5_0_commacat(B,C,D,E,F))
<=> ? [G,H,I] :
( m1_subset_1(G,u1_cat_1(B))
& m1_subset_1(H,u1_cat_1(C))
& m1_subset_1(I,u2_cat_1(D))
& A = k1_domain_1(k2_zfmisc_1(u1_cat_1(B),u1_cat_1(C)),u2_cat_1(D),k1_domain_1(u1_cat_1(B),u1_cat_1(C),G,H),I)
& r2_hidden(I,k6_cat_1(D,k13_cat_1(B,D,E,G),k13_cat_1(C,D,F,H))) ) ) ) ).
fof(fraenkel_a_5_1_commacat,axiom,
! [A,B,C,D,E,F] :
( ( 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,B,D)
& m2_cat_1(F,C,D) )
=> ( r2_hidden(A,a_5_1_commacat(B,C,D,E,F))
<=> ? [G,H,I,J] :
( m1_subset_1(G,u2_cat_1(B))
& m1_subset_1(H,u2_cat_1(C))
& m2_subset_1(I,k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(B),u1_cat_1(C)),u2_cat_1(D)),k1_commacat(B,C,D,E,F))
& m2_subset_1(J,k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(B),u1_cat_1(C)),u2_cat_1(D)),k1_commacat(B,C,D,E,F))
& A = k1_domain_1(k2_zfmisc_1(k1_commacat(B,C,D,E,F),k1_commacat(B,C,D,E,F)),k2_zfmisc_1(u2_cat_1(B),u2_cat_1(C)),k1_domain_1(k1_commacat(B,C,D,E,F),k1_commacat(B,C,D,E,F),I,J),k1_domain_1(u2_cat_1(B),u2_cat_1(C),G,H))
& k2_cat_1(B,G) = k14_domain_1(u1_cat_1(B),u1_cat_1(C),u2_cat_1(D),I)
& k3_cat_1(B,G) = k14_domain_1(u1_cat_1(B),u1_cat_1(C),u2_cat_1(D),J)
& k2_cat_1(C,H) = k15_domain_1(u1_cat_1(B),u1_cat_1(C),u2_cat_1(D),I)
& k3_cat_1(C,H) = k15_domain_1(u1_cat_1(B),u1_cat_1(C),u2_cat_1(D),J)
& k4_cat_1(D,k8_funct_2(u2_cat_1(B),u2_cat_1(D),E,G),k3_domain_1(k2_zfmisc_1(u1_cat_1(B),u1_cat_1(C)),u2_cat_1(D),J)) = k4_cat_1(D,k3_domain_1(k2_zfmisc_1(u1_cat_1(B),u1_cat_1(C)),u2_cat_1(D),I),k8_funct_2(u2_cat_1(C),u2_cat_1(D),F,H)) ) ) ) ).
fof(fraenkel_a_5_2_commacat,axiom,
! [A,B,C,D,E,F] :
( ( 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,B,D)
& m2_cat_1(F,C,D) )
=> ( r2_hidden(A,a_5_2_commacat(B,C,D,E,F))
<=> ? [G,H] :
( m2_subset_1(G,k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(B),u1_cat_1(C)),u2_cat_1(D)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(B),u1_cat_1(C)),u2_cat_1(D)),k1_commacat(B,C,D,E,F),k1_commacat(B,C,D,E,F)),k2_zfmisc_1(u2_cat_1(B),u2_cat_1(C))),k2_commacat(B,C,D,E,F))
& m2_subset_1(H,k2_zfmisc_1(k12_mcart_1(k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(B),u1_cat_1(C)),u2_cat_1(D)),k2_zfmisc_1(k2_zfmisc_1(u1_cat_1(B),u1_cat_1(C)),u2_cat_1(D)),k1_commacat(B,C,D,E,F),k1_commacat(B,C,D,E,F)),k2_zfmisc_1(u2_cat_1(B),u2_cat_1(C))),k2_commacat(B,C,D,E,F))
& A = k1_domain_1(k2_commacat(B,C,D,E,F),k2_commacat(B,C,D,E,F),G,H)
& k3_commacat(B,C,D,E,F,G) = k4_commacat(B,C,D,E,F,H) ) ) ) ).
%------------------------------------------------------------------------------