SET007 Axioms: SET007+330.ax


%------------------------------------------------------------------------------
% File     : SET007+330 : TPTP v7.5.0. 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 unit)
%            Number of atoms       :  296 (  40 equality)
%            Maximal formula depth :   24 (  12 average)
%            Number of connectives :  267 (   3 ~  ;   1  |; 179  &)
%                                         (   5 <=>;  79 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   13 (   1 propositional; 0-3 arity)
%            Number of functors    :   43 (   0 constant; 1-7 arity)
%            Number of variables   :  160 (   0 singleton; 136 !;  24 ?)
%            Maximal term depth    :    9 (   2 average)
% 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) ) ) ) )).
%------------------------------------------------------------------------------