SET007 Axioms: SET007+22.ax


%------------------------------------------------------------------------------
% File     : SET007+22 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Domains and Their Cartesian Products
% 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    : domain_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  140 (  34 unit)
%            Number of atoms       :  731 ( 112 equality)
%            Maximal formula depth :   24 (   9 average)
%            Number of connectives :  813 ( 222 ~  ;   2  |; 295  &)
%                                         (  50 <=>; 244 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   19 (   1 propositional; 0-4 arity)
%            Number of functors    :   95 (  13 constant; 0-9 arity)
%            Number of variables   :  524 (   0 singleton; 453 !;  71 ?)
%            Maximal term depth    :    3 (   1 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(t1_domain_1,axiom,(
    $true )).

fof(t2_domain_1,axiom,(
    $true )).

fof(t3_domain_1,axiom,(
    $true )).

fof(t4_domain_1,axiom,(
    $true )).

fof(t5_domain_1,axiom,(
    $true )).

fof(t6_domain_1,axiom,(
    $true )).

fof(t7_domain_1,axiom,(
    $true )).

fof(t8_domain_1,axiom,(
    $true )).

fof(t9_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ~ ( r2_hidden(A,k2_zfmisc_1(B,C))
              & ! [D] :
                  ( m1_subset_1(D,B)
                 => ! [E] :
                      ( m1_subset_1(E,C)
                     => A != k4_tarski(D,E) ) ) ) ) ) )).

fof(t10_domain_1,axiom,(
    $true )).

fof(t11_domain_1,axiom,(
    $true )).

fof(t12_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,k2_zfmisc_1(A,B))
             => ! [D] :
                  ( m1_subset_1(D,k2_zfmisc_1(A,B))
                 => ( ( k1_mcart_1(C) = k1_mcart_1(D)
                      & k2_mcart_1(C) = k2_mcart_1(D) )
                   => C = D ) ) ) ) ) )).

fof(t13_domain_1,axiom,(
    $true )).

fof(t14_domain_1,axiom,(
    $true )).

fof(t15_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ~ v1_xboole_0(D)
             => ( r2_hidden(A,k3_zfmisc_1(B,C,D))
              <=> ? [E] :
                    ( m1_subset_1(E,B)
                    & ? [F] :
                        ( m1_subset_1(F,C)
                        & ? [G] :
                            ( m1_subset_1(G,D)
                            & A = k3_mcart_1(E,F,G) ) ) ) ) ) ) ) )).

fof(t16_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ( ! [E] :
                        ( r2_hidden(E,A)
                      <=> ? [F] :
                            ( m1_subset_1(F,B)
                            & ? [G] :
                                ( m1_subset_1(G,C)
                                & ? [H] :
                                    ( m1_subset_1(H,D)
                                    & E = k3_mcart_1(F,G,H) ) ) ) )
                   => A = k3_zfmisc_1(B,C,D) ) ) ) ) ) )).

fof(t17_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ( A = k3_zfmisc_1(B,C,D)
                  <=> ! [E] :
                        ( r2_hidden(E,A)
                      <=> ? [F] :
                            ( m1_subset_1(F,B)
                            & ? [G] :
                                ( m1_subset_1(G,C)
                                & ? [H] :
                                    ( m1_subset_1(H,D)
                                    & E = k3_mcart_1(F,G,H) ) ) ) ) ) ) ) ) ) )).

fof(t18_domain_1,axiom,(
    $true )).

fof(t19_domain_1,axiom,(
    $true )).

fof(t20_domain_1,axiom,(
    $true )).

fof(t21_domain_1,axiom,(
    $true )).

fof(t22_domain_1,axiom,(
    $true )).

fof(t23_domain_1,axiom,(
    $true )).

fof(t24_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ~ v1_xboole_0(D)
             => ! [E] :
                  ( m1_subset_1(E,k3_zfmisc_1(B,C,D))
                 => ( A = k5_mcart_1(B,C,D,E)
                  <=> ! [F] :
                        ( m1_subset_1(F,B)
                       => ! [G] :
                            ( m1_subset_1(G,C)
                           => ! [H] :
                                ( m1_subset_1(H,D)
                               => ( E = k4_domain_1(B,C,D,F,G,H)
                                 => A = F ) ) ) ) ) ) ) ) ) )).

fof(t25_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ~ v1_xboole_0(D)
             => ! [E] :
                  ( m1_subset_1(E,k3_zfmisc_1(B,C,D))
                 => ( A = k6_mcart_1(B,C,D,E)
                  <=> ! [F] :
                        ( m1_subset_1(F,B)
                       => ! [G] :
                            ( m1_subset_1(G,C)
                           => ! [H] :
                                ( m1_subset_1(H,D)
                               => ( E = k4_domain_1(B,C,D,F,G,H)
                                 => A = G ) ) ) ) ) ) ) ) ) )).

fof(t26_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ~ v1_xboole_0(D)
             => ! [E] :
                  ( m1_subset_1(E,k3_zfmisc_1(B,C,D))
                 => ( A = k7_mcart_1(B,C,D,E)
                  <=> ! [F] :
                        ( m1_subset_1(F,B)
                       => ! [G] :
                            ( m1_subset_1(G,C)
                           => ! [H] :
                                ( m1_subset_1(H,D)
                               => ( E = k4_domain_1(B,C,D,F,G,H)
                                 => A = H ) ) ) ) ) ) ) ) ) )).

fof(t27_domain_1,axiom,(
    $true )).

fof(t28_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
                 => ! [E] :
                      ( m1_subset_1(E,k3_zfmisc_1(A,B,C))
                     => ( ( k5_mcart_1(A,B,C,D) = k5_mcart_1(A,B,C,E)
                          & k6_mcart_1(A,B,C,D) = k6_mcart_1(A,B,C,E)
                          & k7_mcart_1(A,B,C,D) = k7_mcart_1(A,B,C,E) )
                       => D = E ) ) ) ) ) ) )).

fof(t29_domain_1,axiom,(
    $true )).

fof(t30_domain_1,axiom,(
    $true )).

fof(t31_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ~ v1_xboole_0(D)
             => ! [E] :
                  ( ~ v1_xboole_0(E)
                 => ( r2_hidden(A,k4_zfmisc_1(B,C,D,E))
                  <=> ? [F] :
                        ( m1_subset_1(F,B)
                        & ? [G] :
                            ( m1_subset_1(G,C)
                            & ? [H] :
                                ( m1_subset_1(H,D)
                                & ? [I] :
                                    ( m1_subset_1(I,E)
                                    & A = k4_mcart_1(F,G,H,I) ) ) ) ) ) ) ) ) ) )).

fof(t32_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ! [E] :
                      ( ~ v1_xboole_0(E)
                     => ( ! [F] :
                            ( r2_hidden(F,A)
                          <=> ? [G] :
                                ( m1_subset_1(G,B)
                                & ? [H] :
                                    ( m1_subset_1(H,C)
                                    & ? [I] :
                                        ( m1_subset_1(I,D)
                                        & ? [J] :
                                            ( m1_subset_1(J,E)
                                            & F = k4_mcart_1(G,H,I,J) ) ) ) ) )
                       => A = k4_zfmisc_1(B,C,D,E) ) ) ) ) ) ) )).

fof(t33_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ! [E] :
                      ( ~ v1_xboole_0(E)
                     => ( A = k4_zfmisc_1(B,C,D,E)
                      <=> ! [F] :
                            ( r2_hidden(F,A)
                          <=> ? [G] :
                                ( m1_subset_1(G,B)
                                & ? [H] :
                                    ( m1_subset_1(H,C)
                                    & ? [I] :
                                        ( m1_subset_1(I,D)
                                        & ? [J] :
                                            ( m1_subset_1(J,E)
                                            & F = k4_mcart_1(G,H,I,J) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t34_domain_1,axiom,(
    $true )).

fof(t35_domain_1,axiom,(
    $true )).

fof(t36_domain_1,axiom,(
    $true )).

fof(t37_domain_1,axiom,(
    $true )).

fof(t38_domain_1,axiom,(
    $true )).

fof(t39_domain_1,axiom,(
    $true )).

fof(t40_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ~ v1_xboole_0(D)
             => ! [E] :
                  ( ~ v1_xboole_0(E)
                 => ! [F] :
                      ( m1_subset_1(F,k4_zfmisc_1(B,C,D,E))
                     => ( A = k8_mcart_1(B,C,D,E,F)
                      <=> ! [G] :
                            ( m1_subset_1(G,B)
                           => ! [H] :
                                ( m1_subset_1(H,C)
                               => ! [I] :
                                    ( m1_subset_1(I,D)
                                   => ! [J] :
                                        ( m1_subset_1(J,E)
                                       => ( F = k5_domain_1(B,C,D,E,G,H,I,J)
                                         => A = G ) ) ) ) ) ) ) ) ) ) ) )).

fof(t41_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ~ v1_xboole_0(D)
             => ! [E] :
                  ( ~ v1_xboole_0(E)
                 => ! [F] :
                      ( m1_subset_1(F,k4_zfmisc_1(B,C,D,E))
                     => ( A = k9_mcart_1(B,C,D,E,F)
                      <=> ! [G] :
                            ( m1_subset_1(G,B)
                           => ! [H] :
                                ( m1_subset_1(H,C)
                               => ! [I] :
                                    ( m1_subset_1(I,D)
                                   => ! [J] :
                                        ( m1_subset_1(J,E)
                                       => ( F = k5_domain_1(B,C,D,E,G,H,I,J)
                                         => A = H ) ) ) ) ) ) ) ) ) ) ) )).

fof(t42_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ~ v1_xboole_0(D)
             => ! [E] :
                  ( ~ v1_xboole_0(E)
                 => ! [F] :
                      ( m1_subset_1(F,k4_zfmisc_1(B,C,D,E))
                     => ( A = k10_mcart_1(B,C,D,E,F)
                      <=> ! [G] :
                            ( m1_subset_1(G,B)
                           => ! [H] :
                                ( m1_subset_1(H,C)
                               => ! [I] :
                                    ( m1_subset_1(I,D)
                                   => ! [J] :
                                        ( m1_subset_1(J,E)
                                       => ( F = k5_domain_1(B,C,D,E,G,H,I,J)
                                         => A = I ) ) ) ) ) ) ) ) ) ) ) )).

fof(t43_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ~ v1_xboole_0(C)
         => ! [D] :
              ( ~ v1_xboole_0(D)
             => ! [E] :
                  ( ~ v1_xboole_0(E)
                 => ! [F] :
                      ( m1_subset_1(F,k4_zfmisc_1(B,C,D,E))
                     => ( A = k11_mcart_1(B,C,D,E,F)
                      <=> ! [G] :
                            ( m1_subset_1(G,B)
                           => ! [H] :
                                ( m1_subset_1(H,C)
                               => ! [I] :
                                    ( m1_subset_1(I,D)
                                   => ! [J] :
                                        ( m1_subset_1(J,E)
                                       => ( F = k5_domain_1(B,C,D,E,G,H,I,J)
                                         => A = J ) ) ) ) ) ) ) ) ) ) ) )).

fof(t44_domain_1,axiom,(
    $true )).

fof(t45_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ! [E] :
                      ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
                     => ! [F] :
                          ( m1_subset_1(F,k4_zfmisc_1(A,B,C,D))
                         => ( ( k8_mcart_1(A,B,C,D,E) = k8_mcart_1(A,B,C,D,F)
                              & k9_mcart_1(A,B,C,D,E) = k9_mcart_1(A,B,C,D,F)
                              & k10_mcart_1(A,B,C,D,E) = k10_mcart_1(A,B,C,D,F)
                              & k11_mcart_1(A,B,C,D,E) = k11_mcart_1(A,B,C,D,F) )
                           => E = F ) ) ) ) ) ) ) )).

fof(t46_domain_1,axiom,(
    $true )).

fof(t47_domain_1,axiom,(
    $true )).

fof(dt_k1_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,A)
        & m1_subset_1(D,B) )
     => m1_subset_1(k1_domain_1(A,B,C,D),k2_zfmisc_1(A,B)) ) )).

fof(redefinition_k1_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,A)
        & m1_subset_1(D,B) )
     => k1_domain_1(A,B,C,D) = k4_tarski(C,D) ) )).

fof(dt_k2_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,k2_zfmisc_1(A,B)) )
     => m1_subset_1(k2_domain_1(A,B,C),A) ) )).

fof(redefinition_k2_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,k2_zfmisc_1(A,B)) )
     => k2_domain_1(A,B,C) = k1_mcart_1(C) ) )).

fof(dt_k3_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,k2_zfmisc_1(A,B)) )
     => m1_subset_1(k3_domain_1(A,B,C),B) ) )).

fof(redefinition_k3_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(C,k2_zfmisc_1(A,B)) )
     => k3_domain_1(A,B,C) = k2_mcart_1(C) ) )).

fof(dt_k4_domain_1,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,A)
        & m1_subset_1(E,B)
        & m1_subset_1(F,C) )
     => m1_subset_1(k4_domain_1(A,B,C,D,E,F),k3_zfmisc_1(A,B,C)) ) )).

fof(redefinition_k4_domain_1,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,A)
        & m1_subset_1(E,B)
        & m1_subset_1(F,C) )
     => k4_domain_1(A,B,C,D,E,F) = k3_mcart_1(D,E,F) ) )).

fof(dt_k5_domain_1,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & m1_subset_1(E,A)
        & m1_subset_1(F,B)
        & m1_subset_1(G,C)
        & m1_subset_1(H,D) )
     => m1_subset_1(k5_domain_1(A,B,C,D,E,F,G,H),k4_zfmisc_1(A,B,C,D)) ) )).

fof(redefinition_k5_domain_1,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & m1_subset_1(E,A)
        & m1_subset_1(F,B)
        & m1_subset_1(G,C)
        & m1_subset_1(H,D) )
     => k5_domain_1(A,B,C,D,E,F,G,H) = k4_mcart_1(E,F,G,H) ) )).

fof(dt_k6_domain_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => m1_subset_1(k6_domain_1(A,B),k1_zfmisc_1(A)) ) )).

fof(redefinition_k6_domain_1,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => k6_domain_1(A,B) = k1_tarski(B) ) )).

fof(dt_k7_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => m1_subset_1(k7_domain_1(A,B,C),k1_zfmisc_1(A)) ) )).

fof(commutativity_k7_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k7_domain_1(A,B,C) = k7_domain_1(A,C,B) ) )).

fof(redefinition_k7_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A) )
     => k7_domain_1(A,B,C) = k2_tarski(B,C) ) )).

fof(dt_k8_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A) )
     => m1_subset_1(k8_domain_1(A,B,C,D),k1_zfmisc_1(A)) ) )).

fof(redefinition_k8_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A) )
     => k8_domain_1(A,B,C,D) = k1_enumset1(B,C,D) ) )).

fof(dt_k9_domain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A) )
     => m1_subset_1(k9_domain_1(A,B,C,D,E),k1_zfmisc_1(A)) ) )).

fof(redefinition_k9_domain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A) )
     => k9_domain_1(A,B,C,D,E) = k2_enumset1(B,C,D,E) ) )).

fof(dt_k10_domain_1,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A) )
     => m1_subset_1(k10_domain_1(A,B,C,D,E,F),k1_zfmisc_1(A)) ) )).

fof(redefinition_k10_domain_1,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A) )
     => k10_domain_1(A,B,C,D,E,F) = k3_enumset1(B,C,D,E,F) ) )).

fof(dt_k11_domain_1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A)
        & m1_subset_1(G,A) )
     => m1_subset_1(k11_domain_1(A,B,C,D,E,F,G),k1_zfmisc_1(A)) ) )).

fof(redefinition_k11_domain_1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A)
        & m1_subset_1(G,A) )
     => k11_domain_1(A,B,C,D,E,F,G) = k4_enumset1(B,C,D,E,F,G) ) )).

fof(dt_k12_domain_1,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A)
        & m1_subset_1(G,A)
        & m1_subset_1(H,A) )
     => m1_subset_1(k12_domain_1(A,B,C,D,E,F,G,H),k1_zfmisc_1(A)) ) )).

fof(redefinition_k12_domain_1,axiom,(
    ! [A,B,C,D,E,F,G,H] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A)
        & m1_subset_1(G,A)
        & m1_subset_1(H,A) )
     => k12_domain_1(A,B,C,D,E,F,G,H) = k5_enumset1(B,C,D,E,F,G,H) ) )).

fof(dt_k13_domain_1,axiom,(
    ! [A,B,C,D,E,F,G,H,I] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A)
        & m1_subset_1(G,A)
        & m1_subset_1(H,A)
        & m1_subset_1(I,A) )
     => m1_subset_1(k13_domain_1(A,B,C,D,E,F,G,H,I),k1_zfmisc_1(A)) ) )).

fof(redefinition_k13_domain_1,axiom,(
    ! [A,B,C,D,E,F,G,H,I] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A)
        & m1_subset_1(C,A)
        & m1_subset_1(D,A)
        & m1_subset_1(E,A)
        & m1_subset_1(F,A)
        & m1_subset_1(G,A)
        & m1_subset_1(H,A)
        & m1_subset_1(I,A) )
     => k13_domain_1(A,B,C,D,E,F,G,H,I) = k6_enumset1(B,C,D,E,F,G,H,I) ) )).

fof(dt_k14_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k2_zfmisc_1(k2_zfmisc_1(A,B),C)) )
     => m1_subset_1(k14_domain_1(A,B,C,D),A) ) )).

fof(redefinition_k14_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k2_zfmisc_1(k2_zfmisc_1(A,B),C)) )
     => k14_domain_1(A,B,C,D) = k17_mcart_1(D) ) )).

fof(dt_k15_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k2_zfmisc_1(k2_zfmisc_1(A,B),C)) )
     => m1_subset_1(k15_domain_1(A,B,C,D),B) ) )).

fof(redefinition_k15_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k2_zfmisc_1(k2_zfmisc_1(A,B),C)) )
     => k15_domain_1(A,B,C,D) = k18_mcart_1(D) ) )).

fof(dt_k16_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k2_zfmisc_1(A,k2_zfmisc_1(B,C))) )
     => m1_subset_1(k16_domain_1(A,B,C,D),B) ) )).

fof(redefinition_k16_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k2_zfmisc_1(A,k2_zfmisc_1(B,C))) )
     => k16_domain_1(A,B,C,D) = k19_mcart_1(D) ) )).

fof(dt_k17_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k2_zfmisc_1(A,k2_zfmisc_1(B,C))) )
     => m1_subset_1(k17_domain_1(A,B,C,D),C) ) )).

fof(redefinition_k17_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k2_zfmisc_1(A,k2_zfmisc_1(B,C))) )
     => k17_domain_1(A,B,C,D) = k20_mcart_1(D) ) )).

fof(t48_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => A = a_1_0_domain_1(A) ) )).

fof(t49_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => k2_zfmisc_1(A,B) = a_2_0_domain_1(A,B) ) ) )).

fof(t50_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => k3_zfmisc_1(A,B,C) = a_3_0_domain_1(A,B,C) ) ) ) )).

fof(t51_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => k4_zfmisc_1(A,B,C,D) = a_4_0_domain_1(A,B,C,D) ) ) ) ) )).

fof(t52_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => B = a_2_1_domain_1(A,B) ) ) )).

fof(t53_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(B))
                 => k12_mcart_1(A,B,C,D) = a_4_1_domain_1(A,B,C,D) ) ) ) ) )).

fof(t54_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(B))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(C))
                         => k13_mcart_1(A,B,C,D,E,F) = a_6_0_domain_1(A,B,C,D,E,F) ) ) ) ) ) ) )).

fof(t55_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(B))
                         => ! [G] :
                              ( m1_subset_1(G,k1_zfmisc_1(C))
                             => ! [H] :
                                  ( m1_subset_1(H,k1_zfmisc_1(D))
                                 => k14_mcart_1(A,B,C,D,E,F,G,H) = a_8_0_domain_1(A,B,C,D,E,F,G,H) ) ) ) ) ) ) ) ) )).

fof(t56_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k1_subset_1(A) = a_1_1_domain_1(A) ) )).

fof(t57_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => k3_subset_1(A,B) = a_2_2_domain_1(A,B) ) ) )).

fof(t58_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k5_subset_1(A,B,C) = a_3_1_domain_1(A,B,C) ) ) ) )).

fof(t59_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k4_subset_1(A,B,C) = a_3_2_domain_1(A,B,C) ) ) ) )).

fof(t60_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k6_subset_1(A,B,C) = a_3_3_domain_1(A,B,C) ) ) ) )).

fof(t61_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k7_subset_1(A,B,C) = a_3_4_domain_1(A,B,C) ) ) ) )).

fof(t62_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k7_subset_1(A,B,C) = a_3_5_domain_1(A,B,C) ) ) ) )).

fof(t63_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k7_subset_1(A,B,C) = a_3_6_domain_1(A,B,C) ) ) ) )).

fof(t64_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k7_subset_1(A,B,C) = a_3_7_domain_1(A,B,C) ) ) ) )).

fof(s1_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m1_subset_1(a_1_2_domain_1(A),k1_zfmisc_1(A)) ) )).

fof(s2_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => m1_subset_1(a_2_3_domain_1(A,B),k1_zfmisc_1(k2_zfmisc_1(A,B))) ) ) )).

fof(s3_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => m1_subset_1(a_3_8_domain_1(A,B,C),k1_zfmisc_1(k3_zfmisc_1(A,B,C))) ) ) ) )).

fof(s4_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( ~ v1_xboole_0(D)
                 => m1_subset_1(a_4_2_domain_1(A,B,C,D),k1_zfmisc_1(k4_zfmisc_1(A,B,C,D))) ) ) ) ) )).

fof(s5_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ! [B] :
            ( m1_subset_1(B,A)
           => ( p1_s5_domain_1(B)
             => p2_s5_domain_1(B) ) )
       => r1_tarski(a_1_3_domain_1(A),a_1_4_domain_1(A)) ) ) )).

fof(s6_domain_1,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ! [B] :
            ( m1_subset_1(B,A)
           => ( p1_s6_domain_1(B)
            <=> p2_s6_domain_1(B) ) )
       => a_1_5_domain_1(A) = a_1_6_domain_1(A) ) ) )).

fof(s7_domain_1,axiom,(
    m1_subset_1(a_0_0_domain_1,k1_zfmisc_1(f1_s7_domain_1)) )).

fof(s8_domain_1,axiom,(
    m1_subset_1(a_0_1_domain_1,k1_zfmisc_1(f2_s8_domain_1)) )).

fof(s9_domain_1,axiom,(
    m1_subset_1(a_0_2_domain_1,k1_zfmisc_1(f3_s9_domain_1)) )).

fof(s10_domain_1,axiom,(
    a_0_3_domain_1 = k3_xboole_0(a_0_4_domain_1,a_0_5_domain_1) )).

fof(fraenkel_a_1_0_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_0_domain_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,B)
            & A = C ) ) ) )).

fof(fraenkel_a_2_0_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C) )
     => ( r2_hidden(A,a_2_0_domain_1(B,C))
      <=> ? [D,E] :
            ( m1_subset_1(D,B)
            & m1_subset_1(E,C)
            & A = k1_domain_1(B,C,D,E) ) ) ) )).

fof(fraenkel_a_3_0_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D) )
     => ( r2_hidden(A,a_3_0_domain_1(B,C,D))
      <=> ? [E,F,G] :
            ( m1_subset_1(E,B)
            & m1_subset_1(F,C)
            & m1_subset_1(G,D)
            & A = k4_domain_1(B,C,D,E,F,G) ) ) ) )).

fof(fraenkel_a_4_0_domain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & ~ v1_xboole_0(E) )
     => ( r2_hidden(A,a_4_0_domain_1(B,C,D,E))
      <=> ? [F,G,H,I] :
            ( m1_subset_1(F,B)
            & m1_subset_1(G,C)
            & m1_subset_1(H,D)
            & m1_subset_1(I,E)
            & A = k5_domain_1(B,C,D,E,F,G,H,I) ) ) ) )).

fof(fraenkel_a_2_1_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_2_1_domain_1(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,B)
            & A = D
            & r2_hidden(D,C) ) ) ) )).

fof(fraenkel_a_4_1_domain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k1_zfmisc_1(B))
        & m1_subset_1(E,k1_zfmisc_1(C)) )
     => ( r2_hidden(A,a_4_1_domain_1(B,C,D,E))
      <=> ? [F,G] :
            ( m1_subset_1(F,B)
            & m1_subset_1(G,C)
            & A = k1_domain_1(B,C,F,G)
            & r2_hidden(F,D)
            & r2_hidden(G,E) ) ) ) )).

fof(fraenkel_a_6_0_domain_1,axiom,(
    ! [A,B,C,D,E,F,G] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & m1_subset_1(E,k1_zfmisc_1(B))
        & m1_subset_1(F,k1_zfmisc_1(C))
        & m1_subset_1(G,k1_zfmisc_1(D)) )
     => ( r2_hidden(A,a_6_0_domain_1(B,C,D,E,F,G))
      <=> ? [H,I,J] :
            ( m1_subset_1(H,B)
            & m1_subset_1(I,C)
            & m1_subset_1(J,D)
            & A = k4_domain_1(B,C,D,H,I,J)
            & r2_hidden(H,E)
            & r2_hidden(I,F)
            & r2_hidden(J,G) ) ) ) )).

fof(fraenkel_a_8_0_domain_1,axiom,(
    ! [A,B,C,D,E,F,G,H,I] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & ~ v1_xboole_0(E)
        & m1_subset_1(F,k1_zfmisc_1(B))
        & m1_subset_1(G,k1_zfmisc_1(C))
        & m1_subset_1(H,k1_zfmisc_1(D))
        & m1_subset_1(I,k1_zfmisc_1(E)) )
     => ( r2_hidden(A,a_8_0_domain_1(B,C,D,E,F,G,H,I))
      <=> ? [J,K,L,M] :
            ( m1_subset_1(J,B)
            & m1_subset_1(K,C)
            & m1_subset_1(L,D)
            & m1_subset_1(M,E)
            & A = k5_domain_1(B,C,D,E,J,K,L,M)
            & r2_hidden(J,F)
            & r2_hidden(K,G)
            & r2_hidden(L,H)
            & r2_hidden(M,I) ) ) ) )).

fof(fraenkel_a_1_1_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_1_domain_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,B)
            & A = C
            & ~ $true ) ) ) )).

fof(fraenkel_a_2_2_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_2_2_domain_1(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,B)
            & A = D
            & ~ r2_hidden(D,C) ) ) ) )).

fof(fraenkel_a_3_1_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_1_domain_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & r2_hidden(E,C)
            & r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_2_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_2_domain_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & ( r2_hidden(E,C)
              | r2_hidden(E,D) ) ) ) ) )).

fof(fraenkel_a_3_3_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_3_domain_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & r2_hidden(E,C)
            & ~ r2_hidden(E,D) ) ) ) )).

fof(fraenkel_a_3_4_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_4_domain_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & ( ( r2_hidden(E,C)
                & ~ r2_hidden(E,D) )
              | ( ~ r2_hidden(E,C)
                & r2_hidden(E,D) ) ) ) ) ) )).

fof(fraenkel_a_3_5_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_5_domain_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & ( ~ r2_hidden(E,C)
            <=> r2_hidden(E,D) ) ) ) ) )).

fof(fraenkel_a_3_6_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_6_domain_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & ( r2_hidden(E,C)
            <=> ~ r2_hidden(E,D) ) ) ) ) )).

fof(fraenkel_a_3_7_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_7_domain_1(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & ~ ( r2_hidden(E,C)
              <=> r2_hidden(E,D) ) ) ) ) )).

fof(fraenkel_a_1_2_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_2_domain_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,B)
            & A = C
            & p1_s1_domain_1(C) ) ) ) )).

fof(fraenkel_a_2_3_domain_1,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C) )
     => ( r2_hidden(A,a_2_3_domain_1(B,C))
      <=> ? [D,E] :
            ( m1_subset_1(D,B)
            & m1_subset_1(E,C)
            & A = k1_domain_1(B,C,D,E)
            & p1_s2_domain_1(D,E) ) ) ) )).

fof(fraenkel_a_3_8_domain_1,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D) )
     => ( r2_hidden(A,a_3_8_domain_1(B,C,D))
      <=> ? [E,F,G] :
            ( m1_subset_1(E,B)
            & m1_subset_1(F,C)
            & m1_subset_1(G,D)
            & A = k4_domain_1(B,C,D,E,F,G)
            & p1_s3_domain_1(E,F,G) ) ) ) )).

fof(fraenkel_a_4_2_domain_1,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & ~ v1_xboole_0(D)
        & ~ v1_xboole_0(E) )
     => ( r2_hidden(A,a_4_2_domain_1(B,C,D,E))
      <=> ? [F,G,H,I] :
            ( m1_subset_1(F,B)
            & m1_subset_1(G,C)
            & m1_subset_1(H,D)
            & m1_subset_1(I,E)
            & A = k5_domain_1(B,C,D,E,F,G,H,I)
            & p1_s4_domain_1(F,G,H,I) ) ) ) )).

fof(fraenkel_a_1_3_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_3_domain_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,B)
            & A = C
            & p1_s5_domain_1(C) ) ) ) )).

fof(fraenkel_a_1_4_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_4_domain_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,B)
            & A = C
            & p2_s5_domain_1(C) ) ) ) )).

fof(fraenkel_a_1_5_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_5_domain_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,B)
            & A = C
            & p1_s6_domain_1(C) ) ) ) )).

fof(fraenkel_a_1_6_domain_1,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_6_domain_1(B))
      <=> ? [C] :
            ( m1_subset_1(C,B)
            & A = C
            & p2_s6_domain_1(C) ) ) ) )).

fof(fraenkel_a_0_0_domain_1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_domain_1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s7_domain_1)
          & A = B
          & p1_s7_domain_1(B) ) ) )).

fof(fraenkel_a_0_1_domain_1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_domain_1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s8_domain_1)
          & A = f3_s8_domain_1(B)
          & p1_s8_domain_1(B) ) ) )).

fof(fraenkel_a_0_2_domain_1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_2_domain_1)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s9_domain_1)
          & m1_subset_1(C,f2_s9_domain_1)
          & A = f4_s9_domain_1(B,C)
          & p1_s9_domain_1(B,C) ) ) )).

fof(fraenkel_a_0_3_domain_1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_3_domain_1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s10_domain_1)
          & A = B
          & p1_s10_domain_1(B)
          & p2_s10_domain_1(B) ) ) )).

fof(fraenkel_a_0_4_domain_1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_4_domain_1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s10_domain_1)
          & A = B
          & p1_s10_domain_1(B) ) ) )).

fof(fraenkel_a_0_5_domain_1,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_5_domain_1)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s10_domain_1)
          & A = B
          & p2_s10_domain_1(B) ) ) )).
%------------------------------------------------------------------------------