SET007 Axioms: SET007+34.ax


%------------------------------------------------------------------------------
% File     : SET007+34 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Function Domains and Fraenkel Operator
% 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    : fraenkel [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   98 (  17 unit)
%            Number of atoms       :  419 (  71 equality)
%            Maximal formula depth :   15 (   6 average)
%            Number of connectives :  344 (  23 ~  ;   2  |; 163  &)
%                                         (  50 <=>; 106 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   52 (   1 propositional; 0-4 arity)
%            Number of functors    :  139 ( 103 constant; 0-4 arity)
%            Number of variables   :  215 (   0 singleton; 147 !;  68 ?)
%            Maximal term depth    :    4 (   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(rc1_fraenkel,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_fraenkel(A) ) )).

fof(cc1_fraenkel,axiom,(
    ! [A] :
      ( v1_fraenkel(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( v1_relat_1(B)
            & v1_funct_1(B) ) ) ) )).

fof(fc1_fraenkel,axiom,(
    ! [A,B] : v1_fraenkel(k1_funct_2(A,B)) )).

fof(fc2_fraenkel,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => ( v1_finset_1(k1_funct_2(A,B))
        & v1_fraenkel(k1_funct_2(A,B)) ) ) )).

fof(t1_fraenkel,axiom,(
    $true )).

fof(t2_fraenkel,axiom,(
    $true )).

fof(t3_fraenkel,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,A,B)
                & m2_relset_1(C,A,B) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,A,B)
                    & m2_relset_1(D,A,B) )
                 => ! [E] :
                      ( k7_relat_1(C,E) = k7_relat_1(D,E)
                     => ! [F] :
                          ( m1_subset_1(F,A)
                         => ( r2_hidden(F,E)
                           => k8_funct_2(A,B,C,F) = k8_funct_2(A,B,D,F) ) ) ) ) ) ) ) )).

fof(t4_fraenkel,axiom,(
    $true )).

fof(t5_fraenkel,axiom,(
    ! [A,B] : r1_tarski(k1_funct_2(A,B),k1_zfmisc_1(k2_zfmisc_1(A,B))) )).

fof(t6_fraenkel,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C,D] :
          ( ( r1_tarski(C,B)
            & r1_tarski(D,A) )
         => ( k1_funct_2(C,D) = k1_xboole_0
            | ! [E] :
                ( m1_subset_1(E,k1_funct_2(C,D))
               => ( v1_funct_1(E)
                  & m2_relset_1(E,B,A) ) ) ) ) ) )).

fof(d1_fraenkel,axiom,(
    ! [A] :
      ( v1_fraenkel(A)
    <=> ! [B] :
          ( r2_hidden(B,A)
         => ( v1_relat_1(B)
            & v1_funct_1(B) ) ) ) )).

fof(t7_fraenkel,axiom,(
    $true )).

fof(t8_fraenkel,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => v1_fraenkel(k1_tarski(A)) ) )).

fof(d2_fraenkel,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(C)
        & v1_fraenkel(C) )
     => ( m1_fraenkel(C,A,B)
      <=> ! [D] :
            ( m1_subset_1(D,C)
           => ( v1_funct_1(D)
              & v1_funct_2(D,A,B)
              & m2_relset_1(D,A,B) ) ) ) ) )).

fof(t9_fraenkel,axiom,(
    $true )).

fof(t10_fraenkel,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & v1_funct_2(C,A,B)
        & m2_relset_1(C,A,B) )
     => m1_fraenkel(k2_setwiseo(k1_zfmisc_1(k2_zfmisc_1(A,B)),C),A,B) ) )).

fof(t11_fraenkel,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] : m1_fraenkel(k1_funct_2(B,A),B,A) ) )).

fof(t12_fraenkel,axiom,(
    $true )).

fof(t13_fraenkel,axiom,(
    $true )).

fof(t14_fraenkel,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C,D] :
          ( ( r1_tarski(C,B)
            & r1_tarski(D,A) )
         => ( k1_funct_2(C,D) = k1_xboole_0
            | ! [E] :
                ( m1_subset_1(E,k1_funct_2(C,D))
               => ? [F] :
                    ( m2_fraenkel(F,B,A,k1_fraenkel(B,A))
                    & k7_relat_1(F,C) = E ) ) ) ) ) )).

fof(t15_fraenkel,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C,D] :
          ( m2_fraenkel(D,B,A,k1_fraenkel(B,A))
         => k7_relat_1(D,C) = k7_relat_1(D,k3_xboole_0(B,C)) ) ) )).

fof(t16_fraenkel,axiom,(
    ! [A,B] :
      ( ( v1_finset_1(A)
        & v1_finset_1(B) )
     => v1_finset_1(k1_funct_2(A,B)) ) )).

fof(t17_fraenkel,axiom,(
    ! [A] :
      ( v1_fraenkel(A)
     => ! [B] :
          ( r1_tarski(B,A)
         => v1_fraenkel(B) ) ) )).

fof(s23_fraenkel,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,f1_s23_fraenkel)
         => p1_s23_fraenkel(A,A) )
      & ! [A] :
          ( m1_subset_1(A,f1_s23_fraenkel)
         => ! [B] :
              ( m1_subset_1(B,f1_s23_fraenkel)
             => ! [C] :
                  ( m1_subset_1(C,f1_s23_fraenkel)
                 => ( ( p1_s23_fraenkel(A,B)
                      & p1_s23_fraenkel(B,C) )
                   => p1_s23_fraenkel(A,C) ) ) ) ) )
   => ! [A] :
        ( m1_subset_1(A,f1_s23_fraenkel)
       => ~ ( r2_hidden(A,f2_s23_fraenkel)
            & ! [B] :
                ( m1_subset_1(B,f1_s23_fraenkel)
               => ~ ( r2_hidden(B,f2_s23_fraenkel)
                    & p1_s23_fraenkel(B,A)
                    & ! [C] :
                        ( m1_subset_1(C,f1_s23_fraenkel)
                       => ( ( r2_hidden(C,f2_s23_fraenkel)
                            & p1_s23_fraenkel(C,B) )
                         => p1_s23_fraenkel(B,C) ) ) ) ) ) ) )).

fof(s24_fraenkel,axiom,(
    ? [A] :
      ( m1_subset_1(A,k5_finsub_1(f1_s24_fraenkel))
      & ! [B] :
          ( m1_subset_1(B,f1_s24_fraenkel)
         => ( r2_hidden(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,f2_s24_fraenkel)
                & r2_hidden(C,f3_s24_fraenkel)
                & B = f4_s24_fraenkel(C)
                & p1_s24_fraenkel(B,C) ) ) ) ) )).

fof(s26_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s26_fraenkel)
       => ~ ( r2_hidden(A,f3_s26_fraenkel)
            & ! [B] :
                ( m1_subset_1(B,f2_s26_fraenkel)
               => ~ p1_s26_fraenkel(A,B) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v1_funct_2(A,f1_s26_fraenkel,f2_s26_fraenkel)
        & m2_relset_1(A,f1_s26_fraenkel,f2_s26_fraenkel)
        & ! [B] :
            ( m1_subset_1(B,f1_s26_fraenkel)
           => ( r2_hidden(B,f3_s26_fraenkel)
             => p1_s26_fraenkel(B,k8_funct_2(f1_s26_fraenkel,f2_s26_fraenkel,A,B)) ) ) ) )).

fof(s27_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s27_fraenkel)
       => ~ ( r2_hidden(A,f3_s27_fraenkel)
            & ! [B] :
                ( m1_subset_1(B,f2_s27_fraenkel)
               => ~ p1_s27_fraenkel(A,B) ) ) )
   => ? [A] :
        ( m2_fraenkel(A,f1_s27_fraenkel,f2_s27_fraenkel,k1_fraenkel(f1_s27_fraenkel,f2_s27_fraenkel))
        & ! [B] :
            ( m1_subset_1(B,f1_s27_fraenkel)
           => ( r2_hidden(B,f3_s27_fraenkel)
             => p1_s27_fraenkel(B,k8_funct_2(f1_s27_fraenkel,f2_s27_fraenkel,A,B)) ) ) ) )).

fof(dt_m1_fraenkel,axiom,(
    ! [A,B,C] :
      ( m1_fraenkel(C,A,B)
     => ( ~ v1_xboole_0(C)
        & v1_fraenkel(C) ) ) )).

fof(existence_m1_fraenkel,axiom,(
    ! [A,B] :
    ? [C] : m1_fraenkel(C,A,B) )).

fof(dt_m2_fraenkel,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m1_fraenkel(C,A,B) )
     => ! [D] :
          ( m2_fraenkel(D,A,B,C)
         => ( v1_funct_1(D)
            & v1_funct_2(D,A,B)
            & m2_relset_1(D,A,B) ) ) ) )).

fof(existence_m2_fraenkel,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m1_fraenkel(C,A,B) )
     => ? [D] : m2_fraenkel(D,A,B,C) ) )).

fof(redefinition_m2_fraenkel,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & m1_fraenkel(C,A,B) )
     => ! [D] :
          ( m2_fraenkel(D,A,B,C)
        <=> m1_subset_1(D,C) ) ) )).

fof(dt_k1_fraenkel,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => m1_fraenkel(k1_fraenkel(A,B),A,B) ) )).

fof(redefinition_k1_fraenkel,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => k1_fraenkel(A,B) = k1_funct_2(A,B) ) )).

fof(s1_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s1_fraenkel)
       => ( p1_s1_fraenkel(A)
         => p2_s1_fraenkel(A) ) )
   => r1_tarski(a_0_0_fraenkel,a_0_1_fraenkel) )).

fof(s2_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s2_fraenkel)
       => ! [B] :
            ( m1_subset_1(B,f2_s2_fraenkel)
           => ( p1_s2_fraenkel(A,B)
             => p2_s2_fraenkel(A,B) ) ) )
   => r1_tarski(a_0_2_fraenkel,a_0_3_fraenkel) )).

fof(s3_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s3_fraenkel)
       => ( p1_s3_fraenkel(A)
        <=> p2_s3_fraenkel(A) ) )
   => a_0_4_fraenkel = a_0_5_fraenkel )).

fof(s4_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s4_fraenkel)
       => ! [B] :
            ( m1_subset_1(B,f2_s4_fraenkel)
           => ( p1_s4_fraenkel(A,B)
            <=> p2_s4_fraenkel(A,B) ) ) )
   => a_0_6_fraenkel = a_0_7_fraenkel )).

fof(s5_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s5_fraenkel)
       => f2_s5_fraenkel(A) = f3_s5_fraenkel(A) )
   => a_0_8_fraenkel = a_0_9_fraenkel )).

fof(s6_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s6_fraenkel)
       => ( p1_s6_fraenkel(A)
         => f2_s6_fraenkel(A) = f3_s6_fraenkel(A) ) )
   => a_0_10_fraenkel = a_0_11_fraenkel )).

fof(s7_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s7_fraenkel)
       => ! [B] :
            ( m1_subset_1(B,f2_s7_fraenkel)
           => f3_s7_fraenkel(A,B) = f4_s7_fraenkel(A,B) ) )
   => a_0_12_fraenkel = a_0_13_fraenkel )).

fof(s8_fraenkel,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,f1_s8_fraenkel)
         => ! [B] :
              ( m1_subset_1(B,f2_s8_fraenkel)
             => ( p1_s8_fraenkel(A,B)
              <=> p2_s8_fraenkel(A,B) ) ) )
      & ! [A] :
          ( m1_subset_1(A,f1_s8_fraenkel)
         => ! [B] :
              ( m1_subset_1(B,f2_s8_fraenkel)
             => f3_s8_fraenkel(A,B) = f3_s8_fraenkel(B,A) ) ) )
   => a_0_14_fraenkel = a_0_15_fraenkel )).

fof(s9_fraenkel,axiom,
    ( ( k7_relat_1(f4_s9_fraenkel,f3_s9_fraenkel) = k7_relat_1(f5_s9_fraenkel,f3_s9_fraenkel)
      & ! [A] :
          ( m1_subset_1(A,f1_s9_fraenkel)
         => ( r2_hidden(A,f3_s9_fraenkel)
           => ( p1_s9_fraenkel(A)
            <=> p2_s9_fraenkel(A) ) ) ) )
   => a_0_16_fraenkel = a_0_17_fraenkel )).

fof(s10_fraenkel,axiom,(
    r1_tarski(a_0_18_fraenkel,f1_s10_fraenkel) )).

fof(s11_fraenkel,axiom,
    ( ! [A] :
        ( r2_hidden(A,a_0_19_fraenkel)
       => p2_s11_fraenkel(A) )
   => ! [A] :
        ( m1_subset_1(A,f1_s11_fraenkel)
       => ! [B] :
            ( m1_subset_1(B,f2_s11_fraenkel)
           => ( p1_s11_fraenkel(A,B)
             => p2_s11_fraenkel(f3_s11_fraenkel(A,B)) ) ) ) )).

fof(s12_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s12_fraenkel)
       => ! [B] :
            ( m1_subset_1(B,f2_s12_fraenkel)
           => ( p1_s12_fraenkel(A,B)
             => p2_s12_fraenkel(f3_s12_fraenkel(A,B)) ) ) )
   => ! [A] :
        ( r2_hidden(A,a_0_20_fraenkel)
       => p2_s12_fraenkel(A) ) )).

fof(s13_fraenkel,axiom,(
    a_0_21_fraenkel = a_0_23_fraenkel )).

fof(s14_fraenkel,axiom,(
    a_0_24_fraenkel = a_0_26_fraenkel )).

fof(s15_fraenkel,axiom,(
    a_0_27_fraenkel = a_0_29_fraenkel )).

fof(s16_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s16_fraenkel)
       => ! [B] :
            ( m1_subset_1(B,f2_s16_fraenkel)
           => ~ ( p1_s16_fraenkel(A,B)
                & ! [C] :
                    ( m1_subset_1(C,f1_s16_fraenkel)
                   => ~ ( p2_s16_fraenkel(C,B)
                        & f3_s16_fraenkel(A,B) = f3_s16_fraenkel(C,B) ) ) ) ) )
   => r1_tarski(a_0_30_fraenkel,a_0_31_fraenkel) )).

fof(s17_fraenkel,axiom,(
    r1_tarski(a_0_32_fraenkel,f2_s17_fraenkel) )).

fof(s18_fraenkel,axiom,(
    r1_xboole_0(a_0_33_fraenkel,f2_s18_fraenkel) )).

fof(s19_fraenkel,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s19_fraenkel)
       => ! [B] :
            ( m1_subset_1(B,f2_s19_fraenkel)
           => ( p2_s19_fraenkel(A,B)
            <=> ( B = f4_s19_fraenkel
                & p1_s19_fraenkel(A,B) ) ) ) )
   => a_0_34_fraenkel = a_0_35_fraenkel )).

fof(s20_fraenkel,axiom,(
    a_0_36_fraenkel = a_0_37_fraenkel )).

fof(s21_fraenkel,axiom,
    ( v1_finset_1(f2_s21_fraenkel)
   => v1_finset_1(a_0_38_fraenkel) )).

fof(s22_fraenkel,axiom,
    ( ( v1_finset_1(f3_s22_fraenkel)
      & v1_finset_1(f4_s22_fraenkel) )
   => v1_finset_1(a_0_39_fraenkel) )).

fof(s25_fraenkel,axiom,
    ( ( v1_finset_1(f3_s25_fraenkel)
      & v1_finset_1(f4_s25_fraenkel)
      & ! [A] :
          ( m2_fraenkel(A,f1_s25_fraenkel,f2_s25_fraenkel,k1_fraenkel(f1_s25_fraenkel,f2_s25_fraenkel))
         => ! [B] :
              ( m2_fraenkel(B,f1_s25_fraenkel,f2_s25_fraenkel,k1_fraenkel(f1_s25_fraenkel,f2_s25_fraenkel))
             => ( k7_relat_1(A,f3_s25_fraenkel) = k7_relat_1(B,f3_s25_fraenkel)
               => f5_s25_fraenkel(A) = f5_s25_fraenkel(B) ) ) ) )
   => v1_finset_1(a_0_40_fraenkel) )).

fof(fraenkel_a_0_0_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_fraenkel)
          & A = f2_s1_fraenkel(B)
          & p1_s1_fraenkel(B) ) ) )).

fof(fraenkel_a_0_1_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_fraenkel)
          & A = f2_s1_fraenkel(B)
          & p2_s1_fraenkel(B) ) ) )).

fof(fraenkel_a_0_2_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_2_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s2_fraenkel)
          & m1_subset_1(C,f2_s2_fraenkel)
          & A = f3_s2_fraenkel(B,C)
          & p1_s2_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_3_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_3_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s2_fraenkel)
          & m1_subset_1(C,f2_s2_fraenkel)
          & A = f3_s2_fraenkel(B,C)
          & p2_s2_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_4_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_4_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s3_fraenkel)
          & A = f2_s3_fraenkel(B)
          & p1_s3_fraenkel(B) ) ) )).

fof(fraenkel_a_0_5_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_5_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s3_fraenkel)
          & A = f2_s3_fraenkel(B)
          & p2_s3_fraenkel(B) ) ) )).

fof(fraenkel_a_0_6_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_6_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s4_fraenkel)
          & m1_subset_1(C,f2_s4_fraenkel)
          & A = f3_s4_fraenkel(B,C)
          & p1_s4_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_7_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_7_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s4_fraenkel)
          & m1_subset_1(C,f2_s4_fraenkel)
          & A = f3_s4_fraenkel(B,C)
          & p2_s4_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_8_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_8_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s5_fraenkel)
          & A = f2_s5_fraenkel(B)
          & p1_s5_fraenkel(B) ) ) )).

fof(fraenkel_a_0_9_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_9_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s5_fraenkel)
          & A = f3_s5_fraenkel(B)
          & p1_s5_fraenkel(B) ) ) )).

fof(fraenkel_a_0_10_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_10_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s6_fraenkel)
          & A = f2_s6_fraenkel(B)
          & p1_s6_fraenkel(B) ) ) )).

fof(fraenkel_a_0_11_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_11_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s6_fraenkel)
          & A = f3_s6_fraenkel(B)
          & p1_s6_fraenkel(B) ) ) )).

fof(fraenkel_a_0_12_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_12_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s7_fraenkel)
          & m1_subset_1(C,f2_s7_fraenkel)
          & A = f3_s7_fraenkel(B,C)
          & p1_s7_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_13_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_13_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s7_fraenkel)
          & m1_subset_1(C,f2_s7_fraenkel)
          & A = f4_s7_fraenkel(B,C)
          & p1_s7_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_14_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_14_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s8_fraenkel)
          & m1_subset_1(C,f2_s8_fraenkel)
          & A = f3_s8_fraenkel(B,C)
          & p1_s8_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_15_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_15_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s8_fraenkel)
          & m1_subset_1(C,f2_s8_fraenkel)
          & A = f3_s8_fraenkel(C,B)
          & p2_s8_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_16_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_16_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s9_fraenkel)
          & A = k8_funct_2(f1_s9_fraenkel,f2_s9_fraenkel,f4_s9_fraenkel,B)
          & p1_s9_fraenkel(B)
          & r2_hidden(B,f3_s9_fraenkel) ) ) )).

fof(fraenkel_a_0_17_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_17_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s9_fraenkel)
          & A = k8_funct_2(f1_s9_fraenkel,f2_s9_fraenkel,f5_s9_fraenkel,B)
          & p2_s9_fraenkel(B)
          & r2_hidden(B,f3_s9_fraenkel) ) ) )).

fof(fraenkel_a_0_18_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_18_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s10_fraenkel)
          & A = B
          & p1_s10_fraenkel(B) ) ) )).

fof(fraenkel_a_0_19_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_19_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s11_fraenkel)
          & m1_subset_1(C,f2_s11_fraenkel)
          & A = f3_s11_fraenkel(B,C)
          & p1_s11_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_20_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_20_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s12_fraenkel)
          & m1_subset_1(C,f2_s12_fraenkel)
          & A = f3_s12_fraenkel(B,C)
          & p1_s12_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_21_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_21_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f3_s13_fraenkel)
          & A = B
          & r2_hidden(B,a_0_22_fraenkel)
          & p2_s13_fraenkel(B) ) ) )).

fof(fraenkel_a_0_22_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_22_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s13_fraenkel)
          & m1_subset_1(C,f2_s13_fraenkel)
          & A = f4_s13_fraenkel(B,C)
          & p1_s13_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_23_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_23_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s13_fraenkel)
          & m1_subset_1(C,f2_s13_fraenkel)
          & A = f4_s13_fraenkel(B,C)
          & p1_s13_fraenkel(B,C)
          & p2_s13_fraenkel(f4_s13_fraenkel(B,C)) ) ) )).

fof(fraenkel_a_0_24_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_24_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s14_fraenkel)
          & A = f2_s14_fraenkel(B)
          & r2_hidden(B,a_0_25_fraenkel)
          & p1_s14_fraenkel(B) ) ) )).

fof(fraenkel_a_0_25_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_25_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s14_fraenkel)
          & A = B
          & p2_s14_fraenkel(B) ) ) )).

fof(fraenkel_a_0_26_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_26_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s14_fraenkel)
          & A = f2_s14_fraenkel(B)
          & p2_s14_fraenkel(B)
          & p1_s14_fraenkel(B) ) ) )).

fof(fraenkel_a_0_27_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_27_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s15_fraenkel)
          & m1_subset_1(C,f2_s15_fraenkel)
          & A = f3_s15_fraenkel(B,C)
          & r2_hidden(B,a_0_28_fraenkel)
          & p1_s15_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_28_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_28_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s15_fraenkel)
          & A = B
          & p2_s15_fraenkel(B) ) ) )).

fof(fraenkel_a_0_29_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_29_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s15_fraenkel)
          & m1_subset_1(C,f2_s15_fraenkel)
          & A = f3_s15_fraenkel(B,C)
          & p2_s15_fraenkel(B)
          & p1_s15_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_30_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_30_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s16_fraenkel)
          & m1_subset_1(C,f2_s16_fraenkel)
          & A = f3_s16_fraenkel(B,C)
          & p1_s16_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_31_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_31_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s16_fraenkel)
          & m1_subset_1(C,f2_s16_fraenkel)
          & A = f3_s16_fraenkel(B,C)
          & p2_s16_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_32_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_32_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s17_fraenkel)
          & A = f3_s17_fraenkel(B)
          & r2_hidden(f3_s17_fraenkel(B),f2_s17_fraenkel)
          & p1_s17_fraenkel(B) ) ) )).

fof(fraenkel_a_0_33_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_33_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s18_fraenkel)
          & A = f3_s18_fraenkel(B)
          & p1_s18_fraenkel(B)
          & ~ r2_hidden(f3_s18_fraenkel(B),f2_s18_fraenkel) ) ) )).

fof(fraenkel_a_0_34_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_34_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s19_fraenkel)
          & m1_subset_1(C,f2_s19_fraenkel)
          & A = f3_s19_fraenkel(B,C)
          & p2_s19_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_35_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_35_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s19_fraenkel)
          & A = f3_s19_fraenkel(B,f4_s19_fraenkel)
          & p1_s19_fraenkel(B,f4_s19_fraenkel) ) ) )).

fof(fraenkel_a_0_36_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_36_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s20_fraenkel)
          & m1_subset_1(C,f2_s20_fraenkel)
          & A = f3_s20_fraenkel(B,C)
          & C = f4_s20_fraenkel
          & p1_s20_fraenkel(B,C) ) ) )).

fof(fraenkel_a_0_37_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_37_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s20_fraenkel)
          & A = f3_s20_fraenkel(B,f4_s20_fraenkel)
          & p1_s20_fraenkel(B,f4_s20_fraenkel) ) ) )).

fof(fraenkel_a_0_38_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_38_fraenkel)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s21_fraenkel)
          & A = f3_s21_fraenkel(B)
          & r2_hidden(B,f2_s21_fraenkel) ) ) )).

fof(fraenkel_a_0_39_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_39_fraenkel)
    <=> ? [B,C] :
          ( m1_subset_1(B,f1_s22_fraenkel)
          & m1_subset_1(C,f2_s22_fraenkel)
          & A = f5_s22_fraenkel(B,C)
          & r2_hidden(B,f3_s22_fraenkel)
          & r2_hidden(C,f4_s22_fraenkel) ) ) )).

fof(fraenkel_a_0_40_fraenkel,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_40_fraenkel)
    <=> ? [B] :
          ( m2_fraenkel(B,f1_s25_fraenkel,f2_s25_fraenkel,k1_fraenkel(f1_s25_fraenkel,f2_s25_fraenkel))
          & A = f5_s25_fraenkel(B)
          & r1_tarski(k2_funct_2(f1_s25_fraenkel,f2_s25_fraenkel,B,f3_s25_fraenkel),f4_s25_fraenkel) ) ) )).
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