SET007 Axioms: SET007+502.ax


%------------------------------------------------------------------------------
% File     : SET007+502 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Baire Spaces, Sober Spaces
% 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    : yellow_8 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   60 (   9 unit)
%            Number of atoms       :  333 (  21 equality)
%            Maximal formula depth :   19 (   7 average)
%            Number of connectives :  330 (  57 ~  ;   1  |; 130  &)
%                                         (  15 <=>; 127 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   33 (   1 propositional; 0-3 arity)
%            Number of functors    :   20 (   0 constant; 1-3 arity)
%            Number of variables   :  131 (   0 singleton; 122 !;   9 ?)
%            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(cc1_yellow_8,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_yellow_8(B,A)
           => ~ v1_xboole_0(B) ) ) ) )).

fof(rc1_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
          & ~ v1_xboole_0(B)
          & v2_yellow_8(B,A) ) ) )).

fof(cc2_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v3_compts_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_yellow_8(B,A)
           => v1_realset1(B) ) ) ) )).

fof(cc3_yellow_8,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & v3_compts_1(A) )
       => ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & v3_yellow_8(A) ) ) ) )).

fof(rc2_yellow_8,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v2_pre_topc(A)
      & v3_yellow_8(A) ) )).

fof(cc4_yellow_8,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ( ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & v3_yellow_8(A) )
       => ( ~ v3_struct_0(A)
          & v2_pre_topc(A)
          & v2_t_0topsp(A) ) ) ) )).

fof(fc1_yellow_8,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k1_yellow_8(A))
        & v1_pre_topc(k1_yellow_8(A)) ) ) )).

fof(fc2_yellow_8,axiom,(
    ! [A] :
      ( v1_pre_topc(k1_yellow_8(A))
      & v2_pre_topc(k1_yellow_8(A)) ) )).

fof(fc3_yellow_8,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k1_yellow_8(A))
        & v1_pre_topc(k1_yellow_8(A))
        & v2_pre_topc(k1_yellow_8(A))
        & v1_urysohn1(k1_yellow_8(A)) ) ) )).

fof(fc4_yellow_8,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ( ~ v3_struct_0(k1_yellow_8(A))
        & v1_pre_topc(k1_yellow_8(A))
        & v2_pre_topc(k1_yellow_8(A))
        & v1_urysohn1(k1_yellow_8(A))
        & ~ v3_yellow_8(k1_yellow_8(A)) ) ) )).

fof(rc3_yellow_8,axiom,(
    ? [A] :
      ( l1_pre_topc(A)
      & ~ v3_struct_0(A)
      & v2_pre_topc(A)
      & v1_urysohn1(A)
      & ~ v3_yellow_8(A) ) )).

fof(t1_yellow_8,axiom,(
    ! [A,B,C] :
      ( ( r2_hidden(B,k5_finsub_1(A))
        & r1_tarski(C,B) )
     => r2_hidden(C,k5_finsub_1(A)) ) )).

fof(t2_yellow_8,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( r1_tarski(B,k5_finsub_1(A))
       => r2_hidden(k6_setfam_1(A,B),k5_finsub_1(A)) ) ) )).

fof(d1_yellow_8,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_realset1(A)
      <=> ! [B] :
            ( m1_subset_1(B,A)
           => ! [C] :
                ( m1_subset_1(C,A)
               => B = C ) ) ) ) )).

fof(t3_yellow_8,axiom,(
    $true )).

fof(t4_yellow_8,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => r2_wellord2(B,k7_setfam_1(A,B)) ) )).

fof(t5_yellow_8,axiom,(
    ! [A,B] :
      ( ( r2_wellord2(A,B)
        & v1_card_4(A) )
     => v1_card_4(B) ) )).

fof(t6_yellow_8,axiom,(
    $true )).

fof(t7_yellow_8,axiom,(
    $true )).

fof(t8_yellow_8,axiom,(
    $true )).

fof(t9_yellow_8,axiom,(
    $true )).

fof(t10_yellow_8,axiom,(
    $true )).

fof(t11_yellow_8,axiom,(
    $true )).

fof(t12_yellow_8,axiom,(
    $true )).

fof(t13_yellow_8,axiom,(
    $true )).

fof(t14_yellow_8,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r2_hidden(k3_subset_1(u1_struct_0(A),C),k7_setfam_1(u1_struct_0(A),B))
              <=> r2_hidden(C,B) ) ) ) ) )).

fof(t15_yellow_8,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => k8_setfam_1(u1_struct_0(A),k7_setfam_1(u1_struct_0(A),B)) = k3_subset_1(u1_struct_0(A),k5_setfam_1(u1_struct_0(A),B)) ) ) )).

fof(t16_yellow_8,axiom,(
    ! [A] :
      ( l1_struct_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => k5_setfam_1(u1_struct_0(A),k7_setfam_1(u1_struct_0(A),B)) = k3_subset_1(u1_struct_0(A),k8_setfam_1(u1_struct_0(A),B)) ) ) )).

fof(t17_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( ( r1_tarski(C,B)
                  & v4_pre_topc(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ( ( r1_tarski(C,D)
                          & v4_pre_topc(D,A) )
                       => r1_tarski(B,D) ) ) )
               => B = k6_pre_topc(A,C) ) ) ) ) )).

fof(t19_yellow_8,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_cantor_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( r2_hidden(C,B)
               => v3_pre_topc(C,A) ) ) ) ) )).

fof(d2_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
             => ( m1_yellow_8(C,A,B)
              <=> ( r1_tarski(C,u1_pre_topc(A))
                  & r2_hidden(B,k8_setfam_1(u1_struct_0(A),C))
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ~ ( v3_pre_topc(D,A)
                          & r2_hidden(B,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                             => ~ ( r2_hidden(E,C)
                                  & r1_tarski(E,D) ) ) ) ) ) ) ) ) ) )).

fof(t21_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_yellow_8(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( r2_hidden(D,C)
                   => ( v3_pre_topc(D,A)
                      & r2_hidden(B,D) ) ) ) ) ) ) )).

fof(t22_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_yellow_8(C,A,B)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ~ ( r2_hidden(B,D)
                      & v3_pre_topc(D,A)
                      & ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(A)))
                         => ~ ( r2_hidden(E,C)
                              & r1_tarski(E,D) ) ) ) ) ) ) ) )).

fof(t23_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
         => ( ( r1_tarski(B,u1_pre_topc(A))
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ? [D] :
                      ( m1_yellow_8(D,A,C)
                      & r1_tarski(D,B) ) ) )
           => m1_cantor_1(B,A) ) ) ) )).

fof(d3_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_yellow_8(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( v1_card_4(B)
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                   => ( r2_hidden(C,B)
                     => v1_tops_3(C,A) ) )
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                   => ~ ( C = k8_setfam_1(u1_struct_0(A),B)
                        & v1_tops_1(C,A) ) ) ) ) ) ) )).

fof(t24_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_yellow_8(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ( ( v1_card_4(B)
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                   => ( r2_hidden(C,B)
                     => v3_tops_1(C,A) ) ) )
             => v2_tops_1(k5_setfam_1(u1_struct_0(A),B),A) ) ) ) ) )).

fof(d4_yellow_8,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v2_yellow_8(B,A)
          <=> ( ~ v1_xboole_0(B)
              & v4_pre_topc(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                     => ~ ( v4_pre_topc(C,A)
                          & v4_pre_topc(D,A)
                          & B = k1_finsub_1(k1_zfmisc_1(u1_struct_0(A)),C,D)
                          & C != B
                          & D != B ) ) ) ) ) ) ) )).

fof(d5_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ( r1_yellow_8(A,B,C)
              <=> ( r2_hidden(C,B)
                  & r1_tarski(B,k6_pre_topc(A,k1_struct_0(A,C))) ) ) ) ) ) )).

fof(t25_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v4_pre_topc(B,A)
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( r1_yellow_8(A,B,C)
                 => B = k6_pre_topc(A,k1_struct_0(A,C)) ) ) ) ) ) )).

fof(t26_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => v2_yellow_8(k6_pre_topc(A,k1_struct_0(A,B)),A) ) ) )).

fof(d6_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v3_yellow_8(A)
      <=> ! [B] :
            ( ( v2_yellow_8(B,A)
              & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
           => ? [C] :
                ( m1_subset_1(C,u1_struct_0(A))
                & r1_yellow_8(A,B,C)
                & ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ( r1_yellow_8(A,B,D)
                     => C = D ) ) ) ) ) ) )).

fof(t27_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_yellow_8(A,k6_pre_topc(A,k1_struct_0(A,B)),B) ) ) )).

fof(t28_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => r1_yellow_8(A,k1_struct_0(A,B),B) ) ) )).

fof(t29_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( ( v3_pre_topc(B,A)
                  & v4_pre_topc(C,A) )
               => v4_pre_topc(k2_finsub_1(k1_zfmisc_1(u1_struct_0(A)),C,B),A) ) ) ) ) )).

fof(t30_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v3_compts_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_yellow_8(B,A)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A))) )
         => v1_realset1(B) ) ) )).

fof(t31_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v3_compts_1(A)
        & l1_pre_topc(A) )
     => v3_yellow_8(A) ) )).

fof(t32_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v2_t_0topsp(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,u1_struct_0(A))
               => ( k6_pre_topc(A,k1_struct_0(A,B)) = k6_pre_topc(A,k1_struct_0(A,C))
                 => B = C ) ) ) ) ) )).

fof(t33_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v3_yellow_8(A)
        & l1_pre_topc(A) )
     => v2_t_0topsp(A) ) )).

fof(d7_yellow_8,axiom,(
    ! [A,B] :
      ( ( v1_pre_topc(B)
        & l1_pre_topc(B) )
     => ( B = k1_yellow_8(A)
      <=> ( u1_struct_0(B) = A
          & k7_setfam_1(u1_struct_0(B),u1_pre_topc(B)) = k2_xboole_0(k1_tarski(A),k5_finsub_1(A)) ) ) ) )).

fof(t34_yellow_8,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k1_yellow_8(A))))
         => ( v4_pre_topc(B,k1_yellow_8(A))
          <=> ( B = A
              | v1_finset_1(B) ) ) ) ) )).

fof(t35_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v1_urysohn1(A)
       => ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => k6_pre_topc(A,k1_struct_0(A,B)) = k1_struct_0(A,B) ) ) ) )).

fof(t36_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v4_compts_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,u1_struct_0(A))
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
               => ~ ( r2_hidden(B,k1_tops_1(A,C))
                    & ! [D] :
                        ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                       => ~ ( v4_pre_topc(D,A)
                            & r1_tarski(D,C)
                            & r2_hidden(B,k1_tops_1(A,D)) ) ) ) ) ) ) ) )).

fof(t37_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( v4_compts_1(A)
       => ( v6_waybel_3(A)
        <=> ! [B] :
              ( m1_subset_1(B,u1_struct_0(A))
             => ? [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                  & r2_hidden(B,k1_tops_1(A,C))
                  & v6_compts_1(C,A) ) ) ) ) ) )).

fof(dt_m1_yellow_8,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ! [C] :
          ( m1_yellow_8(C,A,B)
         => m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A)))) ) ) )).

fof(existence_m1_yellow_8,axiom,(
    ! [A,B] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A)
        & m1_subset_1(B,u1_struct_0(A)) )
     => ? [C] : m1_yellow_8(C,A,B) ) )).

fof(dt_k1_yellow_8,axiom,(
    ! [A] :
      ( v1_pre_topc(k1_yellow_8(A))
      & l1_pre_topc(k1_yellow_8(A)) ) )).

fof(t18_yellow_8,axiom,(
    ! [A] :
      ( l1_pre_topc(A)
     => ! [B] :
          ( m1_cantor_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => ( v3_pre_topc(C,A)
               => C = k3_tarski(a_3_0_yellow_8(A,B,C)) ) ) ) ) )).

fof(t20_yellow_8,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_cantor_1(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
             => k1_tops_1(A,C) = k3_tarski(a_3_1_yellow_8(A,B,C)) ) ) ) )).

fof(fraenkel_a_3_0_yellow_8,axiom,(
    ! [A,B,C,D] :
      ( ( l1_pre_topc(B)
        & m1_cantor_1(C,B)
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_0_yellow_8(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
            & A = E
            & r2_hidden(E,C)
            & r1_tarski(E,D) ) ) ) )).

fof(fraenkel_a_3_1_yellow_8,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v3_struct_0(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B)
        & m1_cantor_1(C,B)
        & m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B))) )
     => ( r2_hidden(A,a_3_1_yellow_8(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(B)))
            & A = E
            & r2_hidden(E,C)
            & r1_tarski(E,D) ) ) ) )).
%------------------------------------------------------------------------------