SET007 Axioms: SET007+883.ax


%------------------------------------------------------------------------------
% File     : SET007+883 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On constructing topological spaces and Sorgenfrey line
% 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    : topgen_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   78 (   9 unit)
%            Number of atoms       :  448 (  79 equality)
%            Maximal formula depth :   22 (   7 average)
%            Number of connectives :  448 (  78 ~  ;   3  |; 195  &)
%                                         (  22 <=>; 150 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   43 (   0 propositional; 1-3 arity)
%            Number of functors    :   67 (  14 constant; 0-4 arity)
%            Number of variables   :  199 (   0 singleton; 188 !;  11 ?)
%            Maximal term depth    :    5 (   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(fc1_topgen_3,axiom,
    ( ~ v1_xboole_0(k4_numbers)
    & ~ v1_finset_1(k4_numbers)
    & v1_membered(k4_numbers)
    & v2_membered(k4_numbers)
    & v3_membered(k4_numbers)
    & v4_membered(k4_numbers) )).

fof(fc2_topgen_3,axiom,
    ( ~ v1_xboole_0(k3_numbers)
    & ~ v1_finset_1(k3_numbers)
    & v1_membered(k3_numbers)
    & v2_membered(k3_numbers)
    & v3_membered(k3_numbers) )).

fof(fc3_topgen_3,axiom,
    ( ~ v1_xboole_0(k1_numbers)
    & ~ v1_finset_1(k1_numbers)
    & v1_membered(k1_numbers)
    & v2_membered(k1_numbers) )).

fof(fc4_topgen_3,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ( ~ v1_xboole_0(k1_zfmisc_1(A))
        & ~ v1_finset_1(k1_zfmisc_1(A)) ) ) )).

fof(fc5_topgen_3,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( ~ v1_xboole_0(k3_card_2(np__2,A))
        & v1_ordinal1(k3_card_2(np__2,A))
        & v2_ordinal1(k3_card_2(np__2,A))
        & v3_ordinal1(k3_card_2(np__2,A))
        & ~ v1_finset_1(k3_card_2(np__2,A))
        & v1_card_1(k3_card_2(np__2,A)) ) ) )).

fof(rc1_topgen_3,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ? [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
          & ~ v1_xboole_0(B)
          & ~ v1_finset_1(B) ) ) )).

fof(fc6_topgen_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k5_topgen_3(A))
        & v1_pre_topc(k5_topgen_3(A))
        & v2_pre_topc(k5_topgen_3(A)) ) ) )).

fof(fc7_topgen_3,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k6_topgen_3(A,B))
        & v1_pre_topc(k6_topgen_3(A,B))
        & v2_pre_topc(k6_topgen_3(A,B)) ) ) )).

fof(fc8_topgen_3,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_tex_2(B,k1_zfmisc_1(A))
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => ~ v1_xboole_0(k3_subset_1(A,B)) ) )).

fof(fc9_topgen_3,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(A)
     => ( ~ v3_struct_0(k7_topgen_3(A,B))
        & v1_pre_topc(k7_topgen_3(A,B))
        & v2_pre_topc(k7_topgen_3(A,B)) ) ) )).

fof(d1_topgen_3,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v1_topgen_3(B,A)
      <=> ! [C,D,E] :
            ~ ( r2_hidden(D,B)
              & r2_hidden(E,B)
              & r2_hidden(C,k3_xboole_0(D,E))
              & ! [F] :
                  ( m1_subset_1(F,k1_zfmisc_1(A))
                 => ~ ( r2_hidden(F,B)
                      & r2_hidden(C,F)
                      & r1_tarski(F,k3_xboole_0(D,E)) ) ) ) ) ) )).

fof(t1_topgen_3,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A)))
      <=> ! [C] :
            ~ ( r2_hidden(C,A)
              & ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ~ ( r2_hidden(D,B)
                      & r2_hidden(C,D) ) ) ) ) ) )).

fof(t2_topgen_3,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( ( v1_topgen_3(B,A)
          & v2_abian(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
       => ! [C] :
            ( l1_pre_topc(C)
           => ( ( u1_struct_0(C) = A
                & u1_pre_topc(C) = k1_cantor_1(A,B) )
             => ( v2_pre_topc(C)
                & l1_pre_topc(C)
                & m1_cantor_1(B,C) ) ) ) ) ) )).

fof(t3_topgen_3,axiom,(
    ! [A,B] :
      ( ( v2_relat_1(B)
        & m1_pboole(B,A) )
     => ~ ( r1_tarski(k2_relat_1(B),k1_pcomps_1(k1_pcomps_1(A)))
          & ! [C,D] :
              ( ( r2_hidden(C,A)
                & r2_hidden(D,k1_funct_1(B,C)) )
             => r2_hidden(C,D) )
          & ! [C,D,E] :
              ~ ( r2_hidden(C,E)
                & r2_hidden(E,k1_funct_1(B,D))
                & r2_hidden(D,A)
                & ! [F] :
                    ~ ( r2_hidden(F,k1_funct_1(B,C))
                      & r1_tarski(F,E) ) )
          & ! [C,D,E] :
              ~ ( r2_hidden(C,A)
                & r2_hidden(D,k1_funct_1(B,C))
                & r2_hidden(E,k1_funct_1(B,C))
                & ! [F] :
                    ~ ( r2_hidden(F,k1_funct_1(B,C))
                      & r1_tarski(F,k3_xboole_0(D,E)) ) )
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ~ ( C = k3_card_3(B)
                  & ! [D] :
                      ( l1_pre_topc(D)
                     => ( ( u1_struct_0(D) = A
                          & u1_pre_topc(D) = k1_cantor_1(A,C) )
                       => ( v2_pre_topc(D)
                          & l1_pre_topc(D)
                          & ! [E] :
                              ( ( ~ v3_struct_0(E)
                                & v2_pre_topc(E)
                                & l1_pre_topc(E) )
                             => ( E = D
                               => m1_topgen_2(B,E) ) ) ) ) ) ) ) ) ) )).

fof(t4_topgen_3,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ( ( r2_hidden(k12_arytm_3,B)
          & r2_hidden(A,B)
          & ! [C,D] :
              ( ( r2_hidden(C,B)
                & r2_hidden(D,B) )
             => r2_hidden(k2_xboole_0(C,D),B) )
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
             => ( r1_tarski(C,B)
               => r2_hidden(k8_setfam_1(A,C),B) ) ) )
       => ! [C] :
            ( l1_pre_topc(C)
           => ( ( u1_struct_0(C) = A
                & u1_pre_topc(C) = k7_setfam_1(A,B) )
             => ( v2_pre_topc(C)
                & l1_pre_topc(C)
                & ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
                   => ( v4_pre_topc(D,C)
                    <=> r2_hidden(D,B) ) ) ) ) ) ) ) )).

fof(t5_topgen_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ( ! [C] :
                ( ( v3_pre_topc(C,A)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
              <=> ( v3_pre_topc(C,B)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) ) )
           => g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) ) ) ) )).

fof(t6_topgen_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ( ! [C] :
                ( ( v4_pre_topc(C,A)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A))) )
              <=> ( v4_pre_topc(C,B)
                  & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B))) ) )
           => g1_pre_topc(u1_struct_0(A),u1_pre_topc(A)) = g1_pre_topc(u1_struct_0(B),u1_pre_topc(B)) ) ) ) )).

fof(t7_topgen_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k1_pcomps_1(A),k1_pcomps_1(A))
        & m2_relset_1(B,k1_pcomps_1(A),k1_pcomps_1(A)) )
     => ( ( k1_funct_1(B,k12_arytm_3) = k12_arytm_3
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => r1_tarski(C,k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C)) )
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,k4_subset_1(A,C,D)) = k4_subset_1(A,k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C),k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,D)) ) )
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k8_funct_2(k1_pcomps_1(A),k1_pcomps_1(A),B,k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C)) = k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C) ) )
       => ! [C] :
            ( l1_pre_topc(C)
           => ( ( u1_struct_0(C) = A
                & u1_pre_topc(C) = k7_setfam_1(A,k1_topgen_3(k1_pcomps_1(A),A,B)) )
             => ( v2_pre_topc(C)
                & l1_pre_topc(C)
                & ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
                   => k6_pre_topc(C,D) = k1_funct_1(B,D) ) ) ) ) ) ) )).

fof(t8_topgen_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ( ( u1_struct_0(A) = u1_struct_0(B)
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                     => ( C = D
                       => k6_pre_topc(A,C) = k6_pre_topc(B,D) ) ) ) )
           => u1_pre_topc(A) = u1_pre_topc(B) ) ) ) )).

fof(t9_topgen_3,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k1_pcomps_1(A),k1_pcomps_1(A))
        & m2_relset_1(B,k1_pcomps_1(A),k1_pcomps_1(A)) )
     => ( ( k1_funct_1(B,A) = A
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => r1_tarski(k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C),C) )
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,k5_subset_1(A,C,D)) = k5_subset_1(A,k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C),k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,D)) ) )
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k8_funct_2(k1_pcomps_1(A),k1_pcomps_1(A),B,k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C)) = k8_funct_2(k1_zfmisc_1(A),k1_pcomps_1(A),B,C) ) )
       => ! [C] :
            ( l1_pre_topc(C)
           => ( ( u1_struct_0(C) = A
                & u1_pre_topc(C) = k1_topgen_3(k1_pcomps_1(A),A,B) )
             => ( v2_pre_topc(C)
                & l1_pre_topc(C)
                & ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
                   => k1_tops_1(C,D) = k1_funct_1(B,D) ) ) ) ) ) ) )).

fof(t10_topgen_3,axiom,(
    ! [A] :
      ( ( v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ( ( u1_struct_0(A) = u1_struct_0(B)
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(B)))
                     => ( C = D
                       => k1_tops_1(A,C) = k1_tops_1(B,D) ) ) ) )
           => u1_pre_topc(A) = u1_pre_topc(B) ) ) ) )).

fof(t11_topgen_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( v3_pre_topc(k1_rcomp_2(A,B),k2_topgen_3)
            & m1_subset_1(k1_rcomp_2(A,B),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ) )).

fof(t12_topgen_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( v3_pre_topc(k2_rcomp_1(A,B),k2_topgen_3)
            & m1_subset_1(k2_rcomp_1(A,B),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) ) )).

fof(t13_topgen_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v3_pre_topc(k12_prob_1(A),k2_topgen_3)
        & m1_subset_1(k12_prob_1(A),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) )).

fof(t14_topgen_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v3_pre_topc(k4_limfunc1(A),k2_topgen_3)
        & m1_subset_1(k4_limfunc1(A),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) )).

fof(t15_topgen_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v3_pre_topc(k3_limfunc1(A),k2_topgen_3)
        & m1_subset_1(k3_limfunc1(A),k1_zfmisc_1(u1_struct_0(k2_topgen_3))) ) ) )).

fof(t16_topgen_3,axiom,(
    k1_card_1(k4_numbers) = k3_card_1(np__0) )).

fof(t17_topgen_3,axiom,(
    k1_card_1(k3_numbers) = k3_card_1(np__0) )).

fof(t18_topgen_3,axiom,(
    ! [A] :
      ( ( v4_taxonom2(A)
        & ! [B] :
            ~ ( r2_hidden(B,A)
              & ! [C] :
                  ( v1_xreal_0(C)
                 => ! [D] :
                      ( v1_xreal_0(D)
                     => ~ ( ~ r1_xreal_0(D,C)
                          & r1_tarski(k2_rcomp_1(C,D),B) ) ) ) ) )
     => v1_card_4(A) ) )).

fof(d3_topgen_3,axiom,(
    ! [A,B] :
      ( v1_xreal_0(B)
     => ( r1_topgen_3(A,B)
      <=> ( r2_hidden(B,A)
          & ? [C] :
              ( v1_xreal_0(C)
              & ~ r1_xreal_0(B,C)
              & r1_xboole_0(k2_rcomp_1(C,B),A) ) ) ) ) )).

fof(d4_topgen_3,axiom,(
    k3_topgen_3 = k1_card_1(k1_numbers) )).

fof(d5_topgen_3,axiom,(
    ! [A,B] :
      ( v1_xreal_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,k1_numbers)
            & m2_relset_1(C,k5_numbers,k1_numbers) )
         => ( C = k4_topgen_3(A,B)
          <=> ! [D] :
                ( v4_ordinal2(D)
               => ( ( r2_hidden(D,A)
                    & k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_newton(B,D) )
                  | ( ~ r2_hidden(D,A)
                    & k2_seq_1(k5_numbers,k1_numbers,C,D) = np__0 ) ) ) ) ) ) )).

fof(t21_topgen_3,axiom,(
    ! [A,B] :
      ( v1_xreal_0(B)
     => ~ ( ~ r1_xreal_0(B,np__0)
          & ~ r1_xreal_0(np__1,B)
          & ~ v1_series_1(k4_topgen_3(A,B)) ) ) )).

fof(t22_topgen_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(np__1,A)
              & k2_series_1(k1_seqm_3(k2_prepower(A),B)) != k7_xcmplx_0(k2_newton(A,B),k6_xcmplx_0(np__1,A)) ) ) ) )).

fof(t23_topgen_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_series_1(k1_seqm_3(k2_prepower(k7_xcmplx_0(np__1,np__2)),k1_nat_1(A,np__1))) = k2_newton(k7_xcmplx_0(np__1,np__2),A) ) )).

fof(t24_topgen_3,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ~ ( ~ r1_xreal_0(A,np__0)
            & ~ r1_xreal_0(np__1,A)
            & ~ r1_xreal_0(k2_series_1(k4_topgen_3(B,A)),k2_series_1(k2_prepower(A))) ) ) )).

fof(t25_topgen_3,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => r1_xreal_0(k2_series_1(k1_seqm_3(k4_topgen_3(A,k7_xcmplx_0(np__1,np__2)),k1_nat_1(B,np__1))),k2_newton(k7_xcmplx_0(np__1,np__2),B)) ) )).

fof(t26_topgen_3,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ r1_xreal_0(k2_series_1(k4_topgen_3(A,k7_xcmplx_0(np__1,np__2))),k2_seq_1(k5_numbers,k1_numbers,k1_series_1(k4_topgen_3(A,k7_xcmplx_0(np__1,np__2))),B)) ) ) )).

fof(t27_topgen_3,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & m1_subset_1(A,k1_zfmisc_1(k5_numbers)) )
     => ! [B] :
          ( ( ~ v1_finset_1(B)
            & m1_subset_1(B,k1_zfmisc_1(k5_numbers)) )
         => ( k2_series_1(k4_topgen_3(A,k7_xcmplx_0(np__1,np__2))) = k2_series_1(k4_topgen_3(B,k7_xcmplx_0(np__1,np__2)))
           => A = B ) ) ) )).

fof(t28_topgen_3,axiom,(
    ! [A] :
      ( v1_card_4(A)
     => v1_card_4(k5_finsub_1(A)) ) )).

fof(t29_topgen_3,axiom,(
    k3_topgen_3 = k3_card_2(np__2,k3_card_1(np__0)) )).

fof(t30_topgen_3,axiom,(
    r2_hidden(k3_card_1(np__0),k3_topgen_3) )).

fof(t32_topgen_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_zfmisc_1(k1_numbers)))
     => ~ ( r2_hidden(k1_card_1(A),k3_topgen_3)
          & ! [B] :
              ( v1_xreal_0(B)
             => ! [C] :
                  ( v1_rat_1(C)
                 => ~ ( ~ r1_xreal_0(C,B)
                      & ~ r2_hidden(k1_rcomp_2(B,C),k1_cantor_1(k1_numbers,A)) ) ) ) ) ) )).

fof(t33_topgen_3,axiom,(
    k2_waybel23(k2_topgen_3) = k3_topgen_3 )).

fof(d6_topgen_3,axiom,(
    ! [A,B] :
      ( ( v1_pre_topc(B)
        & v2_pre_topc(B)
        & l1_pre_topc(B) )
     => ( B = k5_topgen_3(A)
      <=> ( u1_struct_0(B) = A
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(B)))
             => ( v4_pre_topc(C,B)
              <=> ( v1_finset_1(C)
                  | C = A ) ) ) ) ) ) )).

fof(t34_topgen_3,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_topgen_3(A))))
     => ( v3_pre_topc(B,k5_topgen_3(A))
      <=> ( B = k12_arytm_3
          | v1_finset_1(k3_subset_1(u1_struct_0(k5_topgen_3(A)),B)) ) ) ) )).

fof(t35_topgen_3,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ~ ( v1_finset_1(B)
              & v1_finset_1(k3_subset_1(A,B)) ) ) ) )).

fof(t36_topgen_3,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v3_pre_topc(B,k5_topgen_3(A))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k5_topgen_3(A)))) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v3_pre_topc(C,k5_topgen_3(A))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k5_topgen_3(A)))) )
             => ~ r2_subset_1(B,C) ) ) ) )).

fof(d7_topgen_3,axiom,(
    ! [A,B,C] :
      ( ( v1_pre_topc(C)
        & v2_pre_topc(C)
        & l1_pre_topc(C) )
     => ( C = k6_topgen_3(A,B)
      <=> ( u1_struct_0(C) = A
          & ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
             => k6_pre_topc(C,D) = k1_cqc_lang(D,k12_arytm_3,D,k2_xboole_0(D,k3_xboole_0(k1_tarski(B),A))) ) ) ) ) )).

fof(t37_topgen_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B)))) )
             => k6_pre_topc(k6_topgen_3(A,B),C) = k2_xboole_0(C,k6_domain_1(A,B)) ) ) ) )).

fof(t38_topgen_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B)))) )
             => ( v4_pre_topc(C,k6_topgen_3(A,B))
              <=> r2_hidden(B,C) ) ) ) ) )).

fof(t39_topgen_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ! [C] :
              ( ( v1_tex_2(C,k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B))))
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B)))) )
             => ( v3_pre_topc(C,k6_topgen_3(A,B))
              <=> ~ r2_hidden(B,C) ) ) ) ) )).

fof(t40_topgen_3,axiom,(
    ! [A,B,C] :
      ( r2_hidden(B,A)
     => ( ( v4_pre_topc(k1_tarski(C),k6_topgen_3(A,B))
          & m1_subset_1(k1_tarski(C),k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B)))) )
      <=> C = B ) ) )).

fof(t41_topgen_3,axiom,(
    ! [A,B,C] :
      ( r2_xboole_0(k1_tarski(B),A)
     => ( ( v3_pre_topc(k1_tarski(C),k6_topgen_3(A,B))
          & m1_subset_1(k1_tarski(C),k1_zfmisc_1(u1_struct_0(k6_topgen_3(A,B)))) )
      <=> ( r2_hidden(C,A)
          & C != B ) ) ) )).

fof(d8_topgen_3,axiom,(
    ! [A,B,C] :
      ( ( v1_pre_topc(C)
        & v2_pre_topc(C)
        & l1_pre_topc(C) )
     => ( C = k7_topgen_3(A,B)
      <=> ( u1_struct_0(C) = A
          & ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(C)))
             => k1_tops_1(C,D) = k1_cqc_lang(D,A,D,k3_xboole_0(D,B)) ) ) ) ) )).

fof(t42_topgen_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( ( v1_tex_2(C,k1_zfmisc_1(u1_struct_0(k7_topgen_3(A,B))))
            & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k7_topgen_3(A,B)))) )
         => k1_tops_1(k7_topgen_3(A,B),C) = k3_xboole_0(C,B) ) ) )).

fof(t43_topgen_3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( ( v1_tex_2(C,k1_zfmisc_1(u1_struct_0(k7_topgen_3(A,B))))
            & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k7_topgen_3(A,B)))) )
         => ( v3_pre_topc(C,k7_topgen_3(A,B))
          <=> r1_tarski(C,B) ) ) ) )).

fof(t44_topgen_3,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => u1_pre_topc(k7_topgen_3(A,B)) = k2_xboole_0(k1_tarski(A),k1_pcomps_1(B)) ) )).

fof(t45_topgen_3,axiom,(
    ! [A] : k2_tex_1(A) = k7_topgen_3(A,k12_arytm_3) )).

fof(t46_topgen_3,axiom,(
    ! [A] : k2_pcomps_1(A) = k7_topgen_3(A,A) )).

fof(s1_topgen_3,axiom,
    ( ( p1_s1_topgen_3(k12_arytm_3)
      & p1_s1_topgen_3(f1_s1_topgen_3)
      & ! [A,B] :
          ( ( p1_s1_topgen_3(A)
            & p1_s1_topgen_3(B) )
         => p1_s1_topgen_3(k2_xboole_0(A,B)) )
      & ! [A] :
          ( m1_subset_1(A,k1_zfmisc_1(k1_zfmisc_1(f1_s1_topgen_3)))
         => ( ! [B] :
                ( r2_hidden(B,A)
               => p1_s1_topgen_3(B) )
           => p1_s1_topgen_3(k8_setfam_1(f1_s1_topgen_3,A)) ) ) )
   => ? [A] :
        ( v1_pre_topc(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & u1_struct_0(A) = f1_s1_topgen_3
        & ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => ( v4_pre_topc(B,A)
            <=> p1_s1_topgen_3(B) ) ) ) )).

fof(s2_topgen_3,axiom,
    ( ( f2_s2_topgen_3(k12_arytm_3) = k12_arytm_3
      & ! [A] :
          ( m1_subset_1(A,k1_zfmisc_1(f1_s2_topgen_3))
         => ( r1_tarski(A,f2_s2_topgen_3(A))
            & r1_tarski(f2_s2_topgen_3(A),f1_s2_topgen_3) ) )
      & ! [A] :
          ( m1_subset_1(A,k1_zfmisc_1(f1_s2_topgen_3))
         => ! [B] :
              ( m1_subset_1(B,k1_zfmisc_1(f1_s2_topgen_3))
             => f2_s2_topgen_3(k4_subset_1(f1_s2_topgen_3,A,B)) = k2_xboole_0(f2_s2_topgen_3(A),f2_s2_topgen_3(B)) ) )
      & ! [A] :
          ( m1_subset_1(A,k1_zfmisc_1(f1_s2_topgen_3))
         => f2_s2_topgen_3(f2_s2_topgen_3(A)) = f2_s2_topgen_3(A) ) )
   => ? [A] :
        ( v1_pre_topc(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & u1_struct_0(A) = f1_s2_topgen_3
        & ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => k6_pre_topc(A,B) = f2_s2_topgen_3(B) ) ) )).

fof(s3_topgen_3,axiom,
    ( ( f2_s3_topgen_3(f1_s3_topgen_3) = f1_s3_topgen_3
      & ! [A] :
          ( m1_subset_1(A,k1_zfmisc_1(f1_s3_topgen_3))
         => r1_tarski(f2_s3_topgen_3(A),A) )
      & ! [A] :
          ( m1_subset_1(A,k1_zfmisc_1(f1_s3_topgen_3))
         => ! [B] :
              ( m1_subset_1(B,k1_zfmisc_1(f1_s3_topgen_3))
             => f2_s3_topgen_3(k5_subset_1(f1_s3_topgen_3,A,B)) = k3_xboole_0(f2_s3_topgen_3(A),f2_s3_topgen_3(B)) ) )
      & ! [A] :
          ( m1_subset_1(A,k1_zfmisc_1(f1_s3_topgen_3))
         => f2_s3_topgen_3(f2_s3_topgen_3(A)) = f2_s3_topgen_3(A) ) )
   => ? [A] :
        ( v1_pre_topc(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A)
        & u1_struct_0(A) = f1_s3_topgen_3
        & ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
           => k1_tops_1(A,B) = f2_s3_topgen_3(B) ) ) )).

fof(dt_k1_topgen_3,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,k1_pcomps_1(B))))
     => m1_subset_1(k1_topgen_3(A,B,C),k1_zfmisc_1(k1_zfmisc_1(B))) ) )).

fof(redefinition_k1_topgen_3,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(k2_zfmisc_1(A,k1_pcomps_1(B))))
     => k1_topgen_3(A,B,C) = k2_relat_1(C) ) )).

fof(dt_k2_topgen_3,axiom,
    ( ~ v3_struct_0(k2_topgen_3)
    & v1_pre_topc(k2_topgen_3)
    & v2_pre_topc(k2_topgen_3)
    & l1_pre_topc(k2_topgen_3) )).

fof(dt_k3_topgen_3,axiom,
    ( ~ v1_finset_1(k3_topgen_3)
    & v1_card_1(k3_topgen_3) )).

fof(dt_k4_topgen_3,axiom,(
    ! [A,B] :
      ( v1_xreal_0(B)
     => ( v1_funct_1(k4_topgen_3(A,B))
        & v1_funct_2(k4_topgen_3(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k4_topgen_3(A,B),k5_numbers,k1_numbers) ) ) )).

fof(dt_k5_topgen_3,axiom,(
    ! [A] :
      ( v1_pre_topc(k5_topgen_3(A))
      & v2_pre_topc(k5_topgen_3(A))
      & l1_pre_topc(k5_topgen_3(A)) ) )).

fof(dt_k6_topgen_3,axiom,(
    ! [A,B] :
      ( v1_pre_topc(k6_topgen_3(A,B))
      & v2_pre_topc(k6_topgen_3(A,B))
      & l1_pre_topc(k6_topgen_3(A,B)) ) )).

fof(dt_k7_topgen_3,axiom,(
    ! [A,B] :
      ( v1_pre_topc(k7_topgen_3(A,B))
      & v2_pre_topc(k7_topgen_3(A,B))
      & l1_pre_topc(k7_topgen_3(A,B)) ) )).

fof(d2_topgen_3,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v1_pre_topc(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ( A = k2_topgen_3
      <=> ( u1_struct_0(A) = k1_numbers
          & ? [B] :
              ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k1_numbers)))
              & u1_pre_topc(A) = k1_cantor_1(k1_numbers,B)
              & B = a_0_0_topgen_3 ) ) ) ) )).

fof(t19_topgen_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => v1_card_4(a_1_0_topgen_3(A)) ) )).

fof(t20_topgen_3,axiom,(
    k1_card_1(a_0_0_topgen_3) = k3_topgen_3 )).

fof(t31_topgen_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_zfmisc_1(k1_numbers)))
     => ( r2_hidden(k1_card_1(A),k3_topgen_3)
       => r2_hidden(k1_card_1(a_1_1_topgen_3(A)),k3_topgen_3) ) ) )).

fof(fraenkel_a_0_0_topgen_3,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_topgen_3)
    <=> ? [B,C] :
          ( m1_subset_1(B,k1_numbers)
          & m1_subset_1(C,k1_numbers)
          & A = k1_rcomp_2(B,C)
          & ~ r1_xreal_0(C,B)
          & v1_rat_1(C) ) ) )).

fof(fraenkel_a_1_0_topgen_3,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
     => ( r2_hidden(A,a_1_0_topgen_3(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_numbers)
            & A = C
            & r1_topgen_3(B,C) ) ) ) )).

fof(fraenkel_a_1_1_topgen_3,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(k1_numbers)))
     => ( r2_hidden(A,a_1_1_topgen_3(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_numbers)
            & A = C
            & ? [D] :
                ( r2_hidden(D,k1_cantor_1(k1_numbers,B))
                & r1_topgen_3(D,C) ) ) ) ) )).
%------------------------------------------------------------------------------