SET007 Axioms: SET007+654.ax


%------------------------------------------------------------------------------
% File     : SET007+654 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Tichonov Theorem
% 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    : yellow17 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   26 (   1 unit)
%            Number of atoms       :  242 (  18 equality)
%            Maximal formula depth :   22 (  12 average)
%            Number of connectives :  259 (  43 ~  ;   4  |;  93  &)
%                                         (   7 <=>; 112 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   1 propositional; 0-3 arity)
%            Number of functors    :   23 (   3 constant; 0-4 arity)
%            Number of variables   :  122 (   0 singleton; 118 !;   4 ?)
%            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(t1_yellow17,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C,D] :
          ( m1_subset_1(D,k1_zfmisc_1(k1_funct_1(A,B)))
         => ( ~ r1_xboole_0(k10_relat_1(k3_pralg_3(A,B),k1_tarski(C)),k10_relat_1(k3_pralg_3(A,B),D))
           => r2_hidden(C,D) ) ) ) )).

fof(t2_yellow17,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C,D] :
              ( ( r2_hidden(D,k1_funct_1(A,C))
                & r2_hidden(B,k4_card_3(A)) )
             => r2_hidden(k2_funct_7(B,C,D),k4_card_3(A)) ) ) ) )).

fof(t3_yellow17,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_hidden(B,k1_relat_1(A))
         => ( k4_card_3(A) = k1_xboole_0
            | k2_relat_1(k3_pralg_3(A,B)) = k1_funct_1(A,B) ) ) ) )).

fof(t4_yellow17,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( r2_hidden(B,k1_relat_1(A))
         => k10_relat_1(k3_pralg_3(A,B),k1_funct_1(A,B)) = k4_card_3(A) ) ) )).

fof(t5_yellow17,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C,D] :
              ( ( r2_hidden(D,k1_funct_1(A,C))
                & r2_hidden(C,k1_relat_1(A))
                & r2_hidden(B,k4_card_3(A)) )
             => r2_hidden(k2_funct_7(B,C,D),k10_relat_1(k3_pralg_3(A,C),k1_tarski(D))) ) ) ) )).

fof(t6_yellow17,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ! [C,D,E,F] :
              ( m1_subset_1(F,k1_zfmisc_1(k1_funct_1(A,D)))
             => ( ( r2_hidden(E,k1_funct_1(A,C))
                  & r2_hidden(C,k1_relat_1(A))
                  & r2_hidden(B,k4_card_3(A)) )
               => ( C = D
                  | ( r2_hidden(B,k10_relat_1(k3_pralg_3(A,D),F))
                  <=> r2_hidden(k2_funct_7(B,C,E),k10_relat_1(k3_pralg_3(A,D),F)) ) ) ) ) ) ) )).

fof(t7_yellow17,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C,D,E] :
          ( m1_subset_1(E,k1_zfmisc_1(k1_funct_1(A,C)))
         => ( ( r2_hidden(D,k1_funct_1(A,B))
              & r2_hidden(B,k1_relat_1(A))
              & r2_hidden(C,k1_relat_1(A)) )
           => ( k4_card_3(A) = k1_xboole_0
              | E = k1_funct_1(A,C)
              | ( r1_tarski(k10_relat_1(k3_pralg_3(A,B),k1_tarski(D)),k10_relat_1(k3_pralg_3(A,C),E))
              <=> ( B = C
                  & r2_hidden(D,E) ) ) ) ) ) ) )).

fof(t8_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k3_waybel18(A,B)))
                 => k1_funct_1(k6_waybel18(A,B,C),D) = k5_waybel18(A,B,D,C) ) ) ) ) )).

fof(t9_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k4_waybel18(A,B,C)))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,C))))
                     => ( ~ r1_xboole_0(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),D)),k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),E))
                       => r2_hidden(D,E) ) ) ) ) ) ) )).

fof(t10_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k2_pre_topc(k4_waybel18(A,B,C))) = k2_pre_topc(k3_waybel18(A,B)) ) ) ) )).

fof(t11_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k4_waybel18(A,B,C)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k3_waybel18(A,B)))
                     => r2_hidden(k2_funct_7(E,C,D),k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),D))) ) ) ) ) ) )).

fof(t12_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k4_waybel18(A,B,C)))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,D))))
                         => ( F != k2_pre_topc(k4_waybel18(A,B,D))
                           => ( r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),E)),k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,D),k6_waybel18(A,B,D),F))
                            <=> ( C = D
                                & r2_hidden(E,F) ) ) ) ) ) ) ) ) ) )).

fof(t13_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k4_waybel18(A,B,C)))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,D))))
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(k3_waybel18(A,B)))
                             => ( C != D
                               => ( r2_hidden(G,k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,D),k6_waybel18(A,B,D),F))
                                <=> r2_hidden(k2_funct_7(G,C,E),k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,D),k6_waybel18(A,B,D),F)) ) ) ) ) ) ) ) ) ) )).

fof(t14_yellow17,axiom,(
    $true )).

fof(t15_yellow17,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_pre_topc(A) )
     => ( v2_compts_1(A)
      <=> ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
           => ~ ( v1_tops_2(B,A)
                & r1_tarski(k2_pre_topc(A),k3_tarski(B))
                & ! [C] :
                    ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(A))))
                   => ~ ( r1_tarski(C,B)
                        & r1_tarski(k2_pre_topc(A),k3_tarski(C))
                        & v1_finset_1(C) ) ) ) ) ) ) )).

fof(t16_yellow17,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m2_cantor_1(B,A)
         => ( v2_compts_1(A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(B))
               => ~ ( r1_tarski(k2_pre_topc(A),k3_tarski(C))
                    & ! [D] :
                        ( ( v1_finset_1(D)
                          & m1_subset_1(D,k1_zfmisc_1(C)) )
                       => ~ r1_tarski(k2_pre_topc(A),k3_tarski(D)) ) ) ) ) ) ) )).

fof(t17_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ~ ( r2_hidden(C,k2_waybel18(A,B))
                & ! [D] :
                    ( m1_subset_1(D,A)
                   => ! [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,D))))
                       => ~ ( v3_pre_topc(E,k4_waybel18(A,B,D))
                            & k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,D),k6_waybel18(A,B,D),E) = C ) ) ) ) ) ) )).

fof(t18_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k4_waybel18(A,B,C)))
                 => ! [E] :
                      ~ ( r2_hidden(E,k2_waybel18(A,B))
                        & r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),D)),E)
                        & E != k2_pre_topc(k3_waybel18(A,B))
                        & ! [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,C))))
                           => ~ ( F != k2_pre_topc(k4_waybel18(A,B,C))
                                & r2_hidden(D,F)
                                & v3_pre_topc(F,k4_waybel18(A,B,C))
                                & E = k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),F) ) ) ) ) ) ) ) )).

fof(t20_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k4_waybel18(A,B,C)))
                 => ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(k2_waybel18(A,B)))
                     => ( ( r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),D)),k3_tarski(E))
                          & ! [F] :
                              ~ ( r2_hidden(F,k2_waybel18(A,B))
                                & r2_hidden(F,E)
                                & r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),D)),F) ) )
                       => r1_tarski(k2_pre_topc(k3_waybel18(A,B)),k3_tarski(E)) ) ) ) ) ) ) )).

fof(t21_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k2_waybel18(A,B)))
                 => ( ! [E] :
                        ( ( v1_finset_1(E)
                          & m1_subset_1(E,k1_zfmisc_1(D)) )
                       => ~ r1_tarski(k2_pre_topc(k3_waybel18(A,B)),k3_tarski(E)) )
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(k4_waybel18(A,B,C)))
                       => ! [F] :
                            ( ( v1_finset_1(F)
                              & m1_subset_1(F,k1_zfmisc_1(D)) )
                           => ~ ( r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),E)),k3_tarski(F))
                                & ! [G] :
                                    ~ ( r2_hidden(G,k2_waybel18(A,B))
                                      & r2_hidden(G,F)
                                      & r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),E)),G) ) ) ) ) ) ) ) ) ) )).

fof(t22_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k2_waybel18(A,B)))
                 => ( ! [E] :
                        ( ( v1_finset_1(E)
                          & m1_subset_1(E,k1_zfmisc_1(D)) )
                       => ~ r1_tarski(k2_pre_topc(k3_waybel18(A,B)),k3_tarski(E)) )
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(k4_waybel18(A,B,C)))
                       => ! [F] :
                            ( ( v1_finset_1(F)
                              & m1_subset_1(F,k1_zfmisc_1(D)) )
                           => ~ ( r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),E)),k3_tarski(F))
                                & ! [G] :
                                    ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,C))))
                                   => ~ ( G != k2_pre_topc(k4_waybel18(A,B,C))
                                        & r2_hidden(E,G)
                                        & r2_hidden(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),G),F)
                                        & v3_pre_topc(G,k4_waybel18(A,B,C)) ) ) ) ) ) ) ) ) ) ) )).

fof(t23_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(k2_waybel18(A,B)))
                 => ~ ( ! [E] :
                          ( m1_subset_1(E,A)
                         => v2_compts_1(k4_waybel18(A,B,E)) )
                      & ! [E] :
                          ( ( v1_finset_1(E)
                            & m1_subset_1(E,k1_zfmisc_1(D)) )
                         => ~ r1_tarski(k2_pre_topc(k3_waybel18(A,B)),k3_tarski(E)) )
                      & ! [E] :
                          ( m1_subset_1(E,u1_struct_0(k4_waybel18(A,B,C)))
                         => ? [F] :
                              ( v1_finset_1(F)
                              & m1_subset_1(F,k1_zfmisc_1(D))
                              & r1_tarski(k5_pre_topc(k3_waybel18(A,B),k4_waybel18(A,B,C),k6_waybel18(A,B,C),k1_struct_0(k4_waybel18(A,B,C),E)),k3_tarski(F)) ) ) ) ) ) ) ) )).

fof(t24_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => v2_compts_1(k4_waybel18(A,B,C)) )
           => v2_compts_1(k3_waybel18(A,B)) ) ) ) )).

fof(s1_yellow17,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s1_yellow17)
       => ? [B] :
            ( m1_subset_1(B,u1_struct_0(k4_waybel18(f1_s1_yellow17,f2_s1_yellow17,A)))
            & p1_s1_yellow17(B,A) ) )
   => ? [A] :
        ( m1_subset_1(A,u1_struct_0(k3_waybel18(f1_s1_yellow17,f2_s1_yellow17)))
        & ! [B] :
            ( m1_subset_1(B,f1_s1_yellow17)
           => p1_s1_yellow17(k5_waybel18(f1_s1_yellow17,f2_s1_yellow17,A,B),B) ) ) )).

fof(t19_yellow17,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v4_waybel_3(B)
            & v1_waybel18(B)
            & m1_pboole(B,A) )
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( ( ~ v1_xboole_0(D)
                    & m1_subset_1(D,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_waybel18(A,B,C))))) )
                 => ( r1_tarski(k2_pre_topc(k4_waybel18(A,B,C)),k3_tarski(D))
                   => r1_tarski(k2_pre_topc(k3_waybel18(A,B)),k3_tarski(a_4_0_yellow17(A,B,C,D))) ) ) ) ) ) )).

fof(fraenkel_a_4_0_yellow17,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & v4_waybel_3(C)
        & v1_waybel18(C)
        & m1_pboole(C,B)
        & m1_subset_1(D,B)
        & ~ v1_xboole_0(E)
        & m1_subset_1(E,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k4_waybel18(B,C,D))))) )
     => ( r2_hidden(A,a_4_0_yellow17(B,C,D,E))
      <=> ? [F] :
            ( m2_subset_1(F,k1_zfmisc_1(u1_struct_0(k4_waybel18(B,C,D))),E)
            & A = k5_pre_topc(k3_waybel18(B,C),k4_waybel18(B,C,D),k6_waybel18(B,C,D),F) ) ) ) )).
%------------------------------------------------------------------------------