SET007 Axioms: SET007+852.ax


%------------------------------------------------------------------------------
% File     : SET007+852 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Continuous Mappings between Topological Spaces
% Version  : [Urb08] axioms.
% English  : Continuous Mappings between Finite and One-Dimensional Finite
%            Topological Spaces

% 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    : fintopo4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   31 (   0 unit)
%            Number of atoms       :  239 (  23 equality)
%            Maximal formula depth :   16 (   9 average)
%            Number of connectives :  250 (  42 ~  ;   2  |;  80  &)
%                                         (   9 <=>; 117 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   0 propositional; 1-4 arity)
%            Number of functors    :   32 (   5 constant; 0-4 arity)
%            Number of variables   :   88 (   0 singleton;  88 !;   0 ?)
%            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(d1_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(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_fintopo4(A,B,C)
              <=> ( r1_xboole_0(k8_fin_topo(A,B),C)
                  & r1_xboole_0(B,k8_fin_topo(A,C)) ) ) ) ) ) )).

fof(t1_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(C,D)
                   => r1_tarski(k7_fintopo3(A,B,C),k7_fintopo3(A,B,D)) ) ) ) ) ) )).

fof(t2_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(C,D)
                   => r1_tarski(k3_fintopo3(A,B,C),k3_fintopo3(A,B,D)) ) ) ) ) ) )).

fof(t3_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(C,D)
                   => r1_tarski(k9_fintopo3(A,B,D),k9_fintopo3(A,B,C)) ) ) ) ) ) )).

fof(t4_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r1_xreal_0(C,D)
                   => r1_tarski(k5_fintopo3(A,B,D),k5_fintopo3(A,B,C)) ) ) ) ) ) )).

fof(t5_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(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_fintopo4(A,B,C)
               => r1_fintopo4(A,C,B) ) ) ) ) )).

fof(t6_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(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_fintopo4(A,B,C)
               => r1_xboole_0(B,C) ) ) ) ) )).

fof(t7_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(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_fin_topo(A)
               => ( r1_fintopo4(A,B,C)
                <=> ( r1_xboole_0(k11_fin_topo(A,B),C)
                    & r1_xboole_0(B,k11_fin_topo(A,C)) ) ) ) ) ) ) )).

fof(t8_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(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_fin_topo(A)
                  & r1_xboole_0(k8_fin_topo(A,B),C) )
               => r1_xboole_0(B,k8_fin_topo(A,C)) ) ) ) ) )).

fof(t9_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(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_fin_topo(A)
                  & r1_xboole_0(B,k8_fin_topo(A,C)) )
               => r1_xboole_0(k8_fin_topo(A,B),C) ) ) ) ) )).

fof(t10_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(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_fin_topo(A)
               => ( r1_fintopo4(A,B,C)
                <=> r1_xboole_0(k8_fin_topo(A,B),C) ) ) ) ) ) )).

fof(t11_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(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_fin_topo(A)
               => ( r1_fintopo4(A,B,C)
                <=> r1_xboole_0(B,k8_fin_topo(A,C)) ) ) ) ) ) )).

fof(t12_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v3_fin_topo(A)
           => ( v6_fin_topo(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)))
                     => ~ ( B = k4_subset_1(u1_struct_0(A),C,D)
                          & r1_fintopo4(A,C,D)
                          & C != B
                          & D != B ) ) ) ) ) ) ) )).

fof(t13_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_fin_topo(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v3_fin_topo(A)
           => ( v6_fin_topo(B,A)
            <=> ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(u1_struct_0(A)))
                 => ~ ( C != k1_xboole_0
                      & k6_subset_1(u1_struct_0(A),B,C) != k1_xboole_0
                      & r1_tarski(C,B)
                      & r1_xboole_0(k8_fin_topo(A,C),k6_subset_1(u1_struct_0(A),B,C)) ) ) ) ) ) ) )).

fof(d2_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_fin_topo(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( r2_fintopo4(A,B,C,D)
                  <=> ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(B))
                           => ( ( r2_hidden(E,u1_struct_0(A))
                                & F = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) )
                             => r1_tarski(k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,k10_fintopo3(A,np__0,E)),k10_fintopo3(B,D,F)) ) ) ) ) ) ) ) ) )).

fof(t14_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_fin_topo(B)
            & l1_fin_topo(B) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(A),u1_struct_0(B))
                    & m2_relset_1(D,u1_struct_0(A),u1_struct_0(B)) )
                 => ( r2_fintopo4(A,B,D,np__0)
                   => r2_fintopo4(A,B,D,C) ) ) ) ) ) )).

fof(t15_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_fin_topo(B)
            & l1_fin_topo(B) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
                        & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
                     => ( ( r2_fintopo4(A,B,E,C)
                          & r1_xreal_0(C,D) )
                       => r2_fintopo4(A,B,E,D) ) ) ) ) ) ) )).

fof(t16_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_fin_topo(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)))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
                        & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
                     => ( ( r2_fintopo4(A,B,E,np__0)
                          & D = k2_funct_2(u1_struct_0(A),u1_struct_0(B),E,C) )
                       => r1_tarski(k2_funct_2(u1_struct_0(A),u1_struct_0(B),E,k8_fin_topo(A,C)),k8_fin_topo(B,D)) ) ) ) ) ) ) )).

fof(t17_fintopo4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_fin_topo(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)))
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(A),u1_struct_0(B))
                        & m2_relset_1(E,u1_struct_0(A),u1_struct_0(B)) )
                     => ( ( v6_fin_topo(C,A)
                          & r2_fintopo4(A,B,E,np__0)
                          & D = k2_funct_2(u1_struct_0(A),u1_struct_0(B),E,C) )
                       => v6_fin_topo(D,B) ) ) ) ) ) ) )).

fof(d3_fintopo4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A)))
            & m2_relset_1(B,k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A))) )
         => ( B = k1_fintopo4(A)
          <=> ( k1_relat_1(B) = k2_finseq_1(A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k2_finseq_1(A))
                   => k1_funct_1(B,C) = k1_enumset1(C,k1_limfunc1(k5_binarith(C,np__1),np__1),k1_rfunct_3(k1_nat_1(C,np__1),A)) ) ) ) ) ) ) )).

fof(d4_fintopo4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => k2_fintopo4(A) = g1_fin_topo(k2_finseq_1(A),k1_fintopo4(A)) ) ) )).

fof(t18_fintopo4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => v2_fin_topo(k2_fintopo4(A)) ) ) )).

fof(t19_fintopo4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => v3_fin_topo(k2_fintopo4(A)) ) ) )).

fof(d5_fintopo4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A)))
            & m2_relset_1(B,k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A))) )
         => ( B = k3_fintopo4(A)
          <=> ( k1_relat_1(B) = k2_finseq_1(A)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r2_hidden(C,k2_finseq_1(A))
                   => ( ~ ( ~ r1_xreal_0(C,np__1)
                          & ~ r1_xreal_0(A,C)
                          & k1_funct_1(B,C) != k1_enumset1(C,k5_binarith(C,np__1),k1_nat_1(C,np__1)) )
                      & ( C = np__1
                       => ( r1_xreal_0(A,C)
                          | k1_funct_1(B,C) = k1_enumset1(C,A,k1_nat_1(C,np__1)) ) )
                      & ( C = A
                       => ( r1_xreal_0(C,np__1)
                          | k1_funct_1(B,C) = k1_enumset1(C,k5_binarith(C,np__1),np__1) ) )
                      & ( ( C = np__1
                          & C = A )
                       => k1_funct_1(B,C) = k1_tarski(C) ) ) ) ) ) ) ) ) )).

fof(d6_fintopo4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => k4_fintopo4(A) = g1_fin_topo(k2_finseq_1(A),k3_fintopo4(A)) ) ) )).

fof(t20_fintopo4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => v2_fin_topo(k4_fintopo4(A)) ) ) )).

fof(t21_fintopo4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => v3_fin_topo(k4_fintopo4(A)) ) ) )).

fof(dt_k1_fintopo4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_funct_1(k1_fintopo4(A))
        & v1_funct_2(k1_fintopo4(A),k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A)))
        & m2_relset_1(k1_fintopo4(A),k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A))) ) ) )).

fof(dt_k2_fintopo4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v3_struct_0(k2_fintopo4(A))
        & l1_fin_topo(k2_fintopo4(A)) ) ) )).

fof(dt_k3_fintopo4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_funct_1(k3_fintopo4(A))
        & v1_funct_2(k3_fintopo4(A),k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A)))
        & m2_relset_1(k3_fintopo4(A),k2_finseq_1(A),k1_zfmisc_1(k2_finseq_1(A))) ) ) )).

fof(dt_k4_fintopo4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v3_struct_0(k4_fintopo4(A))
        & l1_fin_topo(k4_fintopo4(A)) ) ) )).
%------------------------------------------------------------------------------