SET007 Axioms: SET007+909.ax


%------------------------------------------------------------------------------
% File     : SET007+909 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Homeomorphism between Finite Topological Spaces
% Version  : [Urb08] axioms.
% English  : Homeomorphism between Finite Topological Spaces Two-Dimensional
%            Lattice Spaces and a Fixed Point Theorem

% 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    : fintopo5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   27 (   0 unit)
%            Number of atoms       :  196 (  17 equality)
%            Maximal formula depth :   17 (   9 average)
%            Number of connectives :  207 (  38 ~  ;   0  |;  78  &)
%                                         (   6 <=>;  85 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   19 (   0 propositional; 1-4 arity)
%            Number of functors    :   32 (   6 constant; 0-4 arity)
%            Number of variables   :   86 (   0 singleton;  84 !;   2 ?)
%            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_fintopo5,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k5_numbers) )
     => ( ~ v3_struct_0(k2_fintopo5(A,B))
        & v1_fin_topo(k2_fintopo5(A,B)) ) ) )).

fof(fc2_fintopo5,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k5_numbers) )
     => ( ~ v3_struct_0(k4_fintopo5(A,B))
        & v1_fin_topo(k4_fintopo5(A,B)) ) ) )).

fof(t1_fintopo5,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,A,B)
            & m2_relset_1(C,A,B) )
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( v2_funct_1(C)
               => k9_relat_1(k2_funct_1(C),k2_funct_2(A,B,C,D)) = D ) ) ) ) )).

fof(t2_fintopo5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ ( ~ r1_xreal_0(A,np__0)
            & k2_finseq_1(A) = k1_xboole_0 )
        & ~ ( k2_finseq_1(A) != k1_xboole_0
            & r1_xreal_0(A,np__0) ) ) ) )).

fof(d1_fintopo5,axiom,(
    ! [A] :
      ( l1_fin_topo(A)
     => ! [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)) )
             => ( r1_fintopo5(A,B,C)
              <=> ( v2_funct_1(C)
                  & v2_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                  & ! [D] :
                      ( m1_subset_1(D,u1_struct_0(A))
                     => k2_funct_2(u1_struct_0(A),u1_struct_0(B),C,k1_funct_1(u1_fin_topo(A),D)) = k1_funct_1(u1_fin_topo(B),k1_funct_1(C,D)) ) ) ) ) ) ) )).

fof(t3_fintopo5,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)) )
             => ~ ( r1_fintopo5(A,B,C)
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,u1_struct_0(B),u1_struct_0(A))
                        & m2_relset_1(D,u1_struct_0(B),u1_struct_0(A)) )
                     => ~ ( D = k2_funct_1(C)
                          & r1_fintopo5(B,A,D) ) ) ) ) ) ) )).

fof(t4_fintopo5,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)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(B))
                         => ( ( r1_fintopo5(A,B,C)
                              & F = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) )
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(A))
                               => ( r2_hidden(G,k10_fintopo3(A,D,E))
                                <=> r2_hidden(k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,G),k10_fintopo3(B,D,F)) ) ) ) ) ) ) ) ) ) )).

fof(t5_fintopo5,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)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(A))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(B))
                         => ( ( r1_fintopo5(A,B,C)
                              & F = k8_funct_2(u1_struct_0(A),u1_struct_0(B),C,E) )
                           => ! [G] :
                                ( m1_subset_1(G,u1_struct_0(B))
                               => ( r2_hidden(k1_funct_1(k2_funct_1(C),G),k10_fintopo3(A,D,E))
                                <=> r2_hidden(G,k10_fintopo3(B,D,F)) ) ) ) ) ) ) ) ) ) )).

fof(t6_fintopo5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k2_fintopo4(A)),u1_struct_0(k2_fintopo4(A)))
            & m2_relset_1(B,u1_struct_0(k2_fintopo4(A)),u1_struct_0(k2_fintopo4(A))) )
         => ~ ( r2_fintopo4(k2_fintopo4(A),k2_fintopo4(A),B,np__0)
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(k2_fintopo4(A)))
                 => ~ r2_hidden(k8_funct_2(u1_struct_0(k2_fintopo4(A)),u1_struct_0(k2_fintopo4(A)),B,C),k10_fintopo3(k2_fintopo4(A),np__0,C)) ) ) ) ) )).

fof(t7_fintopo5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( v2_fin_topo(A)
               => r1_tarski(k10_fintopo3(A,C,B),k10_fintopo3(A,k1_nat_1(C,np__1),B)) ) ) ) ) )).

fof(t8_fintopo5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_fin_topo(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( v2_fin_topo(A)
               => r1_tarski(k10_fintopo3(A,np__0,B),k10_fintopo3(A,C,B)) ) ) ) ) )).

fof(t9_fintopo5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k2_fintopo4(A)))
                     => ( E = B
                       => ( r2_hidden(C,k10_fintopo3(k2_fintopo4(A),D,E))
                        <=> ( r2_hidden(C,k2_finseq_1(A))
                            & r1_xreal_0(k18_complex1(k6_xcmplx_0(B,C)),k1_nat_1(D,np__1)) ) ) ) ) ) ) ) ) )).

fof(t10_fintopo5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k2_fintopo4(C)),u1_struct_0(k2_fintopo4(C)))
                    & m2_relset_1(D,u1_struct_0(k2_fintopo4(C)),u1_struct_0(k2_fintopo4(C))) )
                 => ~ ( r2_fintopo4(k2_fintopo4(C),k2_fintopo4(C),D,A)
                      & B = k2_int_1(k7_xcmplx_0(A,np__2))
                      & ! [E] :
                          ( m1_subset_1(E,u1_struct_0(k2_fintopo4(C)))
                         => ~ r2_hidden(k8_funct_2(u1_struct_0(k2_fintopo4(C)),u1_struct_0(k2_fintopo4(C)),D,E),k10_fintopo3(k2_fintopo4(C),B,E)) ) ) ) ) ) ) )).

fof(d2_fintopo5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B))))
                & m2_relset_1(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))) )
             => ( C = k1_fintopo5(A,B)
              <=> ! [D] :
                    ( r2_hidden(D,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ( D = k4_tarski(E,F)
                             => k1_funct_1(C,D) = k2_zfmisc_1(k1_funct_1(k1_fintopo4(A),E),k1_funct_1(k1_fintopo4(B),F)) ) ) ) ) ) ) ) ) )).

fof(d3_fintopo5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_fintopo5(A,B) = g1_fin_topo(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_fintopo5(A,B)) ) ) )).

fof(t11_fintopo5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => v2_fin_topo(k2_fintopo5(A,B)) ) ) )).

fof(t12_fintopo5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => v3_fin_topo(k2_fintopo5(A,B)) ) ) )).

fof(t13_fintopo5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,u1_struct_0(k2_fintopo5(A,np__1)),u1_struct_0(k2_fintopo4(A)))
          & m2_relset_1(B,u1_struct_0(k2_fintopo5(A,np__1)),u1_struct_0(k2_fintopo4(A)))
          & r1_fintopo5(k2_fintopo5(A,np__1),k2_fintopo4(A),B) ) ) )).

fof(d4_fintopo5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B))))
                & m2_relset_1(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))) )
             => ( C = k3_fintopo5(A,B)
              <=> ! [D] :
                    ( r2_hidden(D,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ( D = k4_tarski(E,F)
                             => k1_funct_1(C,D) = k2_xboole_0(k2_zfmisc_1(k1_tarski(E),k1_funct_1(k1_fintopo4(B),F)),k2_zfmisc_1(k1_funct_1(k1_fintopo4(A),E),k1_tarski(F))) ) ) ) ) ) ) ) ) )).

fof(d5_fintopo5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k4_fintopo5(A,B) = g1_fin_topo(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k3_fintopo5(A,B)) ) ) )).

fof(t14_fintopo5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => v2_fin_topo(k4_fintopo5(A,B)) ) ) )).

fof(t15_fintopo5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => v3_fin_topo(k4_fintopo5(A,B)) ) ) )).

fof(t16_fintopo5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,u1_struct_0(k4_fintopo5(A,np__1)),u1_struct_0(k2_fintopo4(A)))
          & m2_relset_1(B,u1_struct_0(k4_fintopo5(A,np__1)),u1_struct_0(k2_fintopo4(A)))
          & r1_fintopo5(k4_fintopo5(A,np__1),k2_fintopo4(A),B) ) ) )).

fof(dt_k1_fintopo5,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_funct_1(k1_fintopo5(A,B))
        & v1_funct_2(k1_fintopo5(A,B),k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B))))
        & m2_relset_1(k1_fintopo5(A,B),k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))) ) ) )).

fof(dt_k2_fintopo5,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_fin_topo(k2_fintopo5(A,B))
        & l1_fin_topo(k2_fintopo5(A,B)) ) ) )).

fof(dt_k3_fintopo5,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_funct_1(k3_fintopo5(A,B))
        & v1_funct_2(k3_fintopo5(A,B),k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B))))
        & m2_relset_1(k3_fintopo5(A,B),k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k1_zfmisc_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)))) ) ) )).

fof(dt_k4_fintopo5,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_fin_topo(k4_fintopo5(A,B))
        & l1_fin_topo(k4_fintopo5(A,B)) ) ) )).
%------------------------------------------------------------------------------