SET007 Axioms: SET007+917.ax


%------------------------------------------------------------------------------
% File     : SET007+917 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Fashoda Meet Theorem for Continuous Mappings
% 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    : jgraph_8 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    8 (   0 unit)
%            Number of atoms       :  131 (  23 equality)
%            Maximal formula depth :   43 (  21 average)
%            Number of connectives :  131 (   8 ~  ;   0  |;  64  &)
%                                         (   0 <=>;  59 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   0 propositional; 1-4 arity)
%            Number of functors    :   27 (   8 constant; 0-4 arity)
%            Number of variables   :   60 (   0 singleton;  59 !;   1 ?)
%            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_jgraph_8,axiom,(
    ! [A,B,C,D] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B)
        & v1_xreal_0(C)
        & v1_xreal_0(D) )
     => v1_jordan1(k2_jgraph_6(A,B,C,D),np__2) ) )).

fof(fc2_jgraph_8,axiom,(
    ! [A,B,C,D] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B)
        & v1_xreal_0(C)
        & v1_xreal_0(D) )
     => ( v2_pre_topc(k1_topreala(A,B,C,D))
        & v1_topalg_2(k1_topreala(A,B,C,D),np__2) ) ) )).

fof(t1_jgraph_8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_xreal_0(B)
            & v2_xreal_0(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(A)))
                & v5_pre_topc(C,k22_borsuk_1,k15_euclid(A))
                & m2_relset_1(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(A))) )
             => ? [D] :
                  ( m2_finseq_1(D,k1_numbers)
                  & k1_goboard1(D,np__1) = np__0
                  & k1_goboard1(D,k3_finseq_1(D)) = np__1
                  & r1_xreal_0(np__5,k3_finseq_1(D))
                  & r1_tarski(k5_relset_1(k5_numbers,k1_numbers,D),u1_struct_0(k22_borsuk_1))
                  & v1_goboard1(D)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k22_borsuk_1)))
                         => ! [G] :
                              ( m1_subset_1(G,k1_zfmisc_1(u1_struct_0(k14_euclid(A))))
                             => ~ ( r1_xreal_0(np__1,E)
                                  & ~ r1_xreal_0(k3_finseq_1(D),E)
                                  & F = k1_rcomp_1(k4_finseq_4(k5_numbers,k1_numbers,D,E),k4_finseq_4(k5_numbers,k1_numbers,D,k1_nat_1(E,np__1)))
                                  & G = k4_pre_topc(k22_borsuk_1,k15_euclid(A),C,F)
                                  & r1_xreal_0(B,k2_tbsp_1(k14_euclid(A),G)) ) ) ) ) ) ) ) ) )).

fof(t2_jgraph_8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
                 => ( ( r1_tarski(D,k3_topreal1(A,B,C))
                      & r2_hidden(B,D)
                      & r2_hidden(C,D)
                      & v2_connsp_1(D,k15_euclid(A)) )
                   => D = k3_topreal1(A,B,C) ) ) ) ) ) )).

fof(t3_jgraph_8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(A)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(A)))
             => ! [D] :
                  ( m1_borsuk_2(D,k15_euclid(A),B,C)
                 => ( r1_tarski(k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),D),k3_topreal1(A,B,C))
                   => k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)),D) = k3_topreal1(A,B,C) ) ) ) ) ) )).

fof(t4_jgraph_8,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(np__2)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k15_euclid(np__2)))
                         => ! [G] :
                              ( m1_borsuk_2(G,k15_euclid(np__2),C,D)
                             => ! [H] :
                                  ( m1_borsuk_2(H,k15_euclid(np__2),E,F)
                                 => ~ ( k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),G) = A
                                      & k1_pscomp_1(u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(np__2)),H) = B
                                      & ! [I] :
                                          ( m1_subset_1(I,u1_struct_0(k15_euclid(np__2)))
                                         => ( r2_hidden(I,A)
                                           => ( r1_xreal_0(k21_euclid(C),k21_euclid(I))
                                              & r1_xreal_0(k21_euclid(I),k21_euclid(D)) ) ) )
                                      & ! [I] :
                                          ( m1_subset_1(I,u1_struct_0(k15_euclid(np__2)))
                                         => ( r2_hidden(I,B)
                                           => ( r1_xreal_0(k21_euclid(C),k21_euclid(I))
                                              & r1_xreal_0(k21_euclid(I),k21_euclid(D)) ) ) )
                                      & ! [I] :
                                          ( m1_subset_1(I,u1_struct_0(k15_euclid(np__2)))
                                         => ( r2_hidden(I,A)
                                           => ( r1_xreal_0(k22_euclid(E),k22_euclid(I))
                                              & r1_xreal_0(k22_euclid(I),k22_euclid(F)) ) ) )
                                      & ! [I] :
                                          ( m1_subset_1(I,u1_struct_0(k15_euclid(np__2)))
                                         => ( r2_hidden(I,B)
                                           => ( r1_xreal_0(k22_euclid(E),k22_euclid(I))
                                              & r1_xreal_0(k22_euclid(I),k22_euclid(F)) ) ) )
                                      & r2_subset_1(A,B) ) ) ) ) ) ) ) ) ) )).

fof(t5_jgraph_8,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)))
                        & v5_pre_topc(E,k22_borsuk_1,k15_euclid(np__2))
                        & m2_relset_1(E,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2))) )
                     => ! [F] :
                          ( ( v1_funct_1(F)
                            & v1_funct_2(F,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)))
                            & v5_pre_topc(F,k22_borsuk_1,k15_euclid(np__2))
                            & m2_relset_1(F,u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2))) )
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(k22_borsuk_1))
                             => ! [H] :
                                  ( m1_subset_1(H,u1_struct_0(k22_borsuk_1))
                                 => ~ ( G = np__0
                                      & H = np__1
                                      & k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,G)) = A
                                      & k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,H)) = B
                                      & k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,G)) = C
                                      & k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,H)) = D
                                      & ! [I] :
                                          ( m1_subset_1(I,u1_struct_0(k22_borsuk_1))
                                         => ( r1_xreal_0(A,k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,I)))
                                            & r1_xreal_0(k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,I)),B)
                                            & r1_xreal_0(A,k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,I)))
                                            & r1_xreal_0(k21_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,I)),B)
                                            & r1_xreal_0(C,k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,I)))
                                            & r1_xreal_0(k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E,I)),D)
                                            & r1_xreal_0(C,k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,I)))
                                            & r1_xreal_0(k22_euclid(k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F,I)),D) ) )
                                      & r1_xboole_0(k1_pscomp_1(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),E),k1_pscomp_1(u1_struct_0(k22_borsuk_1),u1_struct_0(k15_euclid(np__2)),F)) ) ) ) ) ) ) ) ) ) )).

fof(t6_jgraph_8,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k1_topreala(A,B,C,D)))
                     => ! [F] :
                          ( m1_subset_1(F,u1_struct_0(k1_topreala(A,B,C,D)))
                         => ! [G] :
                              ( m1_subset_1(G,u1_struct_0(k1_topreala(A,B,C,D)))
                             => ! [H] :
                                  ( m1_subset_1(H,u1_struct_0(k1_topreala(A,B,C,D)))
                                 => ! [I] :
                                      ( m1_borsuk_2(I,k1_topreala(A,B,C,D),E,F)
                                     => ! [J] :
                                          ( m1_borsuk_2(J,k1_topreala(A,B,C,D),H,G)
                                         => ! [K] :
                                              ( m1_subset_1(K,u1_struct_0(k15_euclid(np__2)))
                                             => ! [L] :
                                                  ( m1_subset_1(L,u1_struct_0(k15_euclid(np__2)))
                                                 => ! [M] :
                                                      ( m1_subset_1(M,u1_struct_0(k15_euclid(np__2)))
                                                     => ! [N] :
                                                          ( m1_subset_1(N,u1_struct_0(k15_euclid(np__2)))
                                                         => ~ ( k21_euclid(K) = A
                                                              & k21_euclid(L) = B
                                                              & k22_euclid(M) = C
                                                              & k22_euclid(N) = D
                                                              & E = K
                                                              & F = L
                                                              & G = M
                                                              & H = N
                                                              & ! [O] :
                                                                  ( m1_subset_1(O,u1_struct_0(k22_borsuk_1))
                                                                 => ! [P] :
                                                                      ( m1_subset_1(P,u1_struct_0(k22_borsuk_1))
                                                                     => k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k1_topreala(A,B,C,D)),I,O) != k8_funct_2(u1_struct_0(k22_borsuk_1),u1_struct_0(k1_topreala(A,B,C,D)),J,P) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------