SET007 Axioms: SET007+529.ax


%------------------------------------------------------------------------------
% File     : SET007+529 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Intermediate Value Theorem and Thickness of Simple Closed Curves
% 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    : topreal5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   25 (   5 unit)
%            Number of atoms       :  226 (  29 equality)
%            Maximal formula depth :   29 (  11 average)
%            Number of connectives :  260 (  59 ~  ;   0  |; 118  &)
%                                         (   0 <=>;  83 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   21 (   1 propositional; 0-3 arity)
%            Number of functors    :   30 (   8 constant; 0-4 arity)
%            Number of variables   :   77 (   0 singleton;  73 !;   4 ?)
%            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(t1_topreal5,axiom,(
    $true )).

fof(t2_topreal5,axiom,(
    $true )).

fof(t3_topreal5,axiom,(
    $true )).

fof(t4_topreal5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(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)))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ~ ( v3_pre_topc(C,A)
                      & v3_pre_topc(D,A)
                      & ~ r1_xboole_0(C,B)
                      & ~ r1_xboole_0(D,B)
                      & r1_tarski(B,k4_subset_1(u1_struct_0(A),C,D))
                      & r1_xboole_0(C,D)
                      & v2_connsp_1(B,A) ) ) ) ) ) )).

fof(t5_topreal5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v2_pre_topc(B)
            & l1_pre_topc(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(B))
                & v5_pre_topc(C,A,B)
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(B)) )
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ( v2_connsp_1(D,A)
                   => v2_connsp_1(k4_pre_topc(A,B,C,D),B) ) ) ) ) ) )).

fof(t6_topreal5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
           => v2_connsp_1(k2_pre_topc(k4_topmetr(A,B)),k4_topmetr(A,B)) ) ) ) )).

fof(t7_topreal5,axiom,(
    $true )).

fof(t8_topreal5,axiom,(
    $true )).

fof(t9_topreal5,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k3_topmetr)))
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( ~ r2_hidden(B,A)
              & ? [C] :
                  ( v1_xreal_0(C)
                  & ? [D] :
                      ( v1_xreal_0(D)
                      & r2_hidden(C,A)
                      & r2_hidden(D,A)
                      & ~ r1_xreal_0(B,C)
                      & ~ r1_xreal_0(D,B) ) )
              & v2_connsp_1(A,k3_topmetr) ) ) ) )).

fof(t10_topreal5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => ! [G] :
                              ( ( v1_funct_1(G)
                                & v1_funct_2(G,u1_struct_0(A),u1_struct_0(k3_topmetr))
                                & v5_pre_topc(G,A,k3_topmetr)
                                & m2_relset_1(G,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
                             => ~ ( v1_connsp_1(A)
                                  & k1_funct_1(G,B) = D
                                  & k1_funct_1(G,C) = E
                                  & r1_xreal_0(D,F)
                                  & r1_xreal_0(F,E)
                                  & ! [H] :
                                      ( m1_subset_1(H,u1_struct_0(A))
                                     => k1_funct_1(G,H) != F ) ) ) ) ) ) ) ) ) )).

fof(t11_topreal5,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(A))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(u1_struct_0(A)))
                 => ! [E] :
                      ( m1_subset_1(E,k1_numbers)
                     => ! [F] :
                          ( m1_subset_1(F,k1_numbers)
                         => ! [G] :
                              ( m1_subset_1(G,k1_numbers)
                             => ! [H] :
                                  ( ( v1_funct_1(H)
                                    & v1_funct_2(H,u1_struct_0(A),u1_struct_0(k3_topmetr))
                                    & v5_pre_topc(H,A,k3_topmetr)
                                    & m2_relset_1(H,u1_struct_0(A),u1_struct_0(k3_topmetr)) )
                                 => ~ ( v2_connsp_1(D,A)
                                      & k1_funct_1(H,B) = E
                                      & k1_funct_1(H,C) = F
                                      & r1_xreal_0(E,G)
                                      & r1_xreal_0(G,F)
                                      & r2_hidden(B,D)
                                      & r2_hidden(C,D)
                                      & ! [I] :
                                          ( m1_subset_1(I,u1_struct_0(A))
                                         => ~ ( r2_hidden(I,D)
                                              & k1_funct_1(H,I) = G ) ) ) ) ) ) ) ) ) ) ) )).

fof(t12_topreal5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ~ r1_xreal_0(B,A)
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr))
                          & v5_pre_topc(E,k4_topmetr(A,B),k3_topmetr)
                          & m2_relset_1(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr)) )
                       => ! [F] :
                            ( v1_xreal_0(F)
                           => ~ ( k1_funct_1(E,A) = C
                                & k1_funct_1(E,B) = D
                                & ~ r1_xreal_0(F,C)
                                & ~ r1_xreal_0(D,F)
                                & ! [G] :
                                    ( m1_subset_1(G,k1_numbers)
                                   => ~ ( k1_funct_1(E,G) = F
                                        & ~ r1_xreal_0(G,A)
                                        & ~ r1_xreal_0(B,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t13_topreal5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ~ r1_xreal_0(B,A)
                   => ! [E] :
                        ( ( v1_funct_1(E)
                          & v1_funct_2(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr))
                          & v5_pre_topc(E,k4_topmetr(A,B),k3_topmetr)
                          & m2_relset_1(E,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr)) )
                       => ! [F] :
                            ( v1_xreal_0(F)
                           => ~ ( k1_funct_1(E,A) = C
                                & k1_funct_1(E,B) = D
                                & ~ r1_xreal_0(C,F)
                                & ~ r1_xreal_0(F,D)
                                & ! [G] :
                                    ( m1_subset_1(G,k1_numbers)
                                   => ~ ( k1_funct_1(E,G) = F
                                        & ~ r1_xreal_0(G,A)
                                        & ~ r1_xreal_0(B,G) ) ) ) ) ) ) ) ) ) ) )).

fof(t14_topreal5,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr))
                & v5_pre_topc(C,k4_topmetr(A,B),k3_topmetr)
                & m2_relset_1(C,u1_struct_0(k4_topmetr(A,B)),u1_struct_0(k3_topmetr)) )
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ! [E] :
                      ( v1_xreal_0(E)
                     => ~ ( ~ r1_xreal_0(B,A)
                          & ~ r1_xreal_0(np__0,k3_xcmplx_0(D,E))
                          & D = k1_funct_1(C,A)
                          & E = k1_funct_1(C,B)
                          & ! [F] :
                              ( m1_subset_1(F,k1_numbers)
                             => ~ ( k1_funct_1(C,F) = np__0
                                  & ~ r1_xreal_0(F,A)
                                  & ~ r1_xreal_0(B,F) ) ) ) ) ) ) ) ) )).

fof(t15_topreal5,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr))
        & m2_relset_1(A,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr)) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ~ ( v5_pre_topc(A,k22_borsuk_1,k3_topmetr)
                  & k1_funct_1(A,np__0) != k1_funct_1(A,np__1)
                  & B = k1_funct_1(A,np__0)
                  & C = k1_funct_1(A,np__1)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ~ r1_xreal_0(np__1,D)
                          & k1_funct_1(A,D) = k7_xcmplx_0(k2_xcmplx_0(B,C),np__2) ) ) ) ) ) ) )).

fof(t16_topreal5,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr))
        & m2_relset_1(A,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr)) )
     => ( A = k16_pscomp_1
       => v5_pre_topc(A,k15_euclid(np__2),k3_topmetr) ) ) )).

fof(t17_topreal5,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr))
        & m2_relset_1(A,u1_struct_0(k15_euclid(np__2)),u1_struct_0(k3_topmetr)) )
     => ( A = k17_pscomp_1
       => v5_pre_topc(A,k15_euclid(np__2),k3_topmetr) ) ) )).

fof(t18_topreal5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
            & m2_relset_1(B,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
         => ~ ( v5_pre_topc(B,k22_borsuk_1,k3_pre_topc(k15_euclid(np__2),A))
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr))
                    & m2_relset_1(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr)) )
                 => ~ ( v5_pre_topc(C,k22_borsuk_1,k3_topmetr)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                             => ( ( r2_hidden(D,u1_struct_0(k22_borsuk_1))
                                  & E = k1_funct_1(B,D) )
                               => k21_euclid(E) = k1_funct_1(C,D) ) ) ) ) ) ) ) ) )).

fof(t19_topreal5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A)))
            & m2_relset_1(B,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_pre_topc(k15_euclid(np__2),A))) )
         => ~ ( v5_pre_topc(B,k22_borsuk_1,k3_pre_topc(k15_euclid(np__2),A))
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr))
                    & m2_relset_1(C,u1_struct_0(k22_borsuk_1),u1_struct_0(k3_topmetr)) )
                 => ~ ( v5_pre_topc(C,k22_borsuk_1,k3_topmetr)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,u1_struct_0(k15_euclid(np__2)))
                             => ( ( r2_hidden(D,u1_struct_0(k22_borsuk_1))
                                  & E = k1_funct_1(B,D) )
                               => k22_euclid(E) = k1_funct_1(C,D) ) ) ) ) ) ) ) ) )).

fof(t20_topreal5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( v1_topreal2(A)
       => ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ? [C] :
                ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
                & r2_hidden(C,A)
                & k22_euclid(C) != B ) ) ) ) )).

fof(t21_topreal5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( v1_topreal2(A)
       => ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ? [C] :
                ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
                & r2_hidden(C,A)
                & k21_euclid(C) != B ) ) ) ) )).

fof(t22_topreal5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ ( v1_topreal2(A)
          & r1_xreal_0(k19_pscomp_1(A),k21_pscomp_1(A)) ) ) )).

fof(t23_topreal5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ ( v1_topreal2(A)
          & r1_xreal_0(k20_pscomp_1(A),k18_pscomp_1(A)) ) ) )).

fof(t24_topreal5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ ( v1_topreal2(A)
          & k37_pscomp_1(A) = k33_pscomp_1(A) ) ) )).

fof(t25_topreal5,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ ( v1_topreal2(A)
          & k30_pscomp_1(A) = k34_pscomp_1(A) ) ) )).
%------------------------------------------------------------------------------