SET007 Axioms: SET007+531.ax


%------------------------------------------------------------------------------
% File     : SET007+531 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Lebesgue's Covering Lemma
% Version  : [Urb08] axioms.
% English  : Uniform Continuity and Segmentation of Arcs

% 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    : uniform1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   18 (   2 unit)
%            Number of atoms       :  261 (  24 equality)
%            Maximal formula depth :   36 (  16 average)
%            Number of connectives :  298 (  55 ~  ;   0  |; 148  &)
%                                         (   2 <=>;  93 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   1 propositional; 0-3 arity)
%            Number of functors    :   31 (   6 constant; 0-4 arity)
%            Number of variables   :   85 (   0 singleton;  84 !;   1 ?)
%            Maximal term depth    :    5 (   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_uniform1,axiom,(
    $true )).

fof(t2_uniform1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ ( ~ r1_xreal_0(B,np__0)
                  & ~ r1_xreal_0(A,k6_real_1(np__1,B)) ) ) ) ) )).

fof(d1_uniform1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & l1_metric_1(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)) )
             => ( v1_uniform1(C,A,B)
              <=> ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,np__0)
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ~ ( ~ r1_xreal_0(E,np__0)
                                & ! [F] :
                                    ( m1_subset_1(F,u1_struct_0(A))
                                   => ! [G] :
                                        ( m1_subset_1(G,u1_struct_0(A))
                                       => ~ ( ~ r1_xreal_0(E,k2_metric_1(A,F,G))
                                            & r1_xreal_0(D,k2_metric_1(B,k4_finseq_4(u1_struct_0(A),u1_struct_0(B),C,F),k4_finseq_4(u1_struct_0(A),u1_struct_0(B),C,G))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t3_uniform1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(A),u1_struct_0(k5_pcomps_1(B)))
                & m2_relset_1(C,u1_struct_0(A),u1_struct_0(k5_pcomps_1(B))) )
             => ( v5_pre_topc(C,A,k5_pcomps_1(B))
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(B))
                       => ! [F] :
                            ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(B))))
                           => ( F = k9_metric_1(B,E,D)
                             => v3_pre_topc(k5_pre_topc(A,k5_pcomps_1(B),C,F),A) ) ) ) ) ) ) ) ) )).

fof(t4_uniform1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
                & m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
             => ( ! [D] :
                    ( v1_xreal_0(D)
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(B))
                           => ~ ( ~ r1_xreal_0(D,np__0)
                                & F = k1_funct_1(C,E)
                                & ! [G] :
                                    ( v1_xreal_0(G)
                                   => ~ ( ~ r1_xreal_0(G,np__0)
                                        & ! [H] :
                                            ( m1_subset_1(H,u1_struct_0(A))
                                           => ! [I] :
                                                ( m1_subset_1(I,u1_struct_0(B))
                                               => ~ ( I = k1_funct_1(C,H)
                                                    & ~ r1_xreal_0(G,k4_metric_1(A,E,H))
                                                    & r1_xreal_0(D,k4_metric_1(B,F,I)) ) ) ) ) ) ) ) ) )
               => v5_pre_topc(C,k5_pcomps_1(A),k5_pcomps_1(B)) ) ) ) ) )).

fof(t5_uniform1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(B) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
                & m2_relset_1(C,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
             => ( v5_pre_topc(C,k5_pcomps_1(A),k5_pcomps_1(B))
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => ! [F] :
                            ( m1_subset_1(F,u1_struct_0(B))
                           => ~ ( ~ r1_xreal_0(D,np__0)
                                & F = k1_funct_1(C,E)
                                & ! [G] :
                                    ( m1_subset_1(G,k1_numbers)
                                   => ~ ( ~ r1_xreal_0(G,np__0)
                                        & ! [H] :
                                            ( m1_subset_1(H,u1_struct_0(A))
                                           => ! [I] :
                                                ( m1_subset_1(I,u1_struct_0(B))
                                               => ~ ( I = k1_funct_1(C,H)
                                                    & ~ r1_xreal_0(G,k4_metric_1(A,E,H))
                                                    & r1_xreal_0(D,k4_metric_1(B,F,I)) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t6_uniform1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(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] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
                    & m2_relset_1(D,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
                 => ( ( C = D
                      & v1_uniform1(C,A,B) )
                   => v5_pre_topc(D,k5_pcomps_1(A),k5_pcomps_1(B)) ) ) ) ) ) )).

fof(t7_uniform1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A)))))
         => ~ ( r1_pre_topc(k5_pcomps_1(A),B)
              & v1_tops_2(B,k5_pcomps_1(A))
              & v2_compts_1(k5_pcomps_1(A))
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ! [E] :
                              ( m1_subset_1(E,u1_struct_0(A))
                             => ~ ( ~ r1_xreal_0(C,k4_metric_1(A,D,E))
                                  & ! [F] :
                                      ( m1_subset_1(F,k1_zfmisc_1(u1_struct_0(k5_pcomps_1(A))))
                                     => ~ ( r2_hidden(D,F)
                                          & r2_hidden(E,F)
                                          & r2_hidden(F,B) ) ) ) ) ) ) ) ) ) ) )).

fof(t8_uniform1,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( ~ v3_struct_0(B)
            & v6_metric_1(B)
            & v7_metric_1(B)
            & v8_metric_1(B)
            & v9_metric_1(B)
            & l1_metric_1(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] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B)))
                    & m2_relset_1(D,u1_struct_0(k5_pcomps_1(A)),u1_struct_0(k5_pcomps_1(B))) )
                 => ( ( D = C
                      & v2_compts_1(k5_pcomps_1(A))
                      & v5_pre_topc(D,k5_pcomps_1(A),k5_pcomps_1(B)) )
                   => v1_uniform1(C,A,B) ) ) ) ) ) )).

fof(t9_uniform1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k14_euclid(A)))
                & m2_relset_1(C,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k14_euclid(A))) )
             => ( ( v5_pre_topc(B,k5_topmetr,k15_euclid(A))
                  & C = B )
               => v1_uniform1(C,k2_topmetr(np__0,np__1),k14_euclid(A)) ) ) ) ) )).

fof(t10_uniform1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(A))))
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k14_euclid(A)))) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & v1_funct_2(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B)))
                    & m2_relset_1(D,u1_struct_0(k5_topmetr),u1_struct_0(k3_pre_topc(k15_euclid(A),B))) )
                 => ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k1_topmetr(k14_euclid(A),C)))
                        & m2_relset_1(E,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k1_topmetr(k14_euclid(A),C))) )
                     => ( ( B = C
                          & v5_pre_topc(D,k5_topmetr,k3_pre_topc(k15_euclid(A),B))
                          & E = D )
                       => v1_uniform1(E,k2_topmetr(np__0,np__1),k1_topmetr(k14_euclid(A),C)) ) ) ) ) ) ) )).

fof(t11_uniform1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)))
            & m2_relset_1(B,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) )
         => ? [C] :
              ( v1_funct_1(C)
              & v1_funct_2(C,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k14_euclid(A)))
              & m2_relset_1(C,u1_struct_0(k2_topmetr(np__0,np__1)),u1_struct_0(k14_euclid(A)))
              & C = B ) ) ) )).

fof(t12_uniform1,axiom,(
    $true )).

fof(t13_uniform1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => k18_complex1(k5_real_1(A,B)) = k18_complex1(k5_real_1(B,A)) ) ) )).

fof(t14_uniform1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ( ( r2_hidden(A,k1_rcomp_1(C,D))
                      & r2_hidden(B,k1_rcomp_1(C,D)) )
                   => r1_xreal_0(k18_complex1(k5_real_1(A,B)),k5_real_1(D,C)) ) ) ) ) ) )).

fof(d2_uniform1,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ( v2_uniform1(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ~ ( r2_hidden(B,k4_finseq_1(A))
                    & r2_hidden(C,k4_finseq_1(A))
                    & ~ r1_xreal_0(C,B)
                    & r1_xreal_0(k1_goboard1(A,B),k1_goboard1(A,C)) ) ) ) ) ) )).

fof(t15_uniform1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ~ ( ~ r1_xreal_0(B,np__0)
                          & v5_pre_topc(C,k5_topmetr,k15_euclid(A))
                          & v2_funct_1(C)
                          & k1_funct_1(C,np__0) = D
                          & k1_funct_1(C,np__1) = E
                          & ! [F] :
                              ( m2_finseq_1(F,k1_numbers)
                             => ~ ( k1_goboard1(F,np__1) = np__0
                                  & k1_goboard1(F,k3_finseq_1(F)) = np__1
                                  & r1_xreal_0(np__5,k3_finseq_1(F))
                                  & r1_tarski(k2_relat_1(F),u1_struct_0(k5_topmetr))
                                  & v1_goboard1(F)
                                  & ! [G] :
                                      ( m2_subset_1(G,k1_numbers,k5_numbers)
                                     => ! [H] :
                                          ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
                                         => ! [I] :
                                              ( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(k14_euclid(A))))
                                             => ~ ( r1_xreal_0(np__1,G)
                                                  & ~ r1_xreal_0(k3_finseq_1(F),G)
                                                  & H = k1_rcomp_1(k4_finseq_4(k5_numbers,k1_numbers,F,G),k4_finseq_4(k5_numbers,k1_numbers,F,k1_nat_1(G,np__1)))
                                                  & I = k4_pre_topc(k5_topmetr,k15_euclid(A),C,H)
                                                  & r1_xreal_0(B,k2_tbsp_1(k14_euclid(A),I)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t16_uniform1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A)))
                & m2_relset_1(C,u1_struct_0(k5_topmetr),u1_struct_0(k15_euclid(A))) )
             => ! [D] :
                  ( m1_subset_1(D,u1_struct_0(k15_euclid(A)))
                 => ! [E] :
                      ( m1_subset_1(E,u1_struct_0(k15_euclid(A)))
                     => ~ ( ~ r1_xreal_0(B,np__0)
                          & v5_pre_topc(C,k5_topmetr,k15_euclid(A))
                          & v2_funct_1(C)
                          & k1_funct_1(C,np__0) = D
                          & k1_funct_1(C,np__1) = E
                          & ! [F] :
                              ( m2_finseq_1(F,k1_numbers)
                             => ~ ( k1_goboard1(F,np__1) = np__1
                                  & k1_goboard1(F,k3_finseq_1(F)) = np__0
                                  & r1_xreal_0(np__5,k3_finseq_1(F))
                                  & r1_tarski(k2_relat_1(F),u1_struct_0(k5_topmetr))
                                  & v2_uniform1(F)
                                  & ! [G] :
                                      ( m2_subset_1(G,k1_numbers,k5_numbers)
                                     => ! [H] :
                                          ( m1_subset_1(H,k1_zfmisc_1(u1_struct_0(k5_topmetr)))
                                         => ! [I] :
                                              ( m1_subset_1(I,k1_zfmisc_1(u1_struct_0(k14_euclid(A))))
                                             => ~ ( r1_xreal_0(np__1,G)
                                                  & ~ r1_xreal_0(k3_finseq_1(F),G)
                                                  & H = k1_rcomp_1(k4_finseq_4(k5_numbers,k1_numbers,F,k1_nat_1(G,np__1)),k4_finseq_4(k5_numbers,k1_numbers,F,G))
                                                  & I = k4_pre_topc(k5_topmetr,k15_euclid(A),C,H)
                                                  & r1_xreal_0(B,k2_tbsp_1(k14_euclid(A),I)) ) ) ) ) ) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------