SET007 Axioms: SET007+632.ax


%------------------------------------------------------------------------------
% File     : SET007+632 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Darboux's Theorem
% 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    : integra3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   20 (   0 unit)
%            Number of atoms       :  228 (  23 equality)
%            Maximal formula depth :   31 (  14 average)
%            Number of connectives :  223 (  15 ~  ;   6  |; 102  &)
%                                         (   0 <=>; 100 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   0 propositional; 1-4 arity)
%            Number of functors    :   38 (   5 constant; 0-5 arity)
%            Number of variables   :   87 (   0 singleton;  86 !;   1 ?)
%            Maximal term depth    :    6 (   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_integra3,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m3_integra1(B,A,k8_integra1(A))
         => ~ ( k3_integra1(A) != np__0
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( r2_hidden(C,k4_finseq_1(B))
                      & ~ r1_xreal_0(k3_integra1(k2_integra1(A,k8_integra1(A),B,C)),np__0) ) ) ) ) ) )).

fof(t2_integra3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( m3_integra1(C,B,k8_integra1(B))
             => ~ ( r2_hidden(A,B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( r2_hidden(D,k4_finseq_1(C))
                          & r2_hidden(A,k2_integra1(B,k8_integra1(B),C,D)) ) ) ) ) ) ) )).

fof(t3_integra3,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m3_integra1(B,A,k8_integra1(A))
         => ! [C] :
              ( m3_integra1(C,A,k8_integra1(A))
             => ? [D] :
                  ( m3_integra1(D,A,k8_integra1(A))
                  & r4_integra1(A,k8_integra1(A),B,D)
                  & r4_integra1(A,k8_integra1(A),C,D)
                  & k16_integra1(D) = k4_subset_1(k1_numbers,k16_integra1(B),k16_integra1(C)) ) ) ) ) )).

fof(t4_integra3,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m3_integra1(B,A,k8_integra1(A))
         => ! [C] :
              ( m3_integra1(C,A,k8_integra1(A))
             => ( ~ r1_xreal_0(k3_seq_4(k16_integra1(k14_integra1(A,k5_rfunct_1(A,A),k8_integra1(A),B))),k17_integra1(A,C))
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ! [E] :
                        ( m1_subset_1(E,k1_numbers)
                       => ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ( ( r2_hidden(F,k4_finseq_1(C))
                                & r2_hidden(D,k5_subset_1(k1_numbers,k16_integra1(B),k2_integra1(A,k8_integra1(A),C,F)))
                                & r2_hidden(E,k5_subset_1(k1_numbers,k16_integra1(B),k2_integra1(A,k8_integra1(A),C,F))) )
                             => D = E ) ) ) ) ) ) ) ) )).

fof(t5_integra3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k1_numbers)
         => ( ( k5_relset_1(k5_numbers,k1_numbers,A) = k5_relset_1(k5_numbers,k1_numbers,B)
              & v1_goboard1(A)
              & v1_goboard1(B) )
           => A = B ) ) ) )).

fof(t6_integra3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_integra1(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
             => ! [D] :
                  ( m3_integra1(D,C,k8_integra1(C))
                 => ! [E] :
                      ( m3_integra1(E,C,k8_integra1(C))
                     => ( ( r4_integra1(C,k8_integra1(C),D,E)
                          & r2_hidden(A,k4_finseq_1(D))
                          & r2_hidden(B,k4_finseq_1(D))
                          & r1_xreal_0(A,B) )
                       => ( r1_xreal_0(k18_integra1(C,k8_integra1(C),D,E,A),k18_integra1(C,k8_integra1(C),D,E,B))
                          & r2_hidden(k18_integra1(C,k8_integra1(C),D,E,A),k4_finseq_1(E)) ) ) ) ) ) ) ) )).

fof(t7_integra3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_integra1(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_numbers)) )
             => ! [D] :
                  ( m3_integra1(D,C,k8_integra1(C))
                 => ! [E] :
                      ( m3_integra1(E,C,k8_integra1(C))
                     => ~ ( r4_integra1(C,k8_integra1(C),D,E)
                          & r2_hidden(A,k4_finseq_1(D))
                          & r2_hidden(B,k4_finseq_1(D))
                          & ~ r1_xreal_0(B,A)
                          & r1_xreal_0(k18_integra1(C,k8_integra1(C),D,E,B),k18_integra1(C,k8_integra1(C),D,E,A)) ) ) ) ) ) ) )).

fof(t8_integra3,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m3_integra1(B,A,k8_integra1(A))
         => r1_xreal_0(np__0,k17_integra1(A,B)) ) ) )).

fof(t9_integra3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( m3_integra1(D,B,k8_integra1(B))
                 => ! [E] :
                      ( m3_integra1(E,B,k8_integra1(B))
                     => ( ( r2_hidden(A,k2_integra1(B,k8_integra1(B),D,k3_finseq_1(D)))
                          & r1_xreal_0(np__2,k3_finseq_1(D))
                          & r4_integra1(B,k8_integra1(B),D,E)
                          & k16_integra1(E) = k4_subset_1(k1_numbers,k16_integra1(D),k1_seq_4(A))
                          & r3_rfunct_1(C,B) )
                       => r1_xreal_0(k5_real_1(k15_rvsum_1(k15_integra1(B,C,k8_integra1(B),E)),k15_rvsum_1(k15_integra1(B,C,k8_integra1(B),D))),k4_real_1(k5_real_1(k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C)),k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k17_integra1(B,D))) ) ) ) ) ) ) )).

fof(t10_integra3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( m3_integra1(D,B,k8_integra1(B))
                 => ! [E] :
                      ( m3_integra1(E,B,k8_integra1(B))
                     => ( ( r2_hidden(A,k2_integra1(B,k8_integra1(B),D,k3_finseq_1(D)))
                          & r1_xreal_0(np__2,k3_finseq_1(D))
                          & r4_integra1(B,k8_integra1(B),D,E)
                          & k16_integra1(E) = k4_subset_1(k1_numbers,k16_integra1(D),k1_seq_4(A))
                          & r3_rfunct_1(C,B) )
                       => r1_xreal_0(k5_real_1(k15_rvsum_1(k14_integra1(B,C,k8_integra1(B),D)),k15_rvsum_1(k14_integra1(B,C,k8_integra1(B),E))),k4_real_1(k5_real_1(k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C)),k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k17_integra1(B,D))) ) ) ) ) ) ) )).

fof(t11_integra3,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m3_integra1(B,A,k8_integra1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ~ ( r2_hidden(D,k4_finseq_1(B))
                          & r2_hidden(E,k4_finseq_1(B))
                          & r1_xreal_0(D,E)
                          & ~ r1_xreal_0(k1_goboard1(k1_jordan3(k1_numbers,B,D,E),np__1),C)
                          & ! [F] :
                              ( ( v1_integra1(F)
                                & m1_subset_1(F,k1_zfmisc_1(k1_numbers)) )
                             => ~ ( C = k4_pscomp_1(F)
                                  & k3_pscomp_1(F) = k1_goboard1(k1_jordan3(k1_numbers,B,D,E),k3_finseq_1(k1_jordan3(k1_numbers,B,D,E)))
                                  & k3_finseq_1(k1_jordan3(k1_numbers,B,D,E)) = k3_real_1(k5_real_1(E,D),np__1)
                                  & m1_integra1(k1_jordan3(k1_numbers,B,D,E),F) ) ) ) ) ) ) ) ) )).

fof(t12_integra3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( m3_integra1(D,B,k8_integra1(B))
                 => ! [E] :
                      ( m3_integra1(E,B,k8_integra1(B))
                     => ( ( r2_hidden(A,k2_integra1(B,k8_integra1(B),D,k3_finseq_1(D)))
                          & r4_integra1(B,k8_integra1(B),D,E)
                          & k16_integra1(E) = k4_subset_1(k1_numbers,k16_integra1(D),k1_seq_4(A))
                          & r3_rfunct_1(C,B) )
                       => ( k3_integra1(B) = np__0
                          | r1_xreal_0(A,k4_pscomp_1(B))
                          | r1_xreal_0(k5_real_1(k15_rvsum_1(k15_integra1(B,C,k8_integra1(B),E)),k15_rvsum_1(k15_integra1(B,C,k8_integra1(B),D))),k4_real_1(k5_real_1(k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C)),k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k17_integra1(B,D))) ) ) ) ) ) ) ) )).

fof(t13_integra3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( m3_integra1(D,B,k8_integra1(B))
                 => ! [E] :
                      ( m3_integra1(E,B,k8_integra1(B))
                     => ( ( r2_hidden(A,k2_integra1(B,k8_integra1(B),D,k3_finseq_1(D)))
                          & r4_integra1(B,k8_integra1(B),D,E)
                          & k16_integra1(E) = k4_subset_1(k1_numbers,k16_integra1(D),k1_seq_4(A))
                          & r3_rfunct_1(C,B) )
                       => ( k3_integra1(B) = np__0
                          | r1_xreal_0(A,k4_pscomp_1(B))
                          | r1_xreal_0(k5_real_1(k15_rvsum_1(k14_integra1(B,C,k8_integra1(B),D)),k15_rvsum_1(k14_integra1(B,C,k8_integra1(B),E))),k4_real_1(k5_real_1(k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C)),k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C))),k17_integra1(B,D))) ) ) ) ) ) ) ) )).

fof(t14_integra3,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( m3_integra1(B,A,k8_integra1(A))
         => ! [C] :
              ( m3_integra1(C,A,k8_integra1(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ~ ( r2_hidden(E,k4_finseq_1(B))
                              & r2_hidden(F,k4_finseq_1(B))
                              & r1_xreal_0(E,F)
                              & r4_integra1(A,k8_integra1(A),B,C)
                              & ~ r1_xreal_0(k1_goboard1(k1_jordan3(k1_numbers,C,k18_integra1(A,k8_integra1(A),B,C,E),k18_integra1(A,k8_integra1(A),B,C,F)),np__1),D)
                              & ! [G] :
                                  ( ( v1_integra1(G)
                                    & m1_subset_1(G,k1_zfmisc_1(k1_numbers)) )
                                 => ! [H] :
                                      ( m3_integra1(H,G,k8_integra1(G))
                                     => ! [I] :
                                          ( m3_integra1(I,G,k8_integra1(G))
                                         => ~ ( D = k4_pscomp_1(G)
                                              & k3_pscomp_1(G) = k1_goboard1(I,k3_finseq_1(I))
                                              & k3_pscomp_1(G) = k1_goboard1(H,k3_finseq_1(H))
                                              & r4_integra1(G,k8_integra1(G),H,I)
                                              & H = k1_jordan3(k1_numbers,B,E,F)
                                              & I = k1_jordan3(k1_numbers,C,k18_integra1(A,k8_integra1(A),B,C,E),k18_integra1(A,k8_integra1(A),B,C,F)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t15_integra3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( m3_integra1(C,B,k8_integra1(B))
             => ( r2_hidden(A,k16_integra1(C))
               => ( r1_xreal_0(k1_goboard1(C,np__1),A)
                  & r1_xreal_0(A,k1_goboard1(C,k3_finseq_1(C))) ) ) ) ) ) )).

fof(t16_integra3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k1_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)
                 => ( ( v1_goboard1(A)
                      & r2_hidden(B,k4_finseq_1(A))
                      & r2_hidden(C,k4_finseq_1(A))
                      & r2_hidden(D,k4_finseq_1(A))
                      & r1_xreal_0(k1_goboard1(A,B),k1_goboard1(A,D))
                      & r1_xreal_0(k1_goboard1(A,D),k1_goboard1(A,C)) )
                   => r2_hidden(k1_goboard1(A,D),k5_relset_1(k5_numbers,k1_numbers,k1_jordan3(k1_numbers,A,B,C))) ) ) ) ) ) )).

fof(t17_integra3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( m3_integra1(D,B,k8_integra1(B))
                 => ( ( r3_rfunct_1(C,B)
                      & r2_hidden(A,k4_finseq_1(D)) )
                   => r1_xreal_0(k4_pscomp_1(k5_relset_1(B,k1_numbers,k2_partfun1(B,k1_numbers,C,k2_integra1(B,k8_integra1(B),D,A)))),k3_pscomp_1(k1_pscomp_1(B,k1_numbers,C))) ) ) ) ) ) )).

fof(t18_integra3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_integra1(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,B,k1_numbers)
                & m2_relset_1(C,B,k1_numbers) )
             => ! [D] :
                  ( m3_integra1(D,B,k8_integra1(B))
                 => ( ( r3_rfunct_1(C,B)
                      & r2_hidden(A,k4_finseq_1(D)) )
                   => r1_xreal_0(k4_pscomp_1(k1_pscomp_1(B,k1_numbers,C)),k3_pscomp_1(k5_relset_1(B,k1_numbers,k2_partfun1(B,k1_numbers,C,k2_integra1(B,k8_integra1(B),D,A))))) ) ) ) ) ) )).

fof(t19_integra3,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,k1_numbers)
            & m2_relset_1(B,A,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k8_integra1(A))
                & m2_relset_1(C,k5_numbers,k8_integra1(A)) )
             => ( ( r3_rfunct_1(B,A)
                  & v1_fdiff_1(k2_integra2(A,C)) )
               => ( k3_integra1(A) = np__0
                  | ( v4_seq_2(k4_integra2(A,B,C))
                    & k2_seq_2(k4_integra2(A,B,C)) = k12_integra1(A,B) ) ) ) ) ) ) )).

fof(t20_integra3,axiom,(
    ! [A] :
      ( ( v1_integra1(A)
        & m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,A,k1_numbers)
            & m2_relset_1(B,A,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k8_integra1(A))
                & m2_relset_1(C,k5_numbers,k8_integra1(A)) )
             => ( ( r3_rfunct_1(B,A)
                  & v1_fdiff_1(k2_integra2(A,C)) )
               => ( k3_integra1(A) = np__0
                  | ( v4_seq_2(k3_integra2(A,B,C))
                    & k2_seq_2(k3_integra2(A,B,C)) = k11_integra1(A,B) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------