SET007 Axioms: SET007+683.ax


%------------------------------------------------------------------------------
% File     : SET007+683 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Some Properties of Dyadic Numbers and Intervals
% 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    : urysohn2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   36 (   4 unit)
%            Number of atoms       :  234 (  58 equality)
%            Maximal formula depth :   20 (   8 average)
%            Number of connectives :  249 (  51 ~  ;   6  |;  81  &)
%                                         (   0 <=>; 111 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   16 (   1 propositional; 0-3 arity)
%            Number of functors    :   34 (  14 constant; 0-3 arity)
%            Number of variables   :   77 (   0 singleton;  77 !;   0 ?)
%            Maximal term depth    :    3 (   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(fc1_urysohn2,axiom,(
    ! [A,B] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
        & m1_subset_1(B,k1_numbers) )
     => ( v5_measure5(k1_integra2(A,B))
        & v1_membered(k1_integra2(A,B))
        & v2_membered(k1_integra2(A,B)) ) ) )).

fof(t1_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ( A != k6_measure5
       => ( ( ~ r1_supinf_1(k7_measure6(A),k6_measure6(A))
           => k5_measure5(A) = k4_supinf_2(k7_measure6(A),k6_measure6(A)) )
          & ( k7_measure6(A) = k6_measure6(A)
           => k5_measure5(A) = k1_supinf_2 ) ) ) ) )).

fof(t2_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( B != np__0
           => k1_integra2(k1_integra2(A,B),k2_real_1(B)) = A ) ) ) )).

fof(t3_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( A != np__0
       => ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(k6_supinf_1))
           => ( B = k6_supinf_1
             => k1_integra2(B,A) = B ) ) ) ) )).

fof(t4_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k6_supinf_1))
     => ( A != k6_measure5
       => k1_integra2(A,np__0) = k1_tarski(np__0) ) ) )).

fof(t5_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k1_integra2(k6_measure5,A) = k6_measure5 ) )).

fof(t6_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ~ ( r1_supinf_1(A,B)
              & ~ ( A = k4_measure6
                  & B = k4_measure6 )
              & ~ ( A = k4_measure6
                  & r2_hidden(B,k6_supinf_1) )
              & ~ ( A = k4_measure6
                  & B = k5_measure6 )
              & ~ ( r2_hidden(A,k6_supinf_1)
                  & r2_hidden(B,k6_supinf_1) )
              & ~ ( r2_hidden(A,k6_supinf_1)
                  & B = k5_measure6 )
              & ~ ( A = k5_measure6
                  & B = k5_measure6 ) ) ) ) )).

fof(t7_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ( v5_measure5(k1_measure5(A,A))
        & m1_subset_1(k1_measure5(A,A),k1_zfmisc_1(k6_supinf_1)) ) ) )).

fof(t8_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ( v5_measure5(k1_integra2(A,np__0))
        & m1_subset_1(k1_integra2(A,np__0),k1_zfmisc_1(k6_supinf_1)) ) ) )).

fof(t9_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( v1_measure5(A)
           => ( B = np__0
              | v1_measure5(k1_integra2(A,B)) ) ) ) ) )).

fof(t10_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( v2_measure5(A)
           => ( B = np__0
              | v2_measure5(k1_integra2(A,B)) ) ) ) ) )).

fof(t11_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( v3_measure5(A)
           => ( r1_xreal_0(B,np__0)
              | v3_measure5(k1_integra2(A,B)) ) ) ) ) )).

fof(t12_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( v3_measure5(A)
           => ( r1_xreal_0(np__0,B)
              | v4_measure5(k1_integra2(A,B)) ) ) ) ) )).

fof(t13_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( v4_measure5(A)
           => ( r1_xreal_0(B,np__0)
              | v4_measure5(k1_integra2(A,B)) ) ) ) ) )).

fof(t14_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( v4_measure5(A)
           => ( r1_xreal_0(np__0,B)
              | v3_measure5(k1_integra2(A,B)) ) ) ) ) )).

fof(t15_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ( A != k6_measure5
       => ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ( ~ r1_xreal_0(B,np__0)
             => ! [C] :
                  ( ( v5_measure5(C)
                    & m1_subset_1(C,k1_zfmisc_1(k6_supinf_1)) )
                 => ( ( C = k1_integra2(A,B)
                      & A = k1_measure5(k6_measure6(A),k7_measure6(A)) )
                   => ( C = k1_measure5(k6_measure6(C),k7_measure6(C))
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( ( D = k6_measure6(A)
                                  & E = k7_measure6(A) )
                               => ( k6_measure6(C) = k4_real_1(B,D)
                                  & k7_measure6(C) = k4_real_1(B,E) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t16_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ( A != k6_measure5
       => ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ( ~ r1_xreal_0(B,np__0)
             => ! [C] :
                  ( ( v5_measure5(C)
                    & m1_subset_1(C,k1_zfmisc_1(k6_supinf_1)) )
                 => ( ( C = k1_integra2(A,B)
                      & A = k3_measure5(k6_measure6(A),k7_measure6(A)) )
                   => ( C = k3_measure5(k6_measure6(C),k7_measure6(C))
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( ( D = k6_measure6(A)
                                  & E = k7_measure6(A) )
                               => ( k6_measure6(C) = k4_real_1(B,D)
                                  & k7_measure6(C) = k4_real_1(B,E) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t17_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ( A != k6_measure5
       => ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ( ~ r1_xreal_0(B,np__0)
             => ! [C] :
                  ( ( v5_measure5(C)
                    & m1_subset_1(C,k1_zfmisc_1(k6_supinf_1)) )
                 => ( ( C = k1_integra2(A,B)
                      & A = k2_measure5(k6_measure6(A),k7_measure6(A)) )
                   => ( C = k2_measure5(k6_measure6(C),k7_measure6(C))
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( ( D = k6_measure6(A)
                                  & E = k7_measure6(A) )
                               => ( k6_measure6(C) = k4_real_1(B,D)
                                  & k7_measure6(C) = k4_real_1(B,E) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t18_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ( A != k6_measure5
       => ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ( ~ r1_xreal_0(B,np__0)
             => ! [C] :
                  ( ( v5_measure5(C)
                    & m1_subset_1(C,k1_zfmisc_1(k6_supinf_1)) )
                 => ( ( C = k1_integra2(A,B)
                      & A = k4_measure5(k6_measure6(A),k7_measure6(A)) )
                   => ( C = k4_measure5(k6_measure6(C),k7_measure6(C))
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( ( D = k6_measure6(A)
                                  & E = k7_measure6(A) )
                               => ( k6_measure6(C) = k4_real_1(B,D)
                                  & k7_measure6(C) = k4_real_1(B,E) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t19_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( v5_measure5(k1_integra2(A,B))
            & m1_subset_1(k1_integra2(A,B),k1_zfmisc_1(k6_supinf_1)) ) ) ) )).

fof(t20_urysohn2,axiom,(
    ! [A] :
      ( ( v5_measure5(A)
        & m1_subset_1(A,k1_zfmisc_1(k6_supinf_1)) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r1_xreal_0(np__0,B)
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ( C = k5_measure5(A)
                 => k4_real_1(B,C) = k5_measure5(k1_integra2(A,B)) ) ) ) ) ) )).

fof(t21_urysohn2,axiom,(
    $true )).

fof(t22_urysohn2,axiom,(
    $true )).

fof(t23_urysohn2,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(k4_real_1(k3_newton(np__2,B),A),np__1) ) ) ) )).

fof(t24_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( r1_xreal_0(np__0,A)
              & ~ r1_xreal_0(k5_real_1(B,A),np__1)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( ~ r1_xreal_0(C,A)
                      & ~ r1_xreal_0(B,C) ) ) ) ) ) )).

fof(t25_urysohn2,axiom,(
    $true )).

fof(t26_urysohn2,axiom,(
    $true )).

fof(t27_urysohn2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_tarski(k3_urysohn1(A),k4_urysohn1) ) )).

fof(t28_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ~ r1_xreal_0(B,A)
              & r1_xreal_0(np__0,A)
              & r1_xreal_0(B,np__1)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( r2_hidden(C,k4_urysohn1)
                      & ~ r1_xreal_0(C,A)
                      & ~ r1_xreal_0(B,C) ) ) ) ) ) )).

fof(t29_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ~ ( ~ r1_xreal_0(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( r2_hidden(C,k5_urysohn1)
                      & ~ r1_xreal_0(C,A)
                      & ~ r1_xreal_0(B,C) ) ) ) ) ) )).

fof(t30_urysohn2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(k3_supinf_1)) )
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( r1_tarski(A,k1_measure5(B,C))
               => ( r1_supinf_1(B,k10_supinf_1(A))
                  & r1_supinf_1(k9_supinf_1(A),C) ) ) ) ) ) )).

fof(t31_urysohn2,axiom,
    ( r2_hidden(np__0,k4_urysohn1)
    & r2_hidden(np__1,k4_urysohn1) )).

fof(t32_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ( ( A = np__0
              & B = np__1 )
           => r1_tarski(k4_urysohn1,k1_measure5(A,B)) ) ) ) )).

fof(t33_urysohn2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,B)
           => r1_tarski(k3_urysohn1(A),k3_urysohn1(B)) ) ) ) )).

fof(t34_urysohn2,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)
                 => ~ ( ~ r1_xreal_0(C,A)
                      & ~ r1_xreal_0(B,C)
                      & ~ r1_xreal_0(D,A)
                      & ~ r1_xreal_0(B,D)
                      & r1_xreal_0(k5_real_1(B,A),k18_complex1(k5_real_1(D,C))) ) ) ) ) ) )).

fof(t35_urysohn2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => ! [B] :
            ( m1_subset_1(B,k1_numbers)
           => ~ ( ~ r1_xreal_0(B,np__0)
                & r1_xreal_0(B,np__1)
                & ! [C] :
                    ( m1_subset_1(C,k1_numbers)
                   => ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( r2_hidden(C,k4_subset_1(k1_numbers,k4_urysohn1,k2_urysohn1))
                            & r2_hidden(D,k4_subset_1(k1_numbers,k4_urysohn1,k2_urysohn1))
                            & ~ r1_xreal_0(C,np__0)
                            & ~ r1_xreal_0(B,C)
                            & ~ r1_xreal_0(D,B)
                            & ~ r1_xreal_0(A,k5_real_1(D,C)) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------