SET007 Axioms: SET007+44.ax


%------------------------------------------------------------------------------
% File     : SET007+44 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Strong Arithmetic of Real Numbers
% 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    : axioms [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   31 (  24 unit)
%            Number of atoms       :   63 (   5 equality)
%            Maximal formula depth :   17 (   3 average)
%            Number of connectives :   38 (   6 ~  ;   0  |;  13  &)
%                                         (   1 <=>;  18 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    9 (   1 propositional; 0-3 arity)
%            Number of functors    :    8 (   4 constant; 0-2 arity)
%            Number of variables   :   19 (   0 singleton;  15 !;   4 ?)
%            Maximal term depth    :    2 (   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_axioms,axiom,(
    $true )).

fof(t2_axioms,axiom,(
    $true )).

fof(t3_axioms,axiom,(
    $true )).

fof(t4_axioms,axiom,(
    $true )).

fof(t5_axioms,axiom,(
    $true )).

fof(t6_axioms,axiom,(
    $true )).

fof(t7_axioms,axiom,(
    $true )).

fof(t8_axioms,axiom,(
    $true )).

fof(t9_axioms,axiom,(
    $true )).

fof(t10_axioms,axiom,(
    $true )).

fof(t11_axioms,axiom,(
    $true )).

fof(t12_axioms,axiom,(
    $true )).

fof(t13_axioms,axiom,(
    $true )).

fof(t14_axioms,axiom,(
    $true )).

fof(t15_axioms,axiom,(
    $true )).

fof(t16_axioms,axiom,(
    $true )).

fof(t17_axioms,axiom,(
    $true )).

fof(t18_axioms,axiom,(
    $true )).

fof(t19_axioms,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ? [B] :
          ( v1_xreal_0(B)
          & k2_xcmplx_0(A,B) = np__0 ) ) )).

fof(t20_axioms,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ~ ( A != np__0
          & ! [B] :
              ( v1_xreal_0(B)
             => k3_xcmplx_0(A,B) != np__1 ) ) ) )).

fof(t21_axioms,axiom,(
    $true )).

fof(t22_axioms,axiom,(
    $true )).

fof(t23_axioms,axiom,(
    $true )).

fof(t24_axioms,axiom,(
    $true )).

fof(t25_axioms,axiom,(
    $true )).

fof(t26_axioms,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ~ ( ! [C] :
                  ( v1_xreal_0(C)
                 => ! [D] :
                      ( v1_xreal_0(D)
                     => ( ( r2_hidden(C,A)
                          & r2_hidden(D,B) )
                       => r1_xreal_0(C,D) ) ) )
              & ! [C] :
                  ( v1_xreal_0(C)
                 => ? [D] :
                      ( v1_xreal_0(D)
                      & ? [E] :
                          ( v1_xreal_0(E)
                          & r2_hidden(D,A)
                          & r2_hidden(E,B)
                          & ~ ( r1_xreal_0(D,C)
                              & r1_xreal_0(C,E) ) ) ) ) ) ) ) )).

fof(t27_axioms,axiom,(
    $true )).

fof(t28_axioms,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ( r2_hidden(A,k5_numbers)
              & r2_hidden(B,k5_numbers) )
           => r2_hidden(k2_xcmplx_0(A,B),k5_numbers) ) ) ) )).

fof(t29_axioms,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ( ( r2_hidden(np__0,A)
          & ! [B] :
              ( v1_xreal_0(B)
             => ( r2_hidden(B,A)
               => r2_hidden(k2_xcmplx_0(B,np__1),A) ) ) )
       => r1_tarski(k5_numbers,A) ) ) )).

fof(t30_axioms,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => A = a_1_0_axioms(A) ) )).

fof(fraenkel_a_1_0_axioms,axiom,(
    ! [A,B] :
      ( v4_ordinal2(B)
     => ( r2_hidden(A,a_1_0_axioms(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = C
            & ~ r1_xreal_0(B,C) ) ) ) )).
%------------------------------------------------------------------------------