SET007 Axioms: SET007+40.ax


%------------------------------------------------------------------------------
% File     : SET007+40 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Introduction to 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    : xreal_0 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   48 (   0 unit)
%            Number of atoms       :  295 (   4 equality)
%            Maximal formula depth :   15 (   7 average)
%            Number of connectives :  339 (  92 ~  ;   1  |; 194  &)
%                                         (   6 <=>;  46 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   0 propositional; 1-2 arity)
%            Number of functors    :   12 (   3 constant; 0-2 arity)
%            Number of variables   :   79 (   0 singleton;  71 !;   8 ?)
%            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(cc1_xreal_0,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => v1_xreal_0(A) ) )).

fof(cc2_xreal_0,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => v1_xcmplx_0(A) ) )).

fof(rc1_xreal_0,axiom,(
    ? [A] :
      ( v1_xcmplx_0(A)
      & v1_xreal_0(A) ) )).

fof(fc1_xreal_0,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_xcmplx_0(k4_xcmplx_0(A))
        & v1_xreal_0(k4_xcmplx_0(A)) ) ) )).

fof(fc2_xreal_0,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_xcmplx_0(k5_xcmplx_0(A))
        & v1_xreal_0(k5_xcmplx_0(A)) ) ) )).

fof(fc3_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B)) ) ) )).

fof(fc4_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B)) ) ) )).

fof(fc5_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B)) ) ) )).

fof(fc6_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( v1_xcmplx_0(k7_xcmplx_0(A,B))
        & v1_xreal_0(k7_xcmplx_0(A,B)) ) ) )).

fof(cc3_xreal_0,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A) )
     => ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) )).

fof(cc4_xreal_0,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & v2_xreal_0(A) ) ) )).

fof(cc5_xreal_0,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & v3_xreal_0(A) )
     => ( ~ v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v2_xreal_0(A) ) ) )).

fof(cc6_xreal_0,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_xreal_0(A)
        & ~ v2_xreal_0(A) )
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & v3_xreal_0(A) ) ) )).

fof(cc7_xreal_0,axiom,(
    ! [A] :
      ( ( v1_xboole_0(A)
        & v1_xreal_0(A) )
     => ( v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & ~ v3_xreal_0(A) ) ) )).

fof(cc8_xreal_0,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ( v1_xboole_0(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A) ) ) )).

fof(rc2_xreal_0,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & v2_xreal_0(A)
      & ~ v3_xreal_0(A) ) )).

fof(rc3_xreal_0,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v2_xreal_0(A)
      & v3_xreal_0(A) ) )).

fof(rc4_xreal_0,axiom,(
    ? [A] :
      ( v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v2_xreal_0(A)
      & ~ v3_xreal_0(A) ) )).

fof(fc7_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) )).

fof(fc8_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & ~ v2_xreal_0(k2_xcmplx_0(A,B)) ) ) )).

fof(fc9_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( ~ v1_xboole_0(k2_xcmplx_0(A,B))
        & v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & v2_xreal_0(k2_xcmplx_0(A,B))
        & ~ v3_xreal_0(k2_xcmplx_0(A,B)) ) ) )).

fof(fc10_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( ~ v1_xboole_0(k2_xcmplx_0(B,A))
        & v1_xcmplx_0(k2_xcmplx_0(B,A))
        & v1_xreal_0(k2_xcmplx_0(B,A))
        & v2_xreal_0(k2_xcmplx_0(B,A))
        & ~ v3_xreal_0(k2_xcmplx_0(B,A)) ) ) )).

fof(fc11_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( ~ v1_xboole_0(k2_xcmplx_0(A,B))
        & v1_xcmplx_0(k2_xcmplx_0(A,B))
        & v1_xreal_0(k2_xcmplx_0(A,B))
        & ~ v2_xreal_0(k2_xcmplx_0(A,B))
        & v3_xreal_0(k2_xcmplx_0(A,B)) ) ) )).

fof(fc12_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( ~ v1_xboole_0(k2_xcmplx_0(B,A))
        & v1_xcmplx_0(k2_xcmplx_0(B,A))
        & v1_xreal_0(k2_xcmplx_0(B,A))
        & ~ v2_xreal_0(k2_xcmplx_0(B,A))
        & v3_xreal_0(k2_xcmplx_0(B,A)) ) ) )).

fof(fc13_xreal_0,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A) )
     => ( v1_xcmplx_0(k4_xcmplx_0(A))
        & v1_xreal_0(k4_xcmplx_0(A))
        & ~ v3_xreal_0(k4_xcmplx_0(A)) ) ) )).

fof(fc14_xreal_0,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ( v1_xcmplx_0(k4_xcmplx_0(A))
        & v1_xreal_0(k4_xcmplx_0(A))
        & ~ v2_xreal_0(k4_xcmplx_0(A)) ) ) )).

fof(fc15_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) )).

fof(fc16_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( v1_xcmplx_0(k6_xcmplx_0(B,A))
        & v1_xreal_0(k6_xcmplx_0(B,A))
        & ~ v2_xreal_0(k6_xcmplx_0(B,A)) ) ) )).

fof(fc17_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( ~ v1_xboole_0(k6_xcmplx_0(A,B))
        & v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & v2_xreal_0(k6_xcmplx_0(A,B))
        & ~ v3_xreal_0(k6_xcmplx_0(A,B)) ) ) )).

fof(fc18_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( ~ v1_xboole_0(k6_xcmplx_0(B,A))
        & v1_xcmplx_0(k6_xcmplx_0(B,A))
        & v1_xreal_0(k6_xcmplx_0(B,A))
        & ~ v2_xreal_0(k6_xcmplx_0(B,A))
        & v3_xreal_0(k6_xcmplx_0(B,A)) ) ) )).

fof(fc19_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( ~ v1_xboole_0(k6_xcmplx_0(A,B))
        & v1_xcmplx_0(k6_xcmplx_0(A,B))
        & v1_xreal_0(k6_xcmplx_0(A,B))
        & ~ v2_xreal_0(k6_xcmplx_0(A,B))
        & v3_xreal_0(k6_xcmplx_0(A,B)) ) ) )).

fof(fc20_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( ~ v1_xboole_0(k6_xcmplx_0(B,A))
        & v1_xcmplx_0(k6_xcmplx_0(B,A))
        & v1_xreal_0(k6_xcmplx_0(B,A))
        & v2_xreal_0(k6_xcmplx_0(B,A))
        & ~ v3_xreal_0(k6_xcmplx_0(B,A)) ) ) )).

fof(fc21_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & ~ v2_xreal_0(k3_xcmplx_0(A,B)) ) ) )).

fof(fc22_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(B,A))
        & v1_xreal_0(k3_xcmplx_0(B,A))
        & ~ v2_xreal_0(k3_xcmplx_0(B,A)) ) ) )).

fof(fc23_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) )).

fof(fc24_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( v1_xcmplx_0(k3_xcmplx_0(A,B))
        & v1_xreal_0(k3_xcmplx_0(A,B))
        & ~ v3_xreal_0(k3_xcmplx_0(A,B)) ) ) )).

fof(fc25_xreal_0,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A) )
     => ( v1_xcmplx_0(k5_xcmplx_0(A))
        & v1_xreal_0(k5_xcmplx_0(A))
        & ~ v2_xreal_0(k5_xcmplx_0(A)) ) ) )).

fof(fc26_xreal_0,axiom,(
    ! [A] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A) )
     => ( v1_xcmplx_0(k5_xcmplx_0(A))
        & v1_xreal_0(k5_xcmplx_0(A))
        & ~ v3_xreal_0(k5_xcmplx_0(A)) ) ) )).

fof(fc27_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( v1_xcmplx_0(k7_xcmplx_0(A,B))
        & v1_xreal_0(k7_xcmplx_0(A,B))
        & ~ v2_xreal_0(k7_xcmplx_0(A,B)) ) ) )).

fof(fc28_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( v1_xcmplx_0(k7_xcmplx_0(B,A))
        & v1_xreal_0(k7_xcmplx_0(B,A))
        & ~ v2_xreal_0(k7_xcmplx_0(B,A)) ) ) )).

fof(fc29_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v3_xreal_0(B) )
     => ( v1_xcmplx_0(k7_xcmplx_0(A,B))
        & v1_xreal_0(k7_xcmplx_0(A,B))
        & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) )).

fof(fc30_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & ~ v2_xreal_0(A)
        & v1_xreal_0(B)
        & ~ v2_xreal_0(B) )
     => ( v1_xcmplx_0(k7_xcmplx_0(A,B))
        & v1_xreal_0(k7_xcmplx_0(A,B))
        & ~ v3_xreal_0(k7_xcmplx_0(A,B)) ) ) )).

fof(d1_xreal_0,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
    <=> r2_hidden(A,k1_numbers) ) )).

fof(d2_xreal_0,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( ( ( r2_hidden(A,k2_arytm_2)
                & r2_hidden(B,k2_arytm_2) )
             => ( r1_xreal_0(A,B)
              <=> ? [C] :
                    ( m1_subset_1(C,k2_arytm_2)
                    & ? [D] :
                        ( m1_subset_1(D,k2_arytm_2)
                        & A = C
                        & B = D
                        & r1_arytm_2(C,D) ) ) ) )
            & ( ( r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2))
                & r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
             => ( r1_xreal_0(A,B)
              <=> ? [C] :
                    ( m1_subset_1(C,k2_arytm_2)
                    & ? [D] :
                        ( m1_subset_1(D,k2_arytm_2)
                        & A = k4_tarski(np__0,C)
                        & B = k4_tarski(np__0,D)
                        & r1_arytm_2(D,C) ) ) ) )
            & ~ ( ~ ( r2_hidden(A,k2_arytm_2)
                    & r2_hidden(B,k2_arytm_2) )
                & ~ ( r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2))
                    & r2_hidden(B,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) )
                & ~ ( r1_xreal_0(A,B)
                  <=> ( r2_hidden(B,k2_arytm_2)
                      & r2_hidden(A,k2_zfmisc_1(k1_tarski(np__0),k2_arytm_2)) ) ) ) ) ) ) )).

fof(d3_xreal_0,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v2_xreal_0(A)
      <=> ~ r1_xreal_0(A,np__0) ) ) )).

fof(d4_xreal_0,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v3_xreal_0(A)
      <=> ~ r1_xreal_0(np__0,A) ) ) )).

fof(reflexivity_r1_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => r1_xreal_0(A,A) ) )).

fof(connectedness_r1_xreal_0,axiom,(
    ! [A,B] :
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( r1_xreal_0(A,B)
        | r1_xreal_0(B,A) ) ) )).
%------------------------------------------------------------------------------