SET007 Axioms: SET007+84.ax


%------------------------------------------------------------------------------
% File     : SET007+84 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Zermelo'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    : wellset1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   10 (   3 unit)
%            Number of atoms       :   43 (   6 equality)
%            Maximal formula depth :   15 (   6 average)
%            Number of connectives :   41 (   8 ~  ;   3  |;  17  &)
%                                         (   2 <=>;  11 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    8 (   1 propositional; 0-2 arity)
%            Number of functors    :    9 (   2 constant; 0-2 arity)
%            Number of variables   :   21 (   0 singleton;  19 !;   2 ?)
%            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(t1_wellset1,axiom,(
    ! [A,B] :
      ( v1_relat_1(B)
     => ( r2_hidden(A,k3_relat_1(B))
      <=> ~ ! [C] :
              ( ~ r2_hidden(k4_tarski(A,C),B)
              & ~ r2_hidden(k4_tarski(C,A),B) ) ) ) )).

fof(t2_wellset1,axiom,(
    $true )).

fof(t3_wellset1,axiom,(
    ! [A,B,C] :
      ( v1_relat_1(C)
     => ( C = k2_zfmisc_1(A,B)
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | k3_relat_1(C) = k2_xboole_0(A,B) ) ) ) )).

fof(t4_wellset1,axiom,(
    $true )).

fof(t5_wellset1,axiom,(
    $true )).

fof(t6_wellset1,axiom,(
    ! [A,B,C] :
      ( v1_relat_1(C)
     => ( ( r2_hidden(A,k3_relat_1(C))
          & r2_hidden(B,k3_relat_1(C))
          & v2_wellord1(C) )
       => ( r2_hidden(A,k1_wellord1(C,B))
          | r2_hidden(k4_tarski(B,A),C) ) ) ) )).

fof(t7_wellset1,axiom,(
    ! [A,B,C] :
      ( v1_relat_1(C)
     => ~ ( r2_hidden(A,k3_relat_1(C))
          & r2_hidden(B,k3_relat_1(C))
          & v2_wellord1(C)
          & r2_hidden(A,k1_wellord1(C,B))
          & r2_hidden(k4_tarski(B,A),C) ) ) )).

fof(t8_wellset1,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] :
          ~ ( ! [C] :
                ( r2_hidden(C,B)
               => ( ~ r2_hidden(k1_funct_1(A,C),C)
                  & r2_hidden(k1_funct_1(A,C),k3_tarski(B)) ) )
            & ! [C] :
                ( v1_relat_1(C)
               => ~ ( r1_tarski(k3_relat_1(C),k3_tarski(B))
                    & v2_wellord1(C)
                    & ~ r2_hidden(k3_relat_1(C),B)
                    & ! [D] :
                        ( r2_hidden(D,k3_relat_1(C))
                       => ( r2_hidden(k1_wellord1(C,D),B)
                          & k1_funct_1(A,k1_wellord1(C,D)) = D ) ) ) ) ) ) )).

fof(t9_wellset1,axiom,(
    ! [A] :
    ? [B] :
      ( v1_relat_1(B)
      & v2_wellord1(B)
      & k3_relat_1(B) = A ) )).

fof(s1_wellset1,axiom,(
    ? [A] :
    ! [B] :
      ( v1_relat_1(B)
     => ( r2_hidden(B,A)
      <=> ( r2_hidden(B,f1_s1_wellset1)
          & p1_s1_wellset1(B) ) ) ) )).
%------------------------------------------------------------------------------