%------------------------------------------------------------------------------
% File     : SET960+1 : TPTP v8.2.0. Released v3.2.0.
% Domain   : Set theory
% Problem  : cart_prod(A,B) = empty <=> ( A = empty | B = empty )
% Version  : [Urb06] axioms : Especial.
% English  :

% Refs     : [Byl90] Bylinski (1990), Some Basic Properties of Sets
%          : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb06]
% Names    : zfmisc_1__t113_zfmisc_1 [Urb06]

% Status   : Theorem
% Rating   : 0.39 v8.1.0, 0.36 v7.5.0, 0.34 v7.4.0, 0.30 v7.3.0, 0.34 v7.2.0, 0.31 v7.1.0, 0.35 v7.0.0, 0.23 v6.4.0, 0.35 v6.3.0, 0.38 v6.2.0, 0.40 v6.1.0, 0.43 v6.0.0, 0.48 v5.5.0, 0.41 v5.4.0, 0.46 v5.3.0, 0.48 v5.2.0, 0.30 v5.1.0, 0.29 v5.0.0, 0.33 v4.1.0, 0.35 v4.0.1, 0.39 v4.0.0, 0.38 v3.7.0, 0.25 v3.5.0, 0.26 v3.4.0, 0.32 v3.3.0, 0.29 v3.2.0
% Syntax   : Number of formulae    :   10 (   6 unt;   0 def)
%            Number of atoms       :   18 (   8 equ)
%            Maximal formula atoms :    5 (   1 avg)
%            Number of connectives :   12 (   4   ~;   1   |;   2   &)
%                                         (   4 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   11 (   4 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    3 (   2 usr;   0 prp; 1-2 aty)
%            Number of functors    :    5 (   5 usr;   1 con; 0-2 aty)
%            Number of variables   :   20 (  16   !;   4   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
    ! [A,B] :
      ( in(A,B)
     => ~ in(B,A) ) ).

fof(commutativity_k2_tarski,axiom,
    ! [A,B] : unordered_pair(A,B) = unordered_pair(B,A) ).

fof(d1_xboole_0,axiom,
    ! [A] :
      ( A = empty_set
    <=> ! [B] : ~ in(B,A) ) ).

fof(d2_zfmisc_1,axiom,
    ! [A,B,C] :
      ( C = cartesian_product2(A,B)
    <=> ! [D] :
          ( in(D,C)
        <=> ? [E,F] :
              ( in(E,A)
              & in(F,B)
              & D = ordered_pair(E,F) ) ) ) ).

fof(d5_tarski,axiom,
    ! [A,B] : ordered_pair(A,B) = unordered_pair(unordered_pair(A,B),singleton(A)) ).

fof(fc1_xboole_0,axiom,
    empty(empty_set) ).

fof(fc1_zfmisc_1,axiom,
    ! [A,B] : ~ empty(ordered_pair(A,B)) ).

fof(rc1_xboole_0,axiom,
    ? [A] : empty(A) ).

fof(rc2_xboole_0,axiom,
    ? [A] : ~ empty(A) ).

fof(t113_zfmisc_1,conjecture,
    ! [A,B] :
      ( cartesian_product2(A,B) = empty_set
    <=> ( A = empty_set
        | B = empty_set ) ) ).

%------------------------------------------------------------------------------