%------------------------------------------------------------------------------
% File     : SET967+1 : TPTP v8.2.0. Released v3.2.0.
% Domain   : Set theory
% Problem  : Basic properties of sets, theorem 120
% 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__t120_zfmisc_1 [Urb06]

% Status   : Theorem
% Rating   : 0.97 v8.2.0, 1.00 v7.4.0, 0.90 v7.3.0, 0.86 v7.1.0, 0.83 v7.0.0, 0.93 v6.4.0, 0.92 v6.3.0, 0.96 v6.2.0, 1.00 v3.2.0
% Syntax   : Number of formulae    :   16 (   7 unt;   0 def)
%            Number of atoms       :   33 (  11 equ)
%            Maximal formula atoms :    7 (   2 avg)
%            Number of connectives :   30 (  13   ~;   1   |;   7   &)
%                                         (   4 <=>;   5  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    3 (   2 usr;   0 prp; 1-2 aty)
%            Number of functors    :    5 (   5 usr;   0 con; 1-2 aty)
%            Number of variables   :   48 (  46   !;   2   ?)
% 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(commutativity_k2_xboole_0,axiom,
    ! [A,B] : set_union2(A,B) = set_union2(B,A) ).

fof(d2_xboole_0,axiom,
    ! [A,B,C] :
      ( C = set_union2(A,B)
    <=> ! [D] :
          ( in(D,C)
        <=> ( in(D,A)
            | in(D,B) ) ) ) ).

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

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

fof(fc2_xboole_0,axiom,
    ! [A,B] :
      ( ~ empty(A)
     => ~ empty(set_union2(A,B)) ) ).

fof(fc3_xboole_0,axiom,
    ! [A,B] :
      ( ~ empty(A)
     => ~ empty(set_union2(B,A)) ) ).

fof(idempotence_k2_xboole_0,axiom,
    ! [A,B] : set_union2(A,A) = A ).

fof(l55_zfmisc_1,axiom,
    ! [A,B,C,D] :
      ( in(ordered_pair(A,B),cartesian_product2(C,D))
    <=> ( in(A,C)
        & in(B,D) ) ) ).

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

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

fof(t102_zfmisc_1,axiom,
    ! [A,B,C] :
      ~ ( in(A,cartesian_product2(B,C))
        & ! [D,E] : ordered_pair(D,E) != A ) ).

fof(t107_zfmisc_1,axiom,
    ! [A,B,C,D] :
      ( in(ordered_pair(A,B),cartesian_product2(C,D))
     => in(ordered_pair(B,A),cartesian_product2(D,C)) ) ).

fof(t112_zfmisc_1,axiom,
    ! [A,B] :
      ( ( ! [C] :
            ~ ( in(C,A)
              & ! [D,E] : C != ordered_pair(D,E) )
        & ! [C] :
            ~ ( in(C,B)
              & ! [D,E] : C != ordered_pair(D,E) )
        & ! [C,D] :
            ( in(ordered_pair(C,D),A)
          <=> in(ordered_pair(C,D),B) ) )
     => A = B ) ).

fof(t120_zfmisc_1,conjecture,
    ! [A,B,C] :
      ( cartesian_product2(set_union2(A,B),C) = set_union2(cartesian_product2(A,C),cartesian_product2(B,C))
      & cartesian_product2(C,set_union2(A,B)) = set_union2(cartesian_product2(C,A),cartesian_product2(C,B)) ) ).

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