%------------------------------------------------------------------------------
% File     : SET956+1 : TPTP v9.0.0. Released v3.2.0.
% Domain   : Set theory
% Problem  : Basic properties of sets, theorem 109
% Version  : [Urb06] axioms : Especial.
% English  : ( subset(A,cartesian_product2(B,C)) &
%            ( in(ordered_pair(E,F),A) => in(ordered_pair(E,F),D) ) )
%            => subset(A,D)

% Refs     : [Byl90] Bylinski (1990), Some Basic Properties of Sets
%          : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb06]
% Names    : zfmisc_1__t109_zfmisc_1 [Urb06]

% Status   : Theorem
% Rating   : 0.21 v9.0.0, 0.19 v8.2.0, 0.22 v8.1.0, 0.28 v7.4.0, 0.20 v7.3.0, 0.24 v7.2.0, 0.21 v7.1.0, 0.22 v7.0.0, 0.23 v6.4.0, 0.35 v6.3.0, 0.21 v6.2.0, 0.20 v6.1.0, 0.23 v6.0.0, 0.09 v5.5.0, 0.22 v5.4.0, 0.21 v5.3.0, 0.37 v5.2.0, 0.20 v5.1.0, 0.19 v5.0.0, 0.21 v4.1.0, 0.26 v4.0.0, 0.25 v3.7.0, 0.20 v3.5.0, 0.21 v3.4.0, 0.26 v3.3.0, 0.21 v3.2.0
% Syntax   : Number of formulae    :   10 (   6 unt;   0 def)
%            Number of atoms       :   20 (   3 equ)
%            Maximal formula atoms :    5 (   2 avg)
%            Number of connectives :   15 (   5   ~;   0   |;   5   &)
%                                         (   1 <=>;   4  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   13 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :    4 (   4 usr;   0 con; 1-2 aty)
%            Number of variables   :   27 (  25   !;   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(d3_tarski,axiom,
    ! [A,B] :
      ( subset(A,B)
    <=> ! [C] :
          ( in(C,A)
         => in(C,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(rc1_xboole_0,axiom,
    ? [A] : empty(A) ).

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

fof(reflexivity_r1_tarski,axiom,
    ! [A,B] : subset(A,A) ).

fof(t103_zfmisc_1,axiom,
    ! [A,B,C,D] :
      ~ ( subset(A,cartesian_product2(B,C))
        & in(D,A)
        & ! [E,F] :
            ~ ( in(E,B)
              & in(F,C)
              & D = ordered_pair(E,F) ) ) ).

fof(t109_zfmisc_1,conjecture,
    ! [A,B,C,D] :
      ( ( subset(A,cartesian_product2(B,C))
        & ! [E,F] :
            ( in(ordered_pair(E,F),A)
           => in(ordered_pair(E,F),D) ) )
     => subset(A,D) ) ).

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