%------------------------------------------------------------------------------
% File     : SEU659^2 : TPTP v9.0.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Ordered Pairs - Properties of Pairs
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! B:i.! x:i.! y:i.in (kpair x y) (cartprod A B) -> in x A)

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC161l [Bro08]

% Status   : Theorem
% Rating   : 0.12 v9.0.0, 0.10 v8.2.0, 0.23 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.00 v6.1.0, 0.14 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v4.1.0, 0.00 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :    8 (   2 unt;   5 typ;   2 def)
%            Number of atoms       :   14 (   5 equ;   0 cnn)
%            Maximal formula atoms :    4 (   4 avg)
%            Number of connectives :   29 (   0   ~;   0   |;   2   &;  22   @)
%                                         (   0 <=>;   5  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   5 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    6 (   6   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   5 usr;   2 con; 0-2 aty)
%            Number of variables   :   13 (   0   ^;  11   !;   2   ?;  13   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=337
%------------------------------------------------------------------------------
thf(in_type,type,
    in: $i > $i > $o ).

thf(kpair_type,type,
    kpair: $i > $i > $i ).

thf(cartprod_type,type,
    cartprod: $i > $i > $i ).

thf(cartprodmempair1_type,type,
    cartprodmempair1: $o ).

thf(cartprodmempair1,definition,
    ( cartprodmempair1
    = ( ! [A: $i,B: $i,Xu: $i] :
          ( ( in @ Xu @ ( cartprod @ A @ B ) )
         => ? [Xx: $i] :
              ( ( in @ Xx @ A )
              & ? [Xy: $i] :
                  ( ( in @ Xy @ B )
                  & ( Xu
                    = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).

thf(setukpairinjL_type,type,
    setukpairinjL: $o ).

thf(setukpairinjL,definition,
    ( setukpairinjL
    = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
          ( ( ( kpair @ Xx @ Xy )
            = ( kpair @ Xz @ Xu ) )
         => ( Xx = Xz ) ) ) ) ).

thf(cartprodpairmemEL,conjecture,
    ( cartprodmempair1
   => ( setukpairinjL
     => ! [A: $i,B: $i,Xx: $i,Xy: $i] :
          ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
         => ( in @ Xx @ A ) ) ) ) ).

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