%------------------------------------------------------------------------------
% File     : SEU619^2 : TPTP v8.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Ordered Pairs - Kuratowski Pairs
% Version  : Especial > Reduced > Especial.
% English  : (! x:i.! y:i.iskpair (setadjoin (setadjoin x emptyset) (setadjoin
%            (setadjoin x (setadjoin y emptyset)) emptyset)))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC121l [Bro08]

% Status   : Theorem
% Rating   : 0.00 v8.2.0, 0.08 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 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 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.00 v3.7.0
% Syntax   : Number of formulae    :   11 (   3 unt;   7 typ;   3 def)
%            Number of atoms       :   14 (   4 equ;   0 cnn)
%            Maximal formula atoms :    3 (   3 avg)
%            Number of connectives :   57 (   0   ~;   0   |;   2   &;  53   @)
%                                         (   0 <=>;   2  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   4 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    6 (   6   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    8 (   7 usr;   3 con; 0-2 aty)
%            Number of variables   :    9 (   1   ^;   6   !;   2   ?;   9   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=174
%------------------------------------------------------------------------------
thf(in_type,type,
    in: $i > $i > $o ).

thf(emptyset_type,type,
    emptyset: $i ).

thf(setadjoin_type,type,
    setadjoin: $i > $i > $i ).

thf(setunion_type,type,
    setunion: $i > $i ).

thf(iskpair_type,type,
    iskpair: $i > $o ).

thf(iskpair,definition,
    ( iskpair
    = ( ^ [A: $i] :
        ? [Xx: $i] :
          ( ( in @ Xx @ ( setunion @ A ) )
          & ? [Xy: $i] :
              ( ( in @ Xy @ ( setunion @ A ) )
              & ( A
                = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ) ).

thf(setukpairIL_type,type,
    setukpairIL: $o ).

thf(setukpairIL,definition,
    ( setukpairIL
    = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).

thf(setukpairIR_type,type,
    setukpairIR: $o ).

thf(setukpairIR,definition,
    ( setukpairIR
    = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).

thf(kpairiskpair,conjecture,
    ( setukpairIL
   => ( setukpairIR
     => ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).

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