TPTP Problem File: SEU649^2.p

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% File     : SEU649^2 : TPTP v7.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Ordered Pairs - Properties of Pairs
% Version  : Especial > Reduced > Especial.
% English  : (! x:i.! y:i.x = y -> setadjoin x (setadjoin y emptyset) =
%            setadjoin x emptyset)

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC151l [Bro08]

% Status   : Theorem
% Rating   : 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.29 v6.0.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   14 (   0 unit;   8 type;   5 defn)
%            Number of atoms       :   76 (  12 equality;  33 variable)
%            Maximal formula depth :   13 (   5 average)
%            Number of connectives :   46 (   0   ~;   1   |;   0   &;  32   @)
%                                         (   0 <=>;  13  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    4 (   4   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   8   :;   0   =)
%            Number of variables   :   15 (   0 sgn;  15   !;   0   ?;   0   ^)
%                                         (  15   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=205
%          : 
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thf(in_type,type,(
    in: $i > $i > $o )).

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

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

thf(setext_type,type,(
    setext: $o )).

thf(setext,definition,
    ( setext
    = ( ! [A: $i,B: $i] :
          ( ! [Xx: $i] :
              ( ( in @ Xx @ A )
             => ( in @ Xx @ B ) )
         => ( ! [Xx: $i] :
                ( ( in @ Xx @ B )
               => ( in @ Xx @ A ) )
           => ( A = B ) ) ) ) )).

thf(setadjoinIL_type,type,(
    setadjoinIL: $o )).

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

thf(uniqinunit_type,type,(
    uniqinunit: $o )).

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

thf(eqinunit_type,type,(
    eqinunit: $o )).

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

thf(upairset2E_type,type,(
    upairset2E: $o )).

thf(upairset2E,definition,
    ( upairset2E
    = ( ! [Xx: $i,Xy: $i,Xz: $i] :
          ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
         => ( ( Xz = Xx )
            | ( Xz = Xy ) ) ) ) )).

thf(setukpairinjR11,conjecture,
    ( setext
   => ( setadjoinIL
     => ( uniqinunit
       => ( eqinunit
         => ( upairset2E
           => ! [Xx: $i,Xy: $i] :
                ( ( Xx = Xy )
               => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
                  = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) )).

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