## TPTP Problem File: SEU649^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU649^2 : TPTP v7.1.0. 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   : 0.12 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
%          :
%------------------------------------------------------------------------------
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 ) ) ) ) ) ) ) )).

%------------------------------------------------------------------------------
```