## TPTP Problem File: SEU645^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU645^2 : 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.kfst (kpair x y) = x)

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC147l [Bro08]

% Status   : Theorem
% Rating   : 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v5.1.0, 0.60 v4.1.0, 0.67 v4.0.0, 1.00 v3.7.0
% Syntax   : Number of formulae    :   24 (   0 unit;  14 type;   9 defn)
%            Number of atoms       :  125 (  13 equality;  42 variable)
%            Maximal formula depth :   14 (   6 average)
%            Number of connectives :   89 (   0   ~;   0   |;   3   &;  77   @)
%                                         (   0 <=>;   9  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   14 (  14   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   16 (  14   :;   0   =)
%            Number of variables   :   23 (   0 sgn;  12   !;   3   ?;   8   ^)
%                                         (  23   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=202
%          :
%------------------------------------------------------------------------------
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(dsetconstr_type,type,(
dsetconstr: \$i > ( \$i > \$o ) > \$i )).

thf(dsetconstrER_type,type,(
dsetconstrER: \$o )).

thf(dsetconstrER,definition,
( dsetconstrER
= ( ! [A: \$i,Xphi: \$i > \$o,Xx: \$i] :
( ( in @ Xx
@ ( dsetconstr @ A
@ ^ [Xy: \$i] :
( Xphi @ Xy ) ) )
=> ( Xphi @ Xx ) ) ) )).

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

thf(kpair,definition,
( kpair
= ( ^ [Xx: \$i,Xy: \$i] :
( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) )).

thf(kpairp_type,type,(
kpairp: \$o )).

thf(kpairp,definition,
( kpairp
= ( ! [Xx: \$i,Xy: \$i] :
( iskpair @ ( kpair @ Xx @ Xy ) ) ) )).

thf(singleton_type,type,(
singleton: \$i > \$o )).

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

thf(setukpairinjL1_type,type,(
setukpairinjL1: \$o )).

thf(setukpairinjL1,definition,
( setukpairinjL1
= ( ! [Xx: \$i,Xy: \$i,Xz: \$i] :
( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
=> ( Xx = Xz ) ) ) )).

thf(kfstsingleton_type,type,(
kfstsingleton: \$o )).

thf(kfstsingleton,definition,
( kfstsingleton
= ( ! [Xu: \$i] :
( ( iskpair @ Xu )
=> ( singleton
@ ( dsetconstr @ ( setunion @ Xu )
@ ^ [Xx: \$i] :
( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) )).

thf(theprop_type,type,(
theprop: \$o )).

thf(theprop,definition,
( theprop
= ( ! [X: \$i] :
( ( singleton @ X )
=> ( in @ ( setunion @ X ) @ X ) ) ) )).

thf(kfst_type,type,(
kfst: \$i > \$i )).

thf(kfst,definition,
( kfst
= ( ^ [Xu: \$i] :
( setunion
@ ( dsetconstr @ ( setunion @ Xu )
@ ^ [Xx: \$i] :
( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) )).

thf(kfstpairEq,conjecture,
( dsetconstrER
=> ( kpairp
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ! [Xx: \$i,Xy: \$i] :
( ( kfst @ ( kpair @ Xx @ Xy ) )
= Xx ) ) ) ) ) )).

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