## TPTP Problem File: SEU655^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU655^2 : TPTP v7.1.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Ordered Pairs - Properties of Pairs
% Version  : Especial > Reduced > Especial.
% English  : (! u:i.iskpair u -> singleton (dsetconstr (setunion u)
%            (^ x:i.u = kpair (kfst u) x)))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC157l [Bro08]

% Status   : Theorem
% Rating   : 0.50 v7.1.0, 0.62 v7.0.0, 0.71 v6.4.0, 0.83 v6.3.0, 0.80 v6.2.0, 0.71 v6.1.0, 0.57 v5.5.0, 0.67 v5.4.0, 1.00 v5.2.0, 0.80 v5.0.0, 1.00 v3.7.0
% Syntax   : Number of formulae    :   21 (   0 unit;  13 type;   7 defn)
%            Number of atoms       :  107 (  14 equality;  45 variable)
%            Maximal formula depth :   14 (   6 average)
%            Number of connectives :   71 (   0   ~;   0   |;   3   &;  58   @)
%                                         (   0 <=>;  10  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   18 (  18   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   15 (  13   :;   0   =)
%            Number of variables   :   23 (   0 sgn;  11   !;   3   ?;   9   ^)
%                                         (  23   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

%          :
%------------------------------------------------------------------------------
thf(in_type,type,(
in: \$i > \$i > \$o )).

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

setadjoin: \$i > \$i > \$i )).

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

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

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

thf(ex1_type,type,(
ex1: \$i > ( \$i > \$o ) > \$o )).

thf(ex1,definition,
( ex1
= ( ^ [A: \$i,Xphi: \$i > \$o] :
( singleton
@ ( dsetconstr @ A
@ ^ [Xx: \$i] :
( Xphi @ Xx ) ) ) ) )).

thf(ex1I_type,type,(
ex1I: \$o )).

thf(ex1I,definition,
( ex1I
= ( ! [A: \$i,Xphi: \$i > \$o,Xx: \$i] :
( ( in @ Xx @ A )
=> ( ( Xphi @ Xx )
=> ( ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( ( Xphi @ Xy )
=> ( Xy = Xx ) ) )
=> ( ex1 @ A
@ ^ [Xy: \$i] :
( Xphi @ Xy ) ) ) ) ) ) )).

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

thf(kfstpairEq_type,type,(
kfstpairEq: \$o )).

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

thf(setukpairinjR_type,type,(
setukpairinjR: \$o )).

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

thf(ksndsingleton,conjecture,
( ex1I
=> ( kfstpairEq
=> ( setukpairinjR
=> ! [Xu: \$i] :
( ( iskpair @ Xu )
=> ( singleton
@ ( dsetconstr @ ( setunion @ Xu )
@ ^ [Xx: \$i] :
( Xu
= ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) )).

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