## TPTP Problem File: SEU657^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU657^2 : 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 -> kpair (kfst u) (ksnd u) = u)

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC159l [Bro08]

% Status   : Theorem
% Rating   : 0.12 v7.1.0, 0.25 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   15 (   0 unit;  10 type;   4 defn)
%            Number of atoms       :   63 (   8 equality;  21 variable)
%            Maximal formula depth :   14 (   5 average)
%            Number of connectives :   42 (   0   ~;   0   |;   2   &;  37   @)
%                                         (   0 <=>;   3  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   10 (  10   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   12 (  10   :;   0   =)
%            Number of variables   :   10 (   0 sgn;   5   !;   2   ?;   3   ^)
%                                         (  10   :;   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(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(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(ksnd_type,type,(
ksnd: \$i > \$i )).

thf(ksndpairEq_type,type,(
ksndpairEq: \$o )).

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

thf(kpairsurjEq,conjecture,
( kfstpairEq
=> ( ksndpairEq
=> ! [Xu: \$i] :
( ( iskpair @ Xu )
=> ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
= Xu ) ) ) )).

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