## TPTP Problem File: SEU668^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU668^2 : TPTP v7.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Ordered Pairs - Sets of Pairs
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! B:i.! phi:i>(i>o).! x:i.in x A -> (! y:i.in y B ->
%            in (kpair x y) (dpsetconstr A B (^ z,u:i.phi z u)) -> phi x y))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC170l [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, 0.40 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 0.67 v4.0.1, 1.00 v4.0.0, 0.67 v3.7.0
% Syntax   : Number of formulae    :   13 (   0 unit;   8 type;   4 defn)
%            Number of atoms       :   76 (   9 equality;  44 variable)
%            Maximal formula depth :   18 (   7 average)
%            Number of connectives :   53 (   0   ~;   0   |;   3   &;  41   @)
%                                         (   0 <=>;   9  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   19 (  19   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   8   :;   0   =)
%            Number of variables   :   25 (   0 sgn;  16   !;   2   ?;   7   ^)
%                                         (  25   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

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

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

thf(cartprod_type,type,(
cartprod: \$i > \$i > \$i )).

thf(setukpairinjL_type,type,(
setukpairinjL: \$o )).

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

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(dpsetconstr_type,type,(
dpsetconstr: \$i > \$i > ( \$i > \$i > \$o ) > \$i )).

thf(dpsetconstr,definition,
( dpsetconstr
= ( ^ [A: \$i,B: \$i,Xphi: \$i > \$i > \$o] :
( dsetconstr @ ( cartprod @ A @ B )
@ ^ [Xu: \$i] :
? [Xx: \$i] :
( ( in @ Xx @ A )
& ? [Xy: \$i] :
( ( in @ Xy @ B )
& ( Xphi @ Xx @ Xy )
& ( Xu
= ( kpair @ Xx @ Xy ) ) ) ) ) ) )).

thf(dpsetconstrERa,conjecture,
( dsetconstrER
=> ( setukpairinjL
=> ( setukpairinjR
=> ! [A: \$i,B: \$i,Xphi: \$i > \$i > \$o,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: \$i,Xu: \$i] :
( Xphi @ Xz @ Xu ) ) )
=> ( Xphi @ Xx @ Xy ) ) ) ) ) ) )).

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