## TPTP Problem File: SEU665^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU665^2 : TPTP v8.0.0. 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 ->
%            phi x y -> in (kpair x y) (dpsetconstr A B (^ z,u:i.phi z u))))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC167l [Bro08]

% Status   : Theorem
% Rating   : 0.45 v7.5.0, 0.29 v7.4.0, 0.11 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0, 0.60 v4.1.0, 1.00 v3.7.0
% Syntax   : Number of formulae    :   13 (   4 unt;   8 typ;   4 def)
%            Number of atoms       :   26 (   6 equ;   0 cnn)
%            Maximal formula atoms :    6 (   5 avg)
%            Number of connectives :   70 (   0   ~;   0   |;   3   &;  53   @)
%                                         (   0 <=>;  14  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   4 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :   19 (  19   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   8 usr;   3 con; 0-3 aty)
%            Number of variables   :   25 (   7   ^  16   !;   2   ?;  25   :)
% SPC      : TH0_THM_EQU_NAR

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

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

thf(dsetconstrI_type,type,
dsetconstrI: \$o ).

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

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

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

thf(cartprodpairin_type,type,
cartprodpairin: \$o ).

thf(cartprodpairin,definition,
( cartprodpairin
= ( ! [A: \$i,B: \$i,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ B )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).

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(dpsetconstrI,conjecture,
( dsetconstrI
=> ( setext
=> ( cartprodpairin
=> ! [A: \$i,B: \$i,Xphi: \$i > \$i > \$o,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ B )
=> ( ( Xphi @ Xx @ Xy )
=> ( in @ ( kpair @ Xx @ Xy )
@ ( dpsetconstr @ A @ B
@ ^ [Xz: \$i,Xu: \$i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ) ).

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