## TPTP Problem File: SEU681^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU681^2 : TPTP v7.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Functions - Extensionality and Beta Reduction
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! B:i.! R:i.breln A B R -> (! phi:i>o.(! x:i.in x A ->
%            (! y:i.in y B -> in (kpair x y) R -> phi (kpair x y))) ->
%            (! x:i.in x R -> phi x)))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC183l [Bro08]

% Status   : Theorem
% Rating   : 0.22 v7.2.0, 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, 0.14 v6.1.0, 0.29 v5.5.0, 0.17 v5.4.0, 0.20 v5.3.0, 0.40 v5.2.0, 0.20 v5.1.0, 0.40 v5.0.0, 0.20 v4.1.0, 0.00 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   11 (   0 unit;   7 type;   3 defn)
%            Number of atoms       :   62 (   4 equality;  36 variable)
%            Maximal formula depth :   18 (   7 average)
%            Number of connectives :   50 (   0   ~;   0   |;   2   &;  37   @)
%                                         (   0 <=>;  11  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   12 (  12   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   7   :;   0   =)
%            Number of variables   :   18 (   0 sgn;  13   !;   2   ?;   3   ^)
%                                         (  18   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

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

thf(subset_type,type,(
subset: \$i > \$i > \$o )).

thf(subsetE_type,type,(
subsetE: \$o )).

thf(subsetE,definition,
( subsetE
= ( ! [A: \$i,B: \$i,Xx: \$i] :
( ( subset @ A @ B )
=> ( ( in @ Xx @ A )
=> ( in @ Xx @ B ) ) ) ) )).

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

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

thf(cartprodmempair1_type,type,(
cartprodmempair1: \$o )).

thf(cartprodmempair1,definition,
( cartprodmempair1
= ( ! [A: \$i,B: \$i,Xu: \$i] :
( ( in @ Xu @ ( cartprod @ A @ B ) )
=> ? [Xx: \$i] :
( ( in @ Xx @ A )
& ? [Xy: \$i] :
( ( in @ Xy @ B )
& ( Xu
= ( kpair @ Xx @ Xy ) ) ) ) ) ) )).

thf(breln_type,type,(
breln: \$i > \$i > \$i > \$o )).

thf(breln,definition,
( breln
= ( ^ [A: \$i,B: \$i,C: \$i] :
( subset @ C @ ( cartprod @ A @ B ) ) ) )).

thf(brelnall1,conjecture,
( subsetE
=> ( cartprodmempair1
=> ! [A: \$i,B: \$i,R: \$i] :
( ( breln @ A @ B @ R )
=> ! [Xphi: \$i > \$o] :
( ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ B )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
=> ! [Xx: \$i] :
( ( in @ Xx @ R )
=> ( Xphi @ Xx ) ) ) ) ) )).

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