## TPTP Problem File: SEU627^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU627^2 : TPTP v7.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Ordered Pairs - Cartesian Products
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! B:i.! x:i.in x A -> (! y:i.in y B -> subset (setadjoin
%            (setadjoin y emptyset)) emptyset)) (powerset (binunion A B))))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC129l [Bro08]

% Status   : Theorem
% Rating   : 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.29 v6.0.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% Syntax   : Number of formulae    :   15 (   0 unit;  10 type;   4 defn)
%            Number of atoms       :   83 (   6 equality;  35 variable)
%            Maximal formula depth :   19 (   6 average)
%            Number of connectives :   66 (   0   ~;   1   |;   0   &;  53   @)
%                                         (   0 <=>;  12  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    9 (   9   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   12 (  10   :;   0   =)
%            Number of variables   :   17 (   0 sgn;  17   !;   0   ?;   0   ^)
%                                         (  17   :;   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(powerset_type,type,(
powerset: \$i > \$i )).

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

thf(subsetI2_type,type,(
subsetI2: \$o )).

thf(subsetI2,definition,
( subsetI2
= ( ! [A: \$i,B: \$i] :
( ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( subset @ A @ B ) ) ) )).

thf(binunion_type,type,(
binunion: \$i > \$i > \$i )).

thf(singletoninpowunion_type,type,(
singletoninpowunion: \$o )).

thf(singletoninpowunion,definition,
( singletoninpowunion
= ( ! [A: \$i,B: \$i,Xx: \$i] :
( ( in @ Xx @ A )
=> ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) )).

thf(upairset2E_type,type,(
upairset2E: \$o )).

thf(upairset2E,definition,
( upairset2E
= ( ! [Xx: \$i,Xy: \$i,Xz: \$i] :
( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
=> ( ( Xz = Xx )
| ( Xz = Xy ) ) ) ) )).

thf(upairinpowunion_type,type,(
upairinpowunion: \$o )).

thf(upairinpowunion,definition,
( upairinpowunion
= ( ! [A: \$i,B: \$i,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ B )
=> ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) )).

thf(ubforcartprodlem1,conjecture,
( subsetI2
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairinpowunion
=> ! [A: \$i,B: \$i,Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ B )
=> ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) )).

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