## TPTP Problem File: SEU788^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU788^2 : TPTP v7.2.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Binary Relations on a Set
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! R:i.breln1 A R -> (! S:i.breln1 A S ->
%            binunion R S = binunion S R))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC290l [Bro08]

% Status   : Theorem
% Rating   : 0.56 v7.2.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% Syntax   : Number of formulae    :   18 (   0 unit;  11 type;   6 defn)
%            Number of atoms       :  153 (   8 equality;  85 variable)
%            Maximal formula depth :   17 (   7 average)
%            Number of connectives :  130 (   0   ~;   1   |;   0   &;  96   @)
%                                         (   0 <=>;  33  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   10 (  10   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   13 (  11   :;   0   =)
%            Number of variables   :   28 (   0 sgn;  28   !;   0   ?;   0   ^)
%                                         (  28   :;   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(setextsub_type,type,(
setextsub: \$o )).

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

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

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

thf(breln1_type,type,(
breln1: \$i > \$i > \$o )).

thf(subbreln1_type,type,(
subbreln1: \$o )).

thf(subbreln1,definition,
( subbreln1
= ( ! [A: \$i,R: \$i] :
( ( breln1 @ A @ R )
=> ! [S: \$i] :
( ( breln1 @ A @ S )
=> ( ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
=> ( subset @ R @ S ) ) ) ) ) )).

thf(breln1unionprop_type,type,(
breln1unionprop: \$o )).

thf(breln1unionprop,definition,
( breln1unionprop
= ( ! [A: \$i,R: \$i] :
( ( breln1 @ A @ R )
=> ! [S: \$i] :
( ( breln1 @ A @ S )
=> ( breln1 @ A @ ( binunion @ R @ S ) ) ) ) ) )).

thf(breln1unionIL_type,type,(
breln1unionIL: \$o )).

thf(breln1unionIL,definition,
( breln1unionIL
= ( ! [A: \$i,R: \$i] :
( ( breln1 @ A @ R )
=> ! [S: \$i] :
( ( breln1 @ A @ S )
=> ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) )).

thf(breln1unionIR_type,type,(
breln1unionIR: \$o )).

thf(breln1unionIR,definition,
( breln1unionIR
= ( ! [A: \$i,R: \$i] :
( ( breln1 @ A @ R )
=> ! [S: \$i] :
( ( breln1 @ A @ S )
=> ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ S )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) ) ) ) ) ) ) ) )).

thf(breln1unionE_type,type,(
breln1unionE: \$o )).

thf(breln1unionE,definition,
( breln1unionE
= ( ! [A: \$i,R: \$i] :
( ( breln1 @ A @ R )
=> ! [S: \$i] :
( ( breln1 @ A @ S )
=> ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ ( binunion @ R @ S ) )
=> ( ( in @ ( kpair @ Xx @ Xy ) @ R )
| ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) ) ) ) ) ) )).

thf(breln1unionCommutes,conjecture,
( setextsub
=> ( subbreln1
=> ( breln1unionprop
=> ( breln1unionIL
=> ( breln1unionIR
=> ( breln1unionE
=> ! [A: \$i,R: \$i] :
( ( breln1 @ A @ R )
=> ! [S: \$i] :
( ( breln1 @ A @ S )
=> ( ( binunion @ R @ S )
= ( binunion @ S @ R ) ) ) ) ) ) ) ) ) )).

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