## TPTP Problem File: SEU779^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU779^2 : TPTP v8.0.0. 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 -> (! x:i.in x A ->
%            (! y:i.in y A -> (! z:i.in z A -> in (kpair x z) R ->
%            in (kpair z y) S -> in (kpair x y) (breln1compset A R S))))))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC281l [Bro08]

% Status   : Theorem
% Rating   : 0.45 v7.5.0, 0.29 v7.4.0, 0.22 v7.2.0, 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 v5.2.0, 0.40 v4.1.0, 0.67 v4.0.0, 1.00 v3.7.0
% Syntax   : Number of formulae    :   14 (   4 unt;   9 typ;   4 def)
%            Number of atoms       :   24 (   4 equ;   0 cnn)
%            Maximal formula atoms :    9 (   4 avg)
%            Number of connectives :   73 (   0   ~;   0   |;   2   &;  60   @)
%                                         (   0 <=>;  11  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (   5 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :   23 (  23   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   10 (   9 usr;   1 con; 0-3 aty)
%            Number of variables   :   24 (  12   ^  11   !;   1   ?;  24   :)
% SPC      : TH0_THM_EQU_NAR

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

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

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

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

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

thf(dpsetconstrI_type,type,
dpsetconstrI: \$o ).

thf(dpsetconstrI,definition,
( dpsetconstrI
= ( ! [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 ) ) ) ) ) ) ) ) ).

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

thf(breln1,definition,
( breln1
= ( ^ [A: \$i,R: \$i] : ( breln @ A @ A @ R ) ) ) ).

thf(breln1compset_type,type,
breln1compset: \$i > \$i > \$i > \$i ).

thf(breln1compset,definition,
( breln1compset
= ( ^ [A: \$i,R: \$i,S: \$i] :
( dpsetconstr @ A @ A
@ ^ [Xx: \$i,Xy: \$i] :
? [Xz: \$i] :
( ( in @ Xz @ A )
& ( in @ ( kpair @ Xx @ Xz ) @ R )
& ( in @ ( kpair @ Xz @ Xy ) @ S ) ) ) ) ) ).

thf(breln1compI,conjecture,
( dpsetconstrI
=> ! [A: \$i,R: \$i] :
( ( breln1 @ A @ R )
=> ! [S: \$i] :
( ( breln1 @ A @ S )
=> ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ! [Xy: \$i] :
( ( in @ Xy @ A )
=> ! [Xz: \$i] :
( ( in @ Xz @ A )
=> ( ( in @ ( kpair @ Xx @ Xz ) @ R )
=> ( ( in @ ( kpair @ Xz @ Xy ) @ S )
=> ( in @ ( kpair @ Xx @ Xy ) @ ( breln1compset @ A @ R @ S ) ) ) ) ) ) ) ) ) ) ).

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