%------------------------------------------------------------------------------
% File     : SEU779^2 : TPTP v7.4.0. 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   : 0.29 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 (   0 unit;   9 type;   4 defn)
%            Number of atoms       :   86 (   4 equality;  51 variable)
%            Maximal formula depth :   19 (   7 average)
%            Number of connectives :   73 (   0   ~;   0   |;   2   &;  60   @)
%                                         (   0 <=>;  11  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   23 (  23   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   11 (   9   :;   0   =)
%            Number of variables   :   24 (   0 sgn;  11   !;   1   ?;  12   ^)
%                                         (  24   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=345
%          : 
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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 ) ) ) ) ) ) ) ) ) )).

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