## TPTP Problem File: SEU600^2.p

View Solutions - Solve Problem

```%------------------------------------------------------------------------------
% File     : SEU600^2 : TPTP v7.1.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Preliminary Notions - Ops on Sets - Unions and Intersections
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! B:i.! C:i.binintersect A (binunion B C) = binunion
%            (binintersect A B) (binintersect A C))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC102l [Bro08]

% Status   : Theorem
% Rating   : 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.86 v5.5.0, 0.83 v5.4.0, 0.80 v5.2.0, 1.00 v3.7.0
% Syntax   : Number of formulae    :   21 (   0 unit;  12 type;   8 defn)
%            Number of atoms       :  112 (  10 equality;  56 variable)
%            Maximal formula depth :   17 (   6 average)
%            Number of connectives :   83 (   0   ~;   0   |;   0   &;  60   @)
%                                         (   0 <=>;  23  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :    8 (   8   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   14 (  12   :;   0   =)
%            Number of variables   :   27 (   0 sgn;  27   !;   0   ?;   0   ^)
%                                         (  27   :;   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(subsetI1_type,type,(
subsetI1: \$o )).

thf(subsetI1,definition,
( subsetI1
= ( ! [A: \$i,B: \$i] :
( ! [Xx: \$i] :
( ( in @ Xx @ A )
=> ( in @ Xx @ B ) )
=> ( subset @ A @ B ) ) ) )).

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(binunionIL_type,type,(
binunionIL: \$o )).

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

thf(binunionIR_type,type,(
binunionIR: \$o )).

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

thf(binunionEcases_type,type,(
binunionEcases: \$o )).

thf(binunionEcases,definition,
( binunionEcases
= ( ! [A: \$i,B: \$i,Xx: \$i,Xphi: \$o] :
( ( in @ Xx @ ( binunion @ A @ B ) )
=> ( ( ( in @ Xx @ A )
=> Xphi )
=> ( ( ( in @ Xx @ B )
=> Xphi )
=> Xphi ) ) ) ) )).

thf(binintersect_type,type,(
binintersect: \$i > \$i > \$i )).

thf(binintersectI_type,type,(
binintersectI: \$o )).

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

thf(binintersectEL_type,type,(
binintersectEL: \$o )).

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

thf(binintersectER_type,type,(
binintersectER: \$o )).

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

thf(bs114d,conjecture,
( subsetI1
=> ( setextsub
=> ( binunionIL
=> ( binunionIR
=> ( binunionEcases
=> ( binintersectI
=> ( binintersectEL
=> ( binintersectER
=> ! [A: \$i,B: \$i,C: \$i] :
( ( binintersect @ A @ ( binunion @ B @ C ) )
= ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ) ) ) ) ) )).

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