%------------------------------------------------------------------------------
% File     : SEU590^2 : TPTP v8.2.0. 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.subset C A -> subset C B -> subset C
%            (binintersect A B))

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC092l [Bro08]

% Status   : Theorem
% Rating   : 0.20 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.11 v7.2.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.29 v6.0.0, 0.43 v5.5.0, 0.33 v5.4.0, 0.40 v5.3.0, 0.60 v4.1.0, 0.67 v4.0.1, 1.00 v4.0.0, 0.67 v3.7.0
% Syntax   : Number of formulae    :   12 (   4 unt;   7 typ;   4 def)
%            Number of atoms       :   23 (   4 equ;   0 cnn)
%            Maximal formula atoms :    6 (   4 avg)
%            Number of connectives :   43 (   0   ~;   0   |;   0   &;  32   @)
%                                         (   0 <=>;  11  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   3 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    9 (   9   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    8 (   7 usr;   3 con; 0-2 aty)
%            Number of variables   :   15 (   3   ^;  12   !;   0   ?;  15   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=366
%------------------------------------------------------------------------------
thf(in_type,type,
    in: $i > $i > $o ).

thf(dsetconstr_type,type,
    dsetconstr: $i > ( $i > $o ) > $i ).

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(subsetE_type,type,
    subsetE: $o ).

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

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

thf(binintersect,definition,
    ( binintersect
    = ( ^ [A: $i,B: $i] :
          ( dsetconstr @ A
          @ ^ [Xx: $i] : ( in @ Xx @ B ) ) ) ) ).

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(binintersectSubset5,conjecture,
    ( subsetI1
   => ( subsetE
     => ( binintersectI
       => ! [A: $i,B: $i,C: $i] :
            ( ( subset @ C @ A )
           => ( ( subset @ C @ B )
             => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------