%------------------------------------------------------------------------------
% File     : SEV454^1 : TPTP v9.0.0. Released v7.0.0.
% Domain   : Analysis
% Problem  : DISJOINT_UNION
% Version  : Especial.
% English  :

% Refs     : [Kal16] Kalisyk (2016), Email to Geoff Sutcliffe
% Source   : [Kal16]
% Names    : DISJOINT_UNION_.p [Kal16]

% Status   : Theorem
% Rating   : 0.67 v9.0.0, 0.33 v8.1.0, 0.50 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax   : Number of formulae    :   12 (   6 unt;   5 typ;   0 def)
%            Number of atoms       :   29 (   9 equ;   0 cnn)
%            Maximal formula atoms :    2 (   4 avg)
%            Number of connectives :   61 (   1   ~;   1   |;   2   &;  57   @)
%                                         (   0 <=>;   0  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    7 (   5 avg)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :   30 (  30   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    6 (   5 usr;   0 con; 2-4 aty)
%            Number of variables   :   29 (   0   ^;  24   !;   0   ?;  29   :)
%                                         (   5  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : Exported from core HOL Light.
%------------------------------------------------------------------------------
thf('thf_const_const/sets/UNION',type,
    'const/sets/UNION': 
      !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > A > $o ) ).

thf('thf_const_const/sets/INTER',type,
    'const/sets/INTER': 
      !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > A > $o ) ).

thf('thf_const_const/sets/IN',type,
    'const/sets/IN': 
      !>[A: $tType] : ( A > ( A > $o ) > $o ) ).

thf('thf_const_const/sets/EMPTY',type,
    'const/sets/EMPTY': 
      !>[A: $tType] : ( A > $o ) ).

thf('thf_const_const/sets/DISJOINT',type,
    'const/sets/DISJOINT': 
      !>[A: $tType] : ( ( A > $o ) > ( A > $o ) > $o ) ).

thf('thm/sets/NOT_IN_EMPTY_',axiom,
    ! [A: $tType,A0: A] :
      ~ ( 'const/sets/IN' @ A @ A0 @ ( 'const/sets/EMPTY' @ A ) ) ).

thf('thm/sets/IN_',axiom,
    ! [A: $tType,P: A > $o,A0: A] :
      ( ( 'const/sets/IN' @ A @ A0 @ P )
      = ( P @ A0 ) ) ).

thf('thm/sets/IN_INTER_',axiom,
    ! [A: $tType,A0: A > $o,A1: A > $o,A2: A] :
      ( ( 'const/sets/IN' @ A @ A2 @ ( 'const/sets/INTER' @ A @ A0 @ A1 ) )
      = ( ( 'const/sets/IN' @ A @ A2 @ A0 )
        & ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ).

thf('thm/sets/IN_UNION_',axiom,
    ! [A: $tType,A0: A > $o,A1: A > $o,A2: A] :
      ( ( 'const/sets/IN' @ A @ A2 @ ( 'const/sets/UNION' @ A @ A0 @ A1 ) )
      = ( ( 'const/sets/IN' @ A @ A2 @ A0 )
        | ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ).

thf('thm/sets/EXTENSION_',axiom,
    ! [A: $tType,A0: A > $o,A1: A > $o] :
      ( ( A0 = A1 )
      = ( ! [A2: A] :
            ( ( 'const/sets/IN' @ A @ A2 @ A0 )
            = ( 'const/sets/IN' @ A @ A2 @ A1 ) ) ) ) ).

thf('thm/sets/DISJOINT_',axiom,
    ! [A: $tType,A0: A > $o,A1: A > $o] :
      ( ( 'const/sets/DISJOINT' @ A @ A0 @ A1 )
      = ( ( 'const/sets/INTER' @ A @ A0 @ A1 )
        = ( 'const/sets/EMPTY' @ A ) ) ) ).

thf('thm/sets/DISJOINT_UNION_',conjecture,
    ! [A: $tType,A0: A > $o,A1: A > $o,A2: A > $o] :
      ( ( 'const/sets/DISJOINT' @ A @ ( 'const/sets/UNION' @ A @ A0 @ A1 ) @ A2 )
      = ( ( 'const/sets/DISJOINT' @ A @ A0 @ A2 )
        & ( 'const/sets/DISJOINT' @ A @ A1 @ A2 ) ) ) ).

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