%------------------------------------------------------------------------------
% File     : SET576^7 : TPTP v9.0.0. Released v5.5.0.
% Domain   : Set Theory
% Problem  : Trybulec's 17th Boolean property of sets
% Version  : [Ben12] axioms.
% English  :

% Refs     : [ILF] The ILF Group (1998), The ILF System: A Tool for the Int
%          : [Try90] Trybulec (1990), Tarski Grothendieck Set Theory
%          : [TS89]  Trybulec & Swieczkowska (1989), Boolean Properties of
%          : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe
% Source   : [Ben12]
% Names    : s4-cumul-SET576+3 [Ben12]

% Status   : Theorem
% Rating   : 0.12 v9.0.0, 0.10 v8.2.0, 0.23 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.33 v7.2.0, 0.25 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.14 v5.5.0
% Syntax   : Number of formulae    :   79 (  33 unt;  39 typ;  32 def)
%            Number of atoms       :  138 (  36 equ;   0 cnn)
%            Maximal formula atoms :   10 (   3 avg)
%            Number of connectives :  194 (   5   ~;   5   |;   9   &; 165   @)
%                                         (   0 <=>;  10  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   14 (   2 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :  189 ( 189   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   47 (  45 usr;   8 con; 0-3 aty)
%            Number of variables   :  100 (  59   ^;  34   !;   7   ?; 100   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : 
%------------------------------------------------------------------------------
%----Include axioms for Modal logic S4 under cumulative domains
include('Axioms/LCL015^0.ax').
include('Axioms/LCL013^5.ax').
include('Axioms/LCL015^1.ax').
%------------------------------------------------------------------------------
thf(intersect_type,type,
    intersect: mu > mu > $i > $o ).

thf(disjoint_type,type,
    disjoint: mu > mu > $i > $o ).

thf(member_type,type,
    member: mu > mu > $i > $o ).

thf(intersect_defn,axiom,
    ( mvalid
    @ ( mforall_ind
      @ ^ [B: mu] :
          ( mforall_ind
          @ ^ [C: mu] :
              ( mequiv @ ( intersect @ B @ C )
              @ ( mexists_ind
                @ ^ [D: mu] : ( mand @ ( member @ D @ B ) @ ( member @ D @ C ) ) ) ) ) ) ) ).

thf(disjoint_defn,axiom,
    ( mvalid
    @ ( mforall_ind
      @ ^ [B: mu] :
          ( mforall_ind
          @ ^ [C: mu] : ( mequiv @ ( disjoint @ B @ C ) @ ( mnot @ ( intersect @ B @ C ) ) ) ) ) ) ).

thf(symmetry_of_intersect,axiom,
    ( mvalid
    @ ( mforall_ind
      @ ^ [B: mu] :
          ( mforall_ind
          @ ^ [C: mu] : ( mimplies @ ( intersect @ B @ C ) @ ( intersect @ C @ B ) ) ) ) ) ).

thf(prove_th17,conjecture,
    ( mvalid
    @ ( mforall_ind
      @ ^ [B: mu] :
          ( mforall_ind
          @ ^ [C: mu] :
              ( mimplies
              @ ( mforall_ind
                @ ^ [D: mu] : ( mimplies @ ( member @ D @ B ) @ ( mnot @ ( member @ D @ C ) ) ) )
              @ ( disjoint @ B @ C ) ) ) ) ) ).

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