%------------------------------------------------------------------------------
% File     : SET948+1 : TPTP v9.0.0. Released v3.2.0.
% Domain   : Set theory
% Problem  : Basic properties of sets, theorem 101
% Version  : [Urb06] axioms : Especial.
% English  : ((in(C,union(A,B)) & in(D,union(A,B)) ) => (C=D | disjoint(C,D)) )
%            => union(intersection(A,B)) = intersection(union(A),union(B))

% Refs     : [Byl90] Bylinski (1990), Some Basic Properties of Sets
%          : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb06]
% Names    : zfmisc_1__t101_zfmisc_1 [Urb06]

% Status   : Theorem
% Rating   : 0.94 v9.0.0, 0.92 v8.2.0, 0.94 v7.4.0, 0.93 v7.3.0, 0.97 v7.2.0, 0.93 v7.1.0, 0.96 v7.0.0, 0.93 v6.4.0, 1.00 v3.2.0
% Syntax   : Number of formulae    :   19 (   8 unt;   0 def)
%            Number of atoms       :   43 (  10 equ)
%            Maximal formula atoms :    5 (   2 avg)
%            Number of connectives :   34 (  10   ~;   2   |;   7   &)
%                                         (   8 <=>;   7  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    8 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    5 (   4 usr;   0 prp; 1-2 aty)
%            Number of functors    :    3 (   3 usr;   0 con; 1-2 aty)
%            Number of variables   :   47 (  43   !;   4   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
%------------------------------------------------------------------------------
fof(antisymmetry_r2_hidden,axiom,
    ! [A,B] :
      ( in(A,B)
     => ~ in(B,A) ) ).

fof(commutativity_k2_xboole_0,axiom,
    ! [A,B] : set_union2(A,B) = set_union2(B,A) ).

fof(commutativity_k3_xboole_0,axiom,
    ! [A,B] : set_intersection2(A,B) = set_intersection2(B,A) ).

fof(d10_xboole_0,axiom,
    ! [A,B] :
      ( A = B
    <=> ( subset(A,B)
        & subset(B,A) ) ) ).

fof(d2_xboole_0,axiom,
    ! [A,B,C] :
      ( C = set_union2(A,B)
    <=> ! [D] :
          ( in(D,C)
        <=> ( in(D,A)
            | in(D,B) ) ) ) ).

fof(d3_tarski,axiom,
    ! [A,B] :
      ( subset(A,B)
    <=> ! [C] :
          ( in(C,A)
         => in(C,B) ) ) ).

fof(d3_xboole_0,axiom,
    ! [A,B,C] :
      ( C = set_intersection2(A,B)
    <=> ! [D] :
          ( in(D,C)
        <=> ( in(D,A)
            & in(D,B) ) ) ) ).

fof(d4_tarski,axiom,
    ! [A,B] :
      ( B = union(A)
    <=> ! [C] :
          ( in(C,B)
        <=> ? [D] :
              ( in(C,D)
              & in(D,A) ) ) ) ).

fof(fc2_xboole_0,axiom,
    ! [A,B] :
      ( ~ empty(A)
     => ~ empty(set_union2(A,B)) ) ).

fof(fc3_xboole_0,axiom,
    ! [A,B] :
      ( ~ empty(A)
     => ~ empty(set_union2(B,A)) ) ).

fof(idempotence_k2_xboole_0,axiom,
    ! [A,B] : set_union2(A,A) = A ).

fof(idempotence_k3_xboole_0,axiom,
    ! [A,B] : set_intersection2(A,A) = A ).

fof(rc1_xboole_0,axiom,
    ? [A] : empty(A) ).

fof(rc2_xboole_0,axiom,
    ? [A] : ~ empty(A) ).

fof(reflexivity_r1_tarski,axiom,
    ! [A,B] : subset(A,A) ).

fof(symmetry_r1_xboole_0,axiom,
    ! [A,B] :
      ( disjoint(A,B)
     => disjoint(B,A) ) ).

fof(t101_zfmisc_1,conjecture,
    ! [A,B] :
      ( ! [C,D] :
          ( ( in(C,set_union2(A,B))
            & in(D,set_union2(A,B)) )
         => ( C = D
            | disjoint(C,D) ) )
     => union(set_intersection2(A,B)) = set_intersection2(union(A),union(B)) ) ).

fof(t4_xboole_0,axiom,
    ! [A,B] :
      ( ~ ( ~ disjoint(A,B)
          & ! [C] : ~ in(C,set_intersection2(A,B)) )
      & ~ ( ? [C] : in(C,set_intersection2(A,B))
          & disjoint(A,B) ) ) ).

fof(t97_zfmisc_1,axiom,
    ! [A,B] : subset(union(set_intersection2(A,B)),set_intersection2(union(A),union(B))) ).

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