%------------------------------------------------------------------------------
% File     : SET937+1 : TPTP v9.0.0. Released v3.2.0.
% Domain   : Set theory
% Problem  : subset(pset(diff(A,B)),union(sgtn(empty),diff(pset(A),pset(B))))
% Version  : [Urb06] axioms : Especial.
% English  :

% Refs     : [Byl90] Bylinski (1990), Some Basic Properties of Sets
%          : [Urb06] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb06]
% Names    : zfmisc_1__t84_zfmisc_1 [Urb06]

% Status   : Theorem
% Rating   : 0.73 v9.0.0, 0.69 v8.2.0, 0.75 v8.1.0, 0.69 v7.5.0, 0.84 v7.4.0, 0.77 v7.3.0, 0.83 v6.4.0, 0.88 v6.3.0, 0.92 v6.2.0, 0.96 v6.1.0, 1.00 v6.0.0, 0.96 v5.5.0, 0.93 v5.4.0, 0.96 v5.2.0, 1.00 v5.0.0, 0.96 v4.1.0, 0.91 v4.0.1, 1.00 v4.0.0, 0.96 v3.7.0, 0.95 v3.3.0, 0.93 v3.2.0
% Syntax   : Number of formulae    :   24 (  11 unt;   0 def)
%            Number of atoms       :   46 (  11 equ)
%            Maximal formula atoms :    4 (   1 avg)
%            Number of connectives :   29 (   7   ~;   1   |;   3   &)
%                                         (  10 <=>;   8  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   4 avg)
%            Maximal term depth    :    4 (   1 avg)
%            Number of predicates  :    5 (   4 usr;   0 prp; 1-2 aty)
%            Number of functors    :    6 (   6 usr;   1 con; 0-2 aty)
%            Number of variables   :   53 (  51   !;   2   ?)
% 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(d1_tarski,axiom,
    ! [A,B] :
      ( B = singleton(A)
    <=> ! [C] :
          ( in(C,B)
        <=> C = A ) ) ).

fof(d1_zfmisc_1,axiom,
    ! [A,B] :
      ( B = powerset(A)
    <=> ! [C] :
          ( in(C,B)
        <=> subset(C,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(d4_xboole_0,axiom,
    ! [A,B,C] :
      ( C = set_difference(A,B)
    <=> ! [D] :
          ( in(D,C)
        <=> ( in(D,A)
            & ~ in(D,B) ) ) ) ).

fof(d7_xboole_0,axiom,
    ! [A,B] :
      ( disjoint(A,B)
    <=> set_intersection2(A,B) = empty_set ) ).

fof(fc1_xboole_0,axiom,
    empty(empty_set) ).

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(t1_xboole_1,axiom,
    ! [A,B,C] :
      ( ( subset(A,B)
        & subset(B,C) )
     => subset(A,C) ) ).

fof(t28_xboole_1,axiom,
    ! [A,B] :
      ( subset(A,B)
     => set_intersection2(A,B) = A ) ).

fof(t36_xboole_1,axiom,
    ! [A,B] : subset(set_difference(A,B),A) ).

fof(t63_xboole_1,axiom,
    ! [A,B,C] :
      ( ( subset(A,B)
        & disjoint(B,C) )
     => disjoint(A,C) ) ).

fof(t79_xboole_1,axiom,
    ! [A,B] : disjoint(set_difference(A,B),B) ).

fof(t84_zfmisc_1,conjecture,
    ! [A,B] : subset(powerset(set_difference(A,B)),set_union2(singleton(empty_set),set_difference(powerset(A),powerset(B)))) ).

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