%------------------------------------------------------------------------------
% File     : SET986+1 : TPTP v9.0.0. Released v3.2.0.
% Domain   : Set theory
% Problem  : in(A,B) => union(difference(B,singleton(A)),singleton(A)) = 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__t140_zfmisc_1 [Urb06]

% Status   : Theorem
% Rating   : 0.67 v9.0.0, 0.64 v8.2.0, 0.67 v8.1.0, 0.64 v7.5.0, 0.69 v7.4.0, 0.60 v7.3.0, 0.59 v7.2.0, 0.55 v7.1.0, 0.61 v7.0.0, 0.67 v6.4.0, 0.65 v6.3.0, 0.75 v6.2.0, 0.76 v6.1.0, 0.80 v6.0.0, 0.78 v5.5.0, 0.85 v5.4.0, 0.86 v5.3.0, 0.89 v5.2.0, 0.90 v5.1.0, 0.86 v5.0.0, 0.79 v4.1.0, 0.83 v3.7.0, 0.80 v3.5.0, 0.84 v3.3.0, 0.86 v3.2.0
% Syntax   : Number of formulae    :   15 (   5 unt;   0 def)
%            Number of atoms       :   33 (   9 equ)
%            Maximal formula atoms :    4 (   2 avg)
%            Number of connectives :   26 (   8   ~;   1   |;   3   &)
%                                         (   9 <=>;   5  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   5 avg)
%            Maximal term depth    :    4 (   1 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :    3 (   3 usr;   0 con; 1-2 aty)
%            Number of variables   :   35 (  33   !;   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(d10_xboole_0,axiom,
    ! [A,B] :
      ( A = B
    <=> ( subset(A,B)
        & subset(B,A) ) ) ).

fof(d1_tarski,axiom,
    ! [A,B] :
      ( B = singleton(A)
    <=> ! [C] :
          ( in(C,B)
        <=> 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(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(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(t140_zfmisc_1,conjecture,
    ! [A,B] :
      ( in(A,B)
     => set_union2(set_difference(B,singleton(A)),singleton(A)) = B ) ).

fof(t64_zfmisc_1,axiom,
    ! [A,B,C] :
      ( in(A,set_difference(B,singleton(C)))
    <=> ( in(A,B)
        & A != C ) ) ).

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