%------------------------------------------------------------------------------
% File     : SET943+1 : TPTP v9.0.0. Released v3.2.0.
% Domain   : Set theory
% Problem  : union(union(A,B)) = union(union(A),union(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__t96_zfmisc_1 [Urb06]

% Status   : Theorem
% Rating   : 0.82 v9.0.0, 0.81 v8.2.0, 0.89 v8.1.0, 0.83 v7.5.0, 0.84 v7.4.0, 0.80 v7.3.0, 0.76 v7.2.0, 0.72 v7.1.0, 0.83 v7.0.0, 0.90 v6.4.0, 0.85 v6.3.0, 0.83 v6.2.0, 0.92 v6.1.0, 0.97 v6.0.0, 0.96 v5.5.0, 0.93 v5.2.0, 0.90 v5.0.0, 0.92 v4.1.0, 0.87 v4.0.1, 0.91 v4.0.0, 0.92 v3.7.0, 0.90 v3.5.0, 0.95 v3.4.0, 1.00 v3.2.0
% Syntax   : Number of formulae    :   16 (   7 unt;   0 def)
%            Number of atoms       :   32 (   6 equ)
%            Maximal formula atoms :    4 (   2 avg)
%            Number of connectives :   22 (   6   ~;   1   |;   3   &)
%                                         (   6 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    8 (   5 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :    2 (   2 usr;   0 con; 1-2 aty)
%            Number of variables   :   36 (  33   !;   3   ?)
% 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(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_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(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(t7_xboole_1,axiom,
    ! [A,B] : subset(A,set_union2(A,B)) ).

fof(t8_xboole_1,axiom,
    ! [A,B,C] :
      ( ( subset(A,B)
        & subset(C,B) )
     => subset(set_union2(A,C),B) ) ).

fof(t95_zfmisc_1,axiom,
    ! [A,B] :
      ( subset(A,B)
     => subset(union(A),union(B)) ) ).

fof(t96_zfmisc_1,conjecture,
    ! [A,B] : union(set_union2(A,B)) = set_union2(union(A),union(B)) ).

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