%------------------------------------------------------------------------------
% File     : SEU152+1 : TPTP v9.0.0. Released v3.3.0.
% Domain   : Set theory
% Problem  : MPTP bushy problem l23_zfmisc_1
% Version  : [Urb07] axioms : Especial.
% English  :

% Refs     : [Ban01] Bancerek et al. (2001), On the Characterizations of Co
%          : [Urb07] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb07]
% Names    : bushy-l23_zfmisc_1 [Urb07]

% Status   : Theorem
% Rating   : 0.03 v7.4.0, 0.00 v6.4.0, 0.04 v6.2.0, 0.08 v6.1.0, 0.10 v6.0.0, 0.09 v5.5.0, 0.04 v5.4.0, 0.07 v5.3.0, 0.15 v5.2.0, 0.05 v5.0.0, 0.04 v4.0.0, 0.08 v3.7.0, 0.10 v3.5.0, 0.11 v3.3.0
% Syntax   : Number of formulae    :    9 (   5 unt;   0 def)
%            Number of atoms       :   13 (   4 equ)
%            Maximal formula atoms :    2 (   1 avg)
%            Number of connectives :    5 (   1   ~;   0   |;   0   &)
%                                         (   1 <=>;   3  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    5 (   3 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    4 (   2 usr;   1 prp; 0-2 aty)
%            Number of functors    :    2 (   2 usr;   0 con; 1-2 aty)
%            Number of variables   :   14 (  14   !;   0   ?)
% 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(dt_k1_tarski,axiom,
    $true ).

fof(dt_k2_xboole_0,axiom,
    $true ).

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

fof(l23_zfmisc_1,conjecture,
    ! [A,B] :
      ( in(A,B)
     => set_union2(singleton(A),B) = B ) ).

fof(l2_zfmisc_1,axiom,
    ! [A,B] :
      ( subset(singleton(A),B)
    <=> in(A,B) ) ).

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

fof(t12_xboole_1,axiom,
    ! [A,B] :
      ( subset(A,B)
     => set_union2(A,B) = B ) ).

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