%------------------------------------------------------------------------------
% File     : SEU131+2 : TPTP v8.2.0. Released v3.3.0.
% Domain   : Set theory
% Problem  : MPTP chainy problem l32_xboole_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    : chainy-l32_xboole_1 [Urb07]

% Status   : Theorem
% Rating   : 0.39 v7.5.0, 0.41 v7.4.0, 0.30 v7.3.0, 0.31 v7.2.0, 0.28 v7.1.0, 0.30 v7.0.0, 0.20 v6.4.0, 0.23 v6.3.0, 0.33 v6.2.0, 0.36 v6.1.0, 0.47 v6.0.0, 0.48 v5.4.0, 0.50 v5.3.0, 0.52 v5.2.0, 0.30 v5.1.0, 0.33 v4.1.0, 0.35 v4.0.1, 0.43 v4.0.0, 0.46 v3.7.0, 0.40 v3.5.0, 0.42 v3.4.0, 0.47 v3.3.0
% Syntax   : Number of formulae    :   44 (  19 unt;   0 def)
%            Number of atoms       :   88 (  21 equ)
%            Maximal formula atoms :    6 (   2 avg)
%            Number of connectives :   63 (  19   ~;   1   |;  17   &)
%                                         (  12 <=>;  14  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   4 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of predicates  :    6 (   4 usr;   1 prp; 0-2 aty)
%            Number of functors    :    4 (   4 usr;   1 con; 0-2 aty)
%            Number of variables   :   85 (  81   !;   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(d1_xboole_0,axiom,
    ! [A] :
      ( A = empty_set
    <=> ! [B] : ~ in(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_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(dt_k1_xboole_0,axiom,
    $true ).

fof(dt_k2_xboole_0,axiom,
    $true ).

fof(dt_k3_xboole_0,axiom,
    $true ).

fof(dt_k4_xboole_0,axiom,
    $true ).

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(l32_xboole_1,conjecture,
    ! [A,B] :
      ( set_difference(A,B) = empty_set
    <=> subset(A,B) ) ).

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

fof(t17_xboole_1,lemma,
    ! [A,B] : subset(set_intersection2(A,B),A) ).

fof(t19_xboole_1,lemma,
    ! [A,B,C] :
      ( ( subset(A,B)
        & subset(A,C) )
     => subset(A,set_intersection2(B,C)) ) ).

fof(t1_boole,axiom,
    ! [A] : set_union2(A,empty_set) = A ).

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

fof(t26_xboole_1,lemma,
    ! [A,B,C] :
      ( subset(A,B)
     => subset(set_intersection2(A,C),set_intersection2(B,C)) ) ).

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

fof(t2_boole,axiom,
    ! [A] : set_intersection2(A,empty_set) = empty_set ).

fof(t2_tarski,axiom,
    ! [A,B] :
      ( ! [C] :
          ( in(C,A)
        <=> in(C,B) )
     => A = B ) ).

fof(t2_xboole_1,lemma,
    ! [A] : subset(empty_set,A) ).

fof(t3_boole,axiom,
    ! [A] : set_difference(A,empty_set) = A ).

fof(t3_xboole_0,lemma,
    ! [A,B] :
      ( ~ ( ~ disjoint(A,B)
          & ! [C] :
              ~ ( in(C,A)
                & in(C,B) ) )
      & ~ ( ? [C] :
              ( in(C,A)
              & in(C,B) )
          & disjoint(A,B) ) ) ).

fof(t3_xboole_1,lemma,
    ! [A] :
      ( subset(A,empty_set)
     => A = empty_set ) ).

fof(t4_boole,axiom,
    ! [A] : set_difference(empty_set,A) = empty_set ).

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

fof(t6_boole,axiom,
    ! [A] :
      ( empty(A)
     => A = empty_set ) ).

fof(t7_boole,axiom,
    ! [A,B] :
      ~ ( in(A,B)
        & empty(B) ) ).

fof(t7_xboole_1,lemma,
    ! [A,B] : subset(A,set_union2(A,B)) ).

fof(t8_boole,axiom,
    ! [A,B] :
      ~ ( empty(A)
        & A != B
        & empty(B) ) ).

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

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