%------------------------------------------------------------------------------
% File     : SEU245+1 : TPTP v9.0.0. Released v3.3.0.
% Domain   : Set theory
% Problem  : MPTP bushy problem t16_wellord1
% 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-t16_wellord1 [Urb07]

% Status   : Theorem
% Rating   : 0.21 v9.0.0, 0.25 v8.2.0, 0.22 v7.5.0, 0.25 v7.4.0, 0.10 v7.3.0, 0.14 v7.2.0, 0.10 v7.1.0, 0.04 v7.0.0, 0.13 v6.4.0, 0.23 v6.3.0, 0.21 v6.2.0, 0.24 v6.1.0, 0.37 v6.0.0, 0.30 v5.4.0, 0.32 v5.3.0, 0.41 v5.2.0, 0.20 v5.1.0, 0.24 v5.0.0, 0.25 v4.1.0, 0.30 v4.0.0, 0.33 v3.7.0, 0.30 v3.5.0, 0.32 v3.4.0, 0.42 v3.3.0
% Syntax   : Number of formulae    :   26 (  11 unt;   0 def)
%            Number of atoms       :   53 (   7 equ)
%            Maximal formula atoms :    6 (   2 avg)
%            Number of connectives :   32 (   5   ~;   1   |;  14   &)
%                                         (   3 <=>;   9  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    8 (   4 avg)
%            Maximal term depth    :    3 (   1 avg)
%            Number of predicates  :    8 (   6 usr;   1 prp; 0-2 aty)
%            Number of functors    :    4 (   4 usr;   1 con; 0-2 aty)
%            Number of variables   :   36 (  30   !;   6   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments : Translated by MPTP 0.2 from the original problem in the Mizar
%            library, www.mizar.org
%------------------------------------------------------------------------------
fof(rc3_funct_1,axiom,
    ? [A] :
      ( relation(A)
      & function(A)
      & one_to_one(A) ) ).

fof(dt_k1_xboole_0,axiom,
    $true ).

fof(rc1_funct_1,axiom,
    ? [A] :
      ( relation(A)
      & function(A) ) ).

fof(rc2_funct_1,axiom,
    ? [A] :
      ( relation(A)
      & empty(A)
      & function(A) ) ).

fof(cc2_funct_1,axiom,
    ! [A] :
      ( ( relation(A)
        & empty(A)
        & function(A) )
     => ( relation(A)
        & function(A)
        & one_to_one(A) ) ) ).

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

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

fof(existence_m1_subset_1,axiom,
    ! [A] :
    ? [B] : element(B,A) ).

fof(dt_m1_subset_1,axiom,
    $true ).

fof(cc1_funct_1,axiom,
    ! [A] :
      ( empty(A)
     => function(A) ) ).

fof(rc1_xboole_0,axiom,
    ? [A] : empty(A) ).

fof(rc2_xboole_0,axiom,
    ? [A] : ~ empty(A) ).

fof(t2_subset,axiom,
    ! [A,B] :
      ( element(A,B)
     => ( empty(B)
        | in(A,B) ) ) ).

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

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

fof(commutativity_k3_xboole_0,axiom,
    ! [A,B] : set_intersection2(A,B) = set_intersection2(B,A) ).

fof(idempotence_k3_xboole_0,axiom,
    ! [A,B] : set_intersection2(A,A) = A ).

fof(antisymmetry_r2_hidden,axiom,
    ! [A,B] :
      ( in(A,B)
     => ~ in(B,A) ) ).

fof(dt_k2_wellord1,axiom,
    ! [A,B] :
      ( relation(A)
     => relation(relation_restriction(A,B)) ) ).

fof(dt_k2_zfmisc_1,axiom,
    $true ).

fof(dt_k3_xboole_0,axiom,
    $true ).

fof(t1_subset,axiom,
    ! [A,B] :
      ( in(A,B)
     => element(A,B) ) ).

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

fof(d6_wellord1,axiom,
    ! [A] :
      ( relation(A)
     => ! [B] : relation_restriction(A,B) = set_intersection2(A,cartesian_product2(B,B)) ) ).

fof(t16_wellord1,conjecture,
    ! [A,B,C] :
      ( relation(C)
     => ( in(A,relation_restriction(C,B))
      <=> ( in(A,C)
          & in(A,cartesian_product2(B,B)) ) ) ) ).

fof(d3_xboole_0,axiom,
    ! [A,B,C] :
      ( C = set_intersection2(A,B)
    <=> ! [D] :
          ( in(D,C)
        <=> ( in(D,A)
            & in(D,B) ) ) ) ).

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