%------------------------------------------------------------------------------
% File     : SEU174+1 : TPTP v8.2.0. Released v3.3.0.
% Domain   : Set theory
% Problem  : MPTP bushy problem t46_setfam_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-t46_setfam_1 [Urb07]

% Status   : Theorem
% Rating   : 0.25 v8.2.0, 0.22 v7.5.0, 0.25 v7.4.0, 0.17 v7.2.0, 0.14 v7.1.0, 0.17 v7.0.0, 0.13 v6.4.0, 0.19 v6.3.0, 0.21 v6.2.0, 0.24 v6.1.0, 0.33 v6.0.0, 0.26 v5.5.0, 0.33 v5.4.0, 0.36 v5.3.0, 0.37 v5.2.0, 0.15 v5.1.0, 0.14 v5.0.0, 0.21 v4.1.0, 0.22 v4.0.1, 0.26 v4.0.0, 0.29 v3.7.0, 0.20 v3.5.0, 0.21 v3.4.0, 0.32 v3.3.0
% Syntax   : Number of formulae    :   27 (   9 unt;   0 def)
%            Number of atoms       :   56 (   7 equ)
%            Maximal formula atoms :    6 (   2 avg)
%            Number of connectives :   40 (  11   ~;   1   |;   9   &)
%                                         (   4 <=>;  15  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   4 avg)
%            Maximal term depth    :    3 (   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   :   47 (  42   !;   5   ?)
% 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(d3_tarski,axiom,
    ! [A,B] :
      ( subset(A,B)
    <=> ! [C] :
          ( in(C,A)
         => in(C,B) ) ) ).

fof(d8_setfam_1,axiom,
    ! [A,B] :
      ( element(B,powerset(powerset(A)))
     => ! [C] :
          ( element(C,powerset(powerset(A)))
         => ( C = complements_of_subsets(A,B)
          <=> ! [D] :
                ( element(D,powerset(A))
               => ( in(D,C)
                <=> in(subset_complement(A,D),B) ) ) ) ) ) ).

fof(dt_k1_xboole_0,axiom,
    $true ).

fof(dt_k1_zfmisc_1,axiom,
    $true ).

fof(dt_k3_subset_1,axiom,
    ! [A,B] :
      ( element(B,powerset(A))
     => element(subset_complement(A,B),powerset(A)) ) ).

fof(dt_k7_setfam_1,axiom,
    ! [A,B] :
      ( element(B,powerset(powerset(A)))
     => element(complements_of_subsets(A,B),powerset(powerset(A))) ) ).

fof(dt_m1_subset_1,axiom,
    $true ).

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

fof(fc1_subset_1,axiom,
    ! [A] : ~ empty(powerset(A)) ).

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

fof(involutiveness_k3_subset_1,axiom,
    ! [A,B] :
      ( element(B,powerset(A))
     => subset_complement(A,subset_complement(A,B)) = B ) ).

fof(involutiveness_k7_setfam_1,axiom,
    ! [A,B] :
      ( element(B,powerset(powerset(A)))
     => complements_of_subsets(A,complements_of_subsets(A,B)) = B ) ).

fof(rc1_subset_1,axiom,
    ! [A] :
      ( ~ empty(A)
     => ? [B] :
          ( element(B,powerset(A))
          & ~ empty(B) ) ) ).

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

fof(rc2_subset_1,axiom,
    ! [A] :
    ? [B] :
      ( element(B,powerset(A))
      & empty(B) ) ).

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

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

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

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

fof(t3_subset,axiom,
    ! [A,B] :
      ( element(A,powerset(B))
    <=> subset(A,B) ) ).

fof(t46_setfam_1,conjecture,
    ! [A,B] :
      ( element(B,powerset(powerset(A)))
     => ~ ( B != empty_set
          & complements_of_subsets(A,B) = empty_set ) ) ).

fof(t4_subset,axiom,
    ! [A,B,C] :
      ( ( in(A,B)
        & element(B,powerset(C)) )
     => element(A,C) ) ).

fof(t5_subset,axiom,
    ! [A,B,C] :
      ~ ( in(A,B)
        & element(B,powerset(C))
        & empty(C) ) ).

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

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

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

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