%------------------------------------------------------------------------------
% File     : ITP004+2 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Problem  : HOL4 set theory export of thm_2Epred__set_2EREST__SUBSET.p, bushy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : thm_2Epred__set_2EREST__SUBSET.p [Gau19]
%          : HL401501+2.p [TPAP]

% Status   : Theorem
% Rating   : 0.45 v9.0.0, 0.44 v8.2.0, 0.50 v7.5.0
% Syntax   : Number of formulae    :   27 (   5 unt;   0 def)
%            Number of atoms       :   82 (   8 equ)
%            Maximal formula atoms :    7 (   3 avg)
%            Number of connectives :   57 (   2   ~;   0   |;   2   &)
%                                         (   8 <=>;  45  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   5 avg)
%            Maximal term depth    :    5 (   2 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :   16 (  16 usr;   5 con; 0-2 aty)
%            Number of variables   :   48 (  48   !;   0   ?)
% SPC      : FOF_THM_RFO_SEQ

% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001+2.ax').
%------------------------------------------------------------------------------
fof(mem_c_2Emin_2E_3D_3D_3E,axiom,
    mem(c_2Emin_2E_3D_3D_3E,arr(bool,arr(bool,bool))) ).

fof(ax_imp_p,axiom,
    ! [Q] :
      ( mem(Q,bool)
     => ! [R] :
          ( mem(R,bool)
         => ( p(ap(ap(c_2Emin_2E_3D_3D_3E,Q),R))
          <=> ( p(Q)
             => p(R) ) ) ) ) ).

fof(mem_c_2Epred__set_2ESUBSET,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Epred__set_2ESUBSET(A_27a),arr(arr(A_27a,bool),arr(arr(A_27a,bool),bool))) ) ).

fof(mem_c_2Ebool_2E_7E,axiom,
    mem(c_2Ebool_2E_7E,arr(bool,bool)) ).

fof(ax_neg_p,axiom,
    ! [Q] :
      ( mem(Q,bool)
     => ( p(ap(c_2Ebool_2E_7E,Q))
      <=> ~ p(Q) ) ) ).

fof(mem_c_2Ebool_2E_2F_5C,axiom,
    mem(c_2Ebool_2E_2F_5C,arr(bool,arr(bool,bool))) ).

fof(ax_and_p,axiom,
    ! [Q] :
      ( mem(Q,bool)
     => ! [R] :
          ( mem(R,bool)
         => ( p(ap(ap(c_2Ebool_2E_2F_5C,Q),R))
          <=> ( p(Q)
              & p(R) ) ) ) ) ).

fof(mem_c_2Ebool_2EIN,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Ebool_2EIN(A_27a),arr(A_27a,arr(arr(A_27a,bool),bool))) ) ).

fof(mem_c_2Epred__set_2ECHOICE,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Epred__set_2ECHOICE(A_27a),arr(arr(A_27a,bool),A_27a)) ) ).

fof(mem_c_2Epred__set_2EDELETE,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Epred__set_2EDELETE(A_27a),arr(arr(A_27a,bool),arr(A_27a,arr(A_27a,bool)))) ) ).

fof(mem_c_2Epred__set_2EREST,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Epred__set_2EREST(A_27a),arr(arr(A_27a,bool),arr(A_27a,bool))) ) ).

fof(mem_c_2Emin_2E_3D,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Emin_2E_3D(A_27a),arr(A_27a,arr(A_27a,bool))) ) ).

fof(ax_eq_p,axiom,
    ! [A] :
      ( ne(A)
     => ! [X] :
          ( mem(X,A)
         => ! [Y] :
              ( mem(Y,A)
             => ( p(ap(ap(c_2Emin_2E_3D(A),X),Y))
              <=> X = Y ) ) ) ) ).

fof(mem_c_2Ebool_2E_21,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2Ebool_2E_21(A_27a),arr(arr(A_27a,bool),bool)) ) ).

fof(ax_all_p,axiom,
    ! [A] :
      ( ne(A)
     => ! [Q] :
          ( mem(Q,arr(A,bool))
         => ( p(ap(c_2Ebool_2E_21(A),Q))
          <=> ! [X] :
                ( mem(X,A)
               => p(ap(Q,X)) ) ) ) ) ).

fof(ax_thm_2Epred__set_2ESUBSET__DEF,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0s] :
          ( mem(V0s,arr(A_27a,bool))
         => ! [V1t] :
              ( mem(V1t,arr(A_27a,bool))
             => ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V0s),V1t))
              <=> ! [V2x] :
                    ( mem(V2x,A_27a)
                   => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s))
                     => p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1t)) ) ) ) ) ) ) ).

fof(conj_thm_2Epred__set_2EIN__DELETE,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0s] :
          ( mem(V0s,arr(A_27a,bool))
         => ! [V1x] :
              ( mem(V1x,A_27a)
             => ! [V2y] :
                  ( mem(V2y,A_27a)
                 => ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),V2y)))
                  <=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s))
                      & V1x != V2y ) ) ) ) ) ) ).

fof(ax_thm_2Epred__set_2EREST__DEF,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0s] :
          ( mem(V0s,arr(A_27a,bool))
         => ap(c_2Epred__set_2EREST(A_27a),V0s) = ap(ap(c_2Epred__set_2EDELETE(A_27a),V0s),ap(c_2Epred__set_2ECHOICE(A_27a),V0s)) ) ) ).

fof(conj_thm_2Epred__set_2EREST__SUBSET,conjecture,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0s] :
          ( mem(V0s,arr(A_27a,bool))
         => p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),ap(c_2Epred__set_2EREST(A_27a),V0s)),V0s)) ) ) ).

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