%------------------------------------------------------------------------------
% File     : SET045^7 : TPTP v8.2.0. Released v5.5.0.
% Domain   : Set Theory
% Problem  : No Universal Set
% Version  : [Ben12] axioms.
% English  :

% Refs     : [KM64]  Kalish & Montegue (1964), Logic: Techniques of Formal
%          : [Goe69] Goedel (1969), An Interpretation of the Intuitionistic
%          : [Pel86] Pelletier (1986), Seventy-five Problems for Testing Au
%          : [Ben12] Benzmueller (2012), Email to Geoff Sutcliffe
% Source   : [Ben12]
% Names    : s4-cumul-GSE045+1 [Ben12]

% Status   : Theorem
% Rating   : 0.90 v8.2.0, 0.85 v8.1.0, 0.91 v7.5.0, 0.86 v7.4.0, 0.89 v7.2.0, 0.88 v7.0.0, 0.86 v6.4.0, 0.83 v6.3.0, 0.80 v6.2.0, 0.86 v5.5.0
% Syntax   : Number of formulae    :   75 (  33 unt;  37 typ;  32 def)
%            Number of atoms       :  143 (  36 equ;   0 cnn)
%            Maximal formula atoms :   29 (   3 avg)
%            Number of connectives :  195 (   5   ~;   5   |;   9   &; 166   @)
%                                         (   0 <=>;  10  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   2 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :  183 ( 183   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   45 (  43 usr;   8 con; 0-3 aty)
%            Number of variables   :   95 (  54   ^;  34   !;   7   ?;  95   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : Goedel translation of SET045+1
%------------------------------------------------------------------------------
%----Include axioms for Modal logic S4 under cumulative domains
include('Axioms/LCL015^0.ax').
include('Axioms/LCL013^5.ax').
include('Axioms/LCL015^1.ax').
%------------------------------------------------------------------------------
thf(element_type,type,
    element: mu > mu > $i > $o ).

thf(pel41_1,axiom,
    ( mvalid
    @ ( mbox_s4
      @ ( mforall_ind
        @ ^ [Z: mu] :
            ( mexists_ind
            @ ^ [Y: mu] :
                ( mbox_s4
                @ ( mforall_ind
                  @ ^ [X: mu] : ( mand @ ( mbox_s4 @ ( mimplies @ ( mbox_s4 @ ( element @ X @ Y ) ) @ ( mand @ ( mbox_s4 @ ( element @ X @ Z ) ) @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( element @ X @ X ) ) ) ) ) ) ) @ ( mbox_s4 @ ( mimplies @ ( mand @ ( mbox_s4 @ ( element @ X @ Z ) ) @ ( mbox_s4 @ ( mnot @ ( mbox_s4 @ ( element @ X @ X ) ) ) ) ) @ ( mbox_s4 @ ( element @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ).

thf(pel41,conjecture,
    ( mvalid
    @ ( mbox_s4
      @ ( mnot
        @ ( mexists_ind
          @ ^ [Z: mu] :
              ( mbox_s4
              @ ( mforall_ind
                @ ^ [X: mu] : ( mbox_s4 @ ( element @ X @ Z ) ) ) ) ) ) ) ) ).

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