SET007 Axioms: SET007+122.ax


%------------------------------------------------------------------------------
% File     : SET007+122 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Schemes
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : schems_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   16 (   0 unit)
%            Number of atoms       :   50 (   0 equality)
%            Maximal formula depth :    7 (   5 average)
%            Number of connectives :   43 (   9 ~  ;   3  |;   9  &)
%                                         (   2 <=>;  20 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   25 (   1 propositional; 0-2 arity)
%            Number of functors    :    0 (   0 constant; --- arity)
%            Number of variables   :   55 (   0 singleton;  35 !;  20 ?)
%            Maximal term depth    :    1 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(s1_schems_1,axiom,
    ( ! [A] : p1_s1_schems_1(A)
   => ? [A] : p1_s1_schems_1(A) )).

fof(s2_schems_1,axiom,
    ( ? [A] :
      ! [B] : p1_s2_schems_1(A,B)
   => ! [A] :
      ? [B] : p1_s2_schems_1(B,A) )).

fof(s3_schems_1,axiom,
    ( ! [A] :
        ( p1_s3_schems_1(A)
       => p2_s3_schems_1(A) )
   => ( ! [A] : p1_s3_schems_1(A)
     => ! [A] : p2_s3_schems_1(A) ) )).

fof(s4_schems_1,axiom,
    ( ! [A] :
        ( p1_s4_schems_1(A)
      <=> p2_s4_schems_1(A) )
   => ( ! [A] : p1_s4_schems_1(A)
    <=> ! [A] : p2_s4_schems_1(A) ) )).

fof(s5_schems_1,axiom,
    ( ! [A] :
        ( p1_s5_schems_1(A)
       => p2_s5_schems_1 )
   => ( ! [A] : p1_s5_schems_1(A)
     => p2_s5_schems_1 ) )).

fof(s6_schems_1,axiom,
    ( ~ ( ! [A] : ~ p1_s6_schems_1(A)
        & ~ ! [A] : p2_s6_schems_1(A) )
   => ? [A] :
      ! [B] :
        ( p1_s6_schems_1(A)
        | p2_s6_schems_1(B) ) )).

fof(s7_schems_1,axiom,
    ( ? [A] :
      ! [B] :
        ( p1_s7_schems_1(A)
        | p2_s7_schems_1(B) )
   => ~ ( ! [A] : ~ p1_s7_schems_1(A)
        & ~ ! [A] : p2_s7_schems_1(A) ) )).

fof(s8_schems_1,axiom,
    ( ! [A] :
        ~ ! [B] :
            ( ~ p1_s8_schems_1(B)
            & ~ p2_s8_schems_1(A) )
   => ? [A] :
      ! [B] :
        ( p1_s8_schems_1(A)
        | p2_s8_schems_1(B) ) )).

fof(s9_schems_1,axiom,
    ( ( ? [A] : p1_s9_schems_1(A)
      & ! [A] : p2_s9_schems_1(A) )
   => ! [A] :
      ? [B] :
        ( p1_s9_schems_1(B)
        & p2_s9_schems_1(A) ) )).

fof(s10_schems_1,axiom,
    ( ! [A] :
      ? [B] :
        ( p1_s10_schems_1(B)
        & p2_s10_schems_1(A) )
   => ( ? [A] : p1_s10_schems_1(A)
      & ! [A] : p2_s10_schems_1(A) ) )).

fof(s11_schems_1,axiom,
    ( ! [A] :
      ? [B] :
        ( p1_s11_schems_1(B)
        & p2_s11_schems_1(A) )
   => ? [A] :
      ! [B] :
        ( p1_s11_schems_1(A)
        & p2_s11_schems_1(B) ) )).

fof(s12_schems_1,axiom,
    ( ! [A,B] : p1_s12_schems_1(A,B)
   => ? [A] :
      ! [B] : p1_s12_schems_1(B,A) )).

fof(s13_schems_1,axiom,
    ( ? [A] :
      ! [B] : p1_s13_schems_1(A,B)
   => ? [A] : p1_s13_schems_1(A,A) )).

fof(s14_schems_1,axiom,
    ( ! [A] : p1_s14_schems_1(A,A)
   => ! [A] :
      ? [B] : p1_s14_schems_1(B,A) )).

fof(s15_schems_1,axiom,
    ( ! [A] : p1_s15_schems_1(A,A)
   => ! [A] :
      ? [B] : p1_s15_schems_1(A,B) )).

fof(s16_schems_1,axiom,
    ( ! [A] :
      ? [B] : p1_s16_schems_1(B,A)
   => ? [A,B] : p1_s16_schems_1(A,B) )).
%------------------------------------------------------------------------------