SET007 Axioms: SET007+2.ax


%------------------------------------------------------------------------------
% File     : SET007+2 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Boolean Properties of Sets - Definitions
% 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    : xboole_0 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   38 (  16 unit)
%            Number of atoms       :   89 (  17 equality)
%            Maximal formula depth :   11 (   5 average)
%            Number of connectives :   76 (  25 ~  ;   2  |;  20  &)
%                                         (  21 <=>;   8 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   1 propositional; 0-2 arity)
%            Number of functors    :    8 (   4 constant; 0-2 arity)
%            Number of variables   :   77 (   4 singleton;  72 !;   5 ?)
%            Maximal term depth    :    3 (   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(fc1_xboole_0,axiom,(
    v1_xboole_0(k1_xboole_0) )).

fof(rc1_xboole_0,axiom,(
    ? [A] : v1_xboole_0(A) )).

fof(rc2_xboole_0,axiom,(
    ? [A] : ~ v1_xboole_0(A) )).

fof(fc2_xboole_0,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(A,B)) ) )).

fof(fc3_xboole_0,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(A)
     => ~ v1_xboole_0(k2_xboole_0(B,A)) ) )).

fof(d1_xboole_0,axiom,(
    ! [A] :
      ( A = k1_xboole_0
    <=> ! [B] : ~ r2_hidden(B,A) ) )).

fof(d2_xboole_0,axiom,(
    ! [A,B,C] :
      ( C = k2_xboole_0(A,B)
    <=> ! [D] :
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) )).

fof(d3_xboole_0,axiom,(
    ! [A,B,C] :
      ( C = k3_xboole_0(A,B)
    <=> ! [D] :
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & r2_hidden(D,B) ) ) ) )).

fof(d4_xboole_0,axiom,(
    ! [A,B,C] :
      ( C = k4_xboole_0(A,B)
    <=> ! [D] :
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) )).

fof(d5_xboole_0,axiom,(
    ! [A] :
      ( v1_xboole_0(A)
    <=> A = k1_xboole_0 ) )).

fof(d6_xboole_0,axiom,(
    ! [A,B] : k5_xboole_0(A,B) = k2_xboole_0(k4_xboole_0(A,B),k4_xboole_0(B,A)) )).

fof(d7_xboole_0,axiom,(
    ! [A,B] :
      ( r1_xboole_0(A,B)
    <=> k3_xboole_0(A,B) = k1_xboole_0 ) )).

fof(d8_xboole_0,axiom,(
    ! [A,B] :
      ( r2_xboole_0(A,B)
    <=> ( r1_tarski(A,B)
        & A != B ) ) )).

fof(d9_xboole_0,axiom,(
    ! [A,B] :
      ( r3_xboole_0(A,B)
    <=> ( r1_tarski(A,B)
        | r1_tarski(B,A) ) ) )).

fof(d10_xboole_0,axiom,(
    ! [A,B] :
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) )).

fof(t1_xboole_0,axiom,(
    ! [A,B,C] :
      ( r2_hidden(A,k5_xboole_0(B,C))
    <=> ~ ( r2_hidden(A,B)
        <=> r2_hidden(A,C) ) ) )).

fof(t2_xboole_0,axiom,(
    ! [A,B,C] :
      ( ! [D] :
          ( ~ r2_hidden(D,A)
        <=> ( r2_hidden(D,B)
          <=> r2_hidden(D,C) ) )
     => A = k5_xboole_0(B,C) ) )).

fof(t3_xboole_0,axiom,(
    ! [A,B] :
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] :
              ~ ( r2_hidden(C,A)
                & r2_hidden(C,B) ) )
      & ~ ( ? [C] :
              ( r2_hidden(C,A)
              & r2_hidden(C,B) )
          & r1_xboole_0(A,B) ) ) )).

fof(t4_xboole_0,axiom,(
    ! [A,B] :
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : ~ r2_hidden(C,k3_xboole_0(A,B)) )
      & ~ ( ? [C] : r2_hidden(C,k3_xboole_0(A,B))
          & r1_xboole_0(A,B) ) ) )).

fof(t5_xboole_0,axiom,(
    ! [A,B,C] :
      ~ ( r1_xboole_0(A,B)
        & r2_hidden(C,k2_xboole_0(A,B))
        & ~ ( r2_hidden(C,A)
            & ~ r2_hidden(C,B) )
        & ~ ( r2_hidden(C,B)
            & ~ r2_hidden(C,A) ) ) )).

fof(s1_xboole_0,axiom,(
    ? [A] :
    ! [B] :
      ( r2_hidden(B,A)
    <=> ( r2_hidden(B,f1_s1_xboole_0)
        & p1_s1_xboole_0(B) ) ) )).

fof(s2_xboole_0,axiom,
    ( ( ! [A] :
          ( r2_hidden(A,f1_s2_xboole_0)
        <=> p1_s2_xboole_0(A) )
      & ! [A] :
          ( r2_hidden(A,f2_s2_xboole_0)
        <=> p1_s2_xboole_0(A) ) )
   => f1_s2_xboole_0 = f2_s2_xboole_0 )).

fof(s3_xboole_0,axiom,(
    ! [A,B] :
      ( ( ! [C] :
            ( r2_hidden(C,A)
          <=> p1_s3_xboole_0(C) )
        & ! [C] :
            ( r2_hidden(C,B)
          <=> p1_s3_xboole_0(C) ) )
     => A = B ) )).

fof(symmetry_r1_xboole_0,axiom,(
    ! [A,B] :
      ( r1_xboole_0(A,B)
     => r1_xboole_0(B,A) ) )).

fof(irreflexivity_r2_xboole_0,axiom,(
    ! [A,B] : ~ r2_xboole_0(A,A) )).

fof(antisymmetry_r2_xboole_0,axiom,(
    ! [A,B] :
      ( r2_xboole_0(A,B)
     => ~ r2_xboole_0(B,A) ) )).

fof(symmetry_r3_xboole_0,axiom,(
    ! [A,B] :
      ( r3_xboole_0(A,B)
     => r3_xboole_0(B,A) ) )).

fof(reflexivity_r3_xboole_0,axiom,(
    ! [A,B] : r3_xboole_0(A,A) )).

fof(dt_k1_xboole_0,axiom,(
    $true )).

fof(dt_k2_xboole_0,axiom,(
    $true )).

fof(commutativity_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,B) = k2_xboole_0(B,A) )).

fof(idempotence_k2_xboole_0,axiom,(
    ! [A,B] : k2_xboole_0(A,A) = A )).

fof(dt_k3_xboole_0,axiom,(
    $true )).

fof(commutativity_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,B) = k3_xboole_0(B,A) )).

fof(idempotence_k3_xboole_0,axiom,(
    ! [A,B] : k3_xboole_0(A,A) = A )).

fof(dt_k4_xboole_0,axiom,(
    $true )).

fof(dt_k5_xboole_0,axiom,(
    $true )).

fof(commutativity_k5_xboole_0,axiom,(
    ! [A,B] : k5_xboole_0(A,B) = k5_xboole_0(B,A) )).
%------------------------------------------------------------------------------