SET007 Axioms: SET007+58.ax


%------------------------------------------------------------------------------
% File     : SET007+58 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Contraction Lemma
% 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    : zf_colla [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   21 (   3 unit)
%            Number of atoms       :  132 (  20 equality)
%            Maximal formula depth :   18 (   9 average)
%            Number of connectives :  140 (  29 ~  ;   0  |;  52  &)
%                                         (   9 <=>;  50 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   16 (   1 propositional; 0-3 arity)
%            Number of functors    :   15 (   2 constant; 0-3 arity)
%            Number of variables   :   71 (   0 singleton;  60 !;  11 ?)
%            Maximal term depth    :    5 (   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(t2_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( ! [C] :
                ( m1_subset_1(C,A)
               => ~ r2_hidden(C,B) )
          <=> r2_hidden(B,k1_zf_colla(A,k1_xboole_0)) ) ) ) )).

fof(t3_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ( r1_tarski(k3_xboole_0(C,A),k1_zf_colla(A,B))
              <=> r2_hidden(C,k1_zf_colla(A,k1_ordinal1(B))) ) ) ) ) )).

fof(t4_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ( r1_ordinal1(B,C)
               => r1_tarski(k1_zf_colla(A,B),k1_zf_colla(A,C)) ) ) ) ) )).

fof(t5_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,A)
         => ? [C] :
              ( v3_ordinal1(C)
              & r2_hidden(B,k1_zf_colla(A,C)) ) ) ) )).

fof(t6_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( m1_subset_1(C,A)
             => ! [D] :
                  ( m1_subset_1(D,A)
                 => ( ( r2_hidden(C,D)
                      & r2_hidden(D,k1_zf_colla(A,B)) )
                   => ( r2_hidden(C,k1_zf_colla(A,B))
                      & ? [E] :
                          ( v3_ordinal1(E)
                          & r2_hidden(E,B)
                          & r2_hidden(C,k1_zf_colla(A,E)) ) ) ) ) ) ) ) )).

fof(t7_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => r1_tarski(k1_zf_colla(A,B),A) ) ) )).

fof(t8_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( v3_ordinal1(B)
          & A = k1_zf_colla(A,B) ) ) )).

fof(t9_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] :
          ( v1_relat_1(B)
          & v1_funct_1(B)
          & k1_relat_1(B) = A
          & ! [C] :
              ( m1_subset_1(C,A)
             => k1_funct_1(B,C) = k9_relat_1(B,C) ) ) ) )).

fof(d2_zf_colla,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] :
          ( r1_zf_colla(A,B,C)
        <=> ( k1_relat_1(A) = B
            & k2_relat_1(A) = C
            & v2_funct_1(A)
            & ! [D,E] :
                ( ( r2_hidden(D,B)
                  & r2_hidden(E,B) )
               => ( ~ ( ? [F] :
                          ( F = E
                          & r2_hidden(D,F) )
                      & ! [F] :
                          ~ ( k1_funct_1(A,E) = F
                            & r2_hidden(k1_funct_1(A,D),F) ) )
                  & ~ ( ? [F] :
                          ( k1_funct_1(A,E) = F
                          & r2_hidden(k1_funct_1(A,D),F) )
                      & ! [F] :
                          ~ ( F = E
                            & r2_hidden(D,F) ) ) ) ) ) ) ) )).

fof(d3_zf_colla,axiom,(
    ! [A,B] :
      ( r2_zf_colla(A,B)
    <=> ? [C] :
          ( v1_relat_1(C)
          & v1_funct_1(C)
          & r1_zf_colla(C,A,B) ) ) )).

fof(t10_zf_colla,axiom,(
    $true )).

fof(t11_zf_colla,axiom,(
    $true )).

fof(t12_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k1_relat_1(B) = A
              & ! [C] :
                  ( m1_subset_1(C,A)
                 => k1_funct_1(B,C) = k9_relat_1(B,C) ) )
           => v1_ordinal1(k2_relat_1(B)) ) ) ) )).

fof(t13_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( r2_zf_model(A,k6_zf_model)
       => ! [B] :
            ( m1_subset_1(B,A)
           => ! [C] :
                ( m1_subset_1(C,A)
               => ( ! [D] :
                      ( m1_subset_1(D,A)
                     => ( r2_hidden(D,B)
                      <=> r2_hidden(D,C) ) )
                 => B = C ) ) ) ) ) )).

fof(t14_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ~ ( r2_zf_model(A,k6_zf_model)
          & ! [B] :
              ~ ( v1_ordinal1(B)
                & r2_zf_colla(A,B) ) ) ) )).

fof(dt_k1_zf_colla,axiom,(
    $true )).

fof(d1_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( C = k1_zf_colla(A,B)
            <=> ? [D] :
                  ( v1_relat_1(D)
                  & v1_funct_1(D)
                  & v5_ordinal1(D)
                  & C = a_2_0_zf_colla(A,D)
                  & k1_relat_1(D) = B
                  & ! [E] :
                      ( v3_ordinal1(E)
                     => ( r2_hidden(E,B)
                       => k1_funct_1(D,E) = a_3_0_zf_colla(A,D,E) ) ) ) ) ) ) )).

fof(t1_zf_colla,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => k1_zf_colla(A,B) = a_2_1_zf_colla(A,B) ) ) )).

fof(fraenkel_a_2_0_zf_colla,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_relat_1(C)
        & v1_funct_1(C)
        & v5_ordinal1(C) )
     => ( r2_hidden(A,a_2_0_zf_colla(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,B)
            & A = D
            & ! [E] :
                ( m1_subset_1(E,B)
               => ~ ( r2_hidden(E,D)
                    & ! [F] :
                        ( v3_ordinal1(F)
                       => ~ ( r2_hidden(F,k1_relat_1(C))
                            & r2_hidden(E,k3_tarski(k1_tarski(k1_funct_1(C,F)))) ) ) ) ) ) ) ) )).

fof(fraenkel_a_3_0_zf_colla,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & v1_relat_1(C)
        & v1_funct_1(C)
        & v5_ordinal1(C)
        & v3_ordinal1(D) )
     => ( r2_hidden(A,a_3_0_zf_colla(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & ! [F] :
                ( m1_subset_1(F,B)
               => ~ ( r2_hidden(F,E)
                    & ! [G] :
                        ( v3_ordinal1(G)
                       => ~ ( r2_hidden(G,k1_relat_1(k2_ordinal1(C,D)))
                            & r2_hidden(F,k3_tarski(k1_tarski(k1_funct_1(k2_ordinal1(C,D),G)))) ) ) ) ) ) ) ) )).

fof(fraenkel_a_2_1_zf_colla,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v3_ordinal1(C) )
     => ( r2_hidden(A,a_2_1_zf_colla(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,B)
            & A = D
            & ! [E] :
                ( m1_subset_1(E,B)
               => ~ ( r2_hidden(E,D)
                    & ! [F] :
                        ( v3_ordinal1(F)
                       => ~ ( r2_hidden(F,C)
                            & r2_hidden(E,k1_zf_colla(B,F)) ) ) ) ) ) ) ) )).
%------------------------------------------------------------------------------