SET007 Axioms: SET007+153.ax


%------------------------------------------------------------------------------
% File     : SET007+153 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On Powers of Cardinals
% 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    : card_5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   66 (  13 unit)
%            Number of atoms       :  326 (  43 equality)
%            Maximal formula depth :   16 (   6 average)
%            Number of connectives :  328 (  68 ~  ;   2  |; 160  &)
%                                         (   6 <=>;  92 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   1 propositional; 0-3 arity)
%            Number of functors    :   34 (   6 constant; 0-2 arity)
%            Number of variables   :   98 (   0 singleton;  91 !;   7 ?)
%            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(fc1_card_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ( v1_ordinal1(k3_card_3(A))
        & v2_ordinal1(k3_card_3(A))
        & v3_ordinal1(k3_card_3(A)) ) ) )).

fof(rc1_card_5,axiom,(
    ? [A] : ~ v1_finset_1(A) )).

fof(rc2_card_5,axiom,(
    ? [A] :
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & ~ v1_finset_1(A)
      & v1_card_1(A) ) )).

fof(cc1_card_5,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ~ v1_xboole_0(A) ) )).

fof(fc2_card_5,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( ~ v1_xboole_0(k3_card_1(A))
        & v1_ordinal1(k3_card_1(A))
        & v2_ordinal1(k3_card_1(A))
        & v3_ordinal1(k3_card_1(A))
        & ~ v1_finset_1(k3_card_1(A))
        & v1_card_1(k3_card_1(A)) ) ) )).

fof(fc3_card_5,axiom,(
    ! [A,B] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & v1_card_1(B) )
     => ( ~ v1_xboole_0(k1_card_2(A,B))
        & v1_ordinal1(k1_card_2(A,B))
        & v2_ordinal1(k1_card_2(A,B))
        & v3_ordinal1(k1_card_2(A,B))
        & ~ v1_finset_1(k1_card_2(A,B))
        & v1_card_1(k1_card_2(A,B)) ) ) )).

fof(fc4_card_5,axiom,(
    ! [A,B] :
      ( ( v1_card_1(A)
        & ~ v1_finset_1(B)
        & v1_card_1(B) )
     => ( ~ v1_xboole_0(k1_card_2(A,B))
        & v1_ordinal1(k1_card_2(A,B))
        & v2_ordinal1(k1_card_2(A,B))
        & v3_ordinal1(k1_card_2(A,B))
        & ~ v1_finset_1(k1_card_2(A,B))
        & v1_card_1(k1_card_2(A,B)) ) ) )).

fof(fc5_card_5,axiom,(
    ! [A,B] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_finset_1(B)
        & v1_card_1(B) )
     => ( ~ v1_xboole_0(k2_card_2(A,B))
        & v1_ordinal1(k2_card_2(A,B))
        & v2_ordinal1(k2_card_2(A,B))
        & v3_ordinal1(k2_card_2(A,B))
        & ~ v1_finset_1(k2_card_2(A,B))
        & v1_card_1(k2_card_2(A,B)) ) ) )).

fof(fc6_card_5,axiom,(
    ! [A,B] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_finset_1(B)
        & v1_card_1(B) )
     => ( ~ v1_xboole_0(k3_card_2(A,B))
        & v1_ordinal1(k3_card_2(A,B))
        & v2_ordinal1(k3_card_2(A,B))
        & v3_ordinal1(k3_card_2(A,B))
        & ~ v1_finset_1(k3_card_2(A,B))
        & v1_card_1(k3_card_2(A,B)) ) ) )).

fof(fc7_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( ~ v1_xboole_0(k2_card_1(A))
        & v1_ordinal1(k2_card_1(A))
        & v2_ordinal1(k2_card_1(A))
        & v3_ordinal1(k2_card_1(A))
        & ~ v1_finset_1(k2_card_1(A))
        & v1_card_1(k2_card_1(A)) ) ) )).

fof(cc2_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ! [B] :
          ( m1_subset_1(B,A)
         => ( v1_ordinal1(B)
            & v2_ordinal1(B)
            & v3_ordinal1(B) ) ) ) )).

fof(fc8_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( ~ v1_xboole_0(k1_card_5(A))
        & v1_ordinal1(k1_card_5(A))
        & v2_ordinal1(k1_card_5(A))
        & v3_ordinal1(k1_card_5(A))
        & ~ v1_finset_1(k1_card_5(A))
        & v1_card_1(k1_card_5(A)) ) ) )).

fof(t1_card_5,axiom,
    ( np__1 = k1_tarski(np__0)
    & np__2 = k2_tarski(np__0,np__1) )).

fof(t2_card_5,axiom,(
    $true )).

fof(t3_card_5,axiom,(
    $true )).

fof(t4_card_5,axiom,(
    $true )).

fof(t5_card_5,axiom,(
    $true )).

fof(t6_card_5,axiom,(
    $true )).

fof(t7_card_5,axiom,(
    $true )).

fof(t8_card_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_finseq_1(A) = k4_xboole_0(k1_nat_1(A,np__1),k1_tarski(np__0)) ) )).

fof(t9_card_5,axiom,(
    ! [A] : k2_card_1(k1_card_1(A)) = k2_card_1(A) )).

fof(t10_card_5,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k3_card_3(B))
      <=> ? [C] :
            ( r2_hidden(C,k1_relat_1(B))
            & r2_hidden(A,k1_funct_1(B,C)) ) ) ) )).

fof(t11_card_5,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ~ v1_finset_1(k3_card_1(A)) ) )).

fof(t12_card_5,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ~ ( ~ v1_finset_1(A)
          & ! [B] :
              ( v3_ordinal1(B)
             => A != k3_card_1(B) ) ) ) )).

fof(t13_card_5,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ~ ( ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => A != k1_card_1(B) )
          & ! [B] :
              ( v3_ordinal1(B)
             => A != k3_card_1(B) ) ) ) )).

fof(t14_card_5,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ~ ( r1_tarski(B,A)
            & ! [C] :
                ( ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v5_ordinal1(C)
                  & v1_ordinal2(C) )
               => ~ ( C = k3_wellord1(k1_wellord2(k2_wellord2(k1_wellord2(B))),k1_wellord2(B))
                    & v2_ordinal2(C)
                    & k1_relat_1(C) = k2_wellord2(k1_wellord2(B))
                    & k2_relat_1(C) = B ) ) ) ) )).

fof(t15_card_5,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r1_tarski(B,A)
         => r4_zfrefle1(k7_ordinal2(B),k2_wellord2(k1_wellord2(B))) ) ) )).

fof(t16_card_5,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( r1_tarski(B,A)
         => k1_card_1(B) = k1_card_1(k2_wellord2(k1_wellord2(B))) ) ) )).

fof(t17_card_5,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ? [B] :
          ( v3_ordinal1(B)
          & r1_ordinal1(B,k1_card_1(A))
          & r4_zfrefle1(A,B) ) ) )).

fof(t18_card_5,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ? [B] :
          ( v1_card_1(B)
          & r1_tarski(B,k1_card_1(A))
          & r4_zfrefle1(A,B)
          & ! [C] :
              ( v3_ordinal1(C)
             => ( r4_zfrefle1(A,C)
               => r1_ordinal1(B,C) ) ) ) ) )).

fof(t19_card_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ( ( k2_relat_1(A) = k2_relat_1(B)
              & v2_ordinal2(A)
              & v2_ordinal2(B) )
           => A = B ) ) ) )).

fof(t20_card_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ( v2_ordinal2(A)
       => v2_funct_1(A) ) ) )).

fof(t21_card_5,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => k2_ordinal1(k1_ordinal4(A,B),k1_relat_1(A)) = A ) ) )).

fof(t23_card_5,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => r2_hidden(A,k3_card_2(np__2,A)) ) )).

fof(d1_card_5,axiom,(
    $true )).

fof(d2_card_5,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( B = k1_card_5(A)
          <=> ( r4_zfrefle1(A,B)
              & ! [C] :
                  ( v1_card_1(C)
                 => ( r4_zfrefle1(A,C)
                   => r1_tarski(B,C) ) ) ) ) ) ) )).

fof(d3_card_5,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_card_3(C) )
             => ( C = k2_card_5(A,B)
              <=> ( ! [D] :
                      ( r2_hidden(D,k1_relat_1(C))
                    <=> ( r2_hidden(D,A)
                        & v1_card_1(D) ) )
                  & ! [D] :
                      ( v1_card_1(D)
                     => ( r2_hidden(D,A)
                       => k1_funct_1(C,D) = k3_card_2(D,B) ) ) ) ) ) ) ) )).

fof(t24_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ? [B] :
          ( v3_ordinal1(B)
          & A = k3_card_1(B) ) ) )).

fof(t25_card_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_finset_1(B)
            & v1_card_1(B) )
         => ( B != np__0
            & B != np__1
            & B != np__2
            & B != k1_card_1(A)
            & r2_hidden(k1_card_1(A),B)
            & r1_tarski(k3_card_1(np__0),B) ) ) ) )).

fof(t26_card_5,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( ( ~ v1_finset_1(B)
            & v1_card_1(B) )
         => ( ( r1_tarski(B,A)
              | r2_hidden(B,A) )
           => ( ~ v1_finset_1(A)
              & v1_card_1(A) ) ) ) ) )).

fof(t27_card_5,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( ( ~ v1_finset_1(B)
            & v1_card_1(B) )
         => ( ( r1_tarski(B,A)
              | r2_hidden(B,A) )
           => ( k1_card_2(B,A) = A
              & k1_card_2(A,B) = A
              & k2_card_2(B,A) = A
              & k2_card_2(A,B) = A ) ) ) ) )).

fof(t28_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( k1_card_2(A,A) = A
        & k2_card_2(A,A) = A ) ) )).

fof(t29_card_5,axiom,(
    $true )).

fof(t30_card_5,axiom,(
    $true )).

fof(t31_card_5,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( ( ~ v1_finset_1(B)
            & v1_card_1(B) )
         => r1_tarski(A,k3_card_2(A,B)) ) ) )).

fof(t32_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => k3_tarski(A) = A ) )).

fof(d4_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v1_card_5(A)
      <=> k1_card_5(A) = A ) ) )).

fof(t33_card_5,axiom,(
    $true )).

fof(t34_card_5,axiom,(
    k1_card_5(k3_card_1(np__0)) = k3_card_1(np__0) )).

fof(t35_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => k1_card_5(k2_card_1(A)) = k2_card_1(A) ) )).

fof(t36_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => r1_tarski(k3_card_1(np__0),k1_card_5(A)) ) )).

fof(t37_card_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( k1_card_5(np__0) = np__0
        & k1_card_5(k1_card_1(k1_nat_1(A,np__1))) = np__1 ) ) )).

fof(t38_card_5,axiom,(
    ! [A,B] :
      ( v1_card_1(B)
     => ( ( r1_tarski(A,B)
          & r2_hidden(k1_card_1(A),k1_card_5(B)) )
       => ( r2_hidden(k7_ordinal2(A),B)
          & r2_hidden(k3_tarski(A),B) ) ) ) )).

fof(t39_card_5,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C)
                & v1_ordinal2(C) )
             => ( ( k1_relat_1(C) = A
                  & r1_tarski(k2_relat_1(C),B)
                  & r2_hidden(A,k1_card_5(B)) )
               => ( r2_hidden(k8_ordinal2(C),B)
                  & r2_hidden(k3_card_3(C),B) ) ) ) ) ) )).

fof(t40_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( r2_hidden(k1_card_5(A),A)
       => v2_card_1(A) ) ) )).

fof(t41_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ~ ( r2_hidden(k1_card_5(A),A)
          & ! [B] :
              ( ( v1_relat_1(B)
                & v1_funct_1(B)
                & v5_ordinal1(B)
                & v1_ordinal2(B) )
             => ~ ( k1_relat_1(B) = k1_card_5(A)
                  & r1_tarski(k2_relat_1(B),A)
                  & v2_ordinal2(B)
                  & A = k8_ordinal2(B)
                  & v1_relat_1(B)
                  & v1_funct_1(B)
                  & v1_card_3(B)
                  & ~ r2_hidden(np__0,k2_relat_1(B)) ) ) ) ) )).

fof(t42_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v1_card_5(k3_card_1(np__0))
        & v1_card_5(k2_card_1(A)) ) ) )).

fof(t43_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ! [B] :
          ( ( ~ v1_finset_1(B)
            & v1_card_1(B) )
         => ( r1_tarski(A,B)
           => k3_card_2(A,B) = k3_card_2(np__2,B) ) ) ) )).

fof(t44_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ! [B] :
          ( ( ~ v1_finset_1(B)
            & v1_card_1(B) )
         => k3_card_2(k2_card_1(A),B) = k2_card_2(k3_card_2(A,B),k2_card_1(A)) ) ) )).

fof(t45_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ! [B] :
          ( ( ~ v1_finset_1(B)
            & v1_card_1(B) )
         => r1_tarski(k6_card_3(k2_card_5(B,A)),k3_card_2(B,A)) ) ) )).

fof(t46_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ! [B] :
          ( ( ~ v1_finset_1(B)
            & v1_card_1(B) )
         => ( ( v2_card_1(A)
              & r2_hidden(B,k1_card_5(A)) )
           => k3_card_2(A,B) = k6_card_3(k2_card_5(A,B)) ) ) ) )).

fof(t47_card_5,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ! [B] :
          ( ( ~ v1_finset_1(B)
            & v1_card_1(B) )
         => ( ( r1_tarski(k1_card_5(A),B)
              & r2_hidden(B,A) )
           => k3_card_2(A,B) = k3_card_2(k6_card_3(k2_card_5(A,B)),k1_card_5(A)) ) ) ) )).

fof(dt_k1_card_5,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => v1_card_1(k1_card_5(A)) ) )).

fof(dt_k2_card_5,axiom,(
    ! [A,B] :
      ( ( v1_card_1(A)
        & v1_card_1(B) )
     => ( v1_relat_1(k2_card_5(A,B))
        & v1_funct_1(k2_card_5(A,B))
        & v1_card_3(k2_card_5(A,B)) ) ) )).

fof(t22_card_5,axiom,(
    ! [A,B] :
      ( v1_card_1(B)
     => ( A != k1_xboole_0
       => r1_tarski(k1_card_1(a_2_0_card_5(A,B)),k2_card_2(B,k3_card_2(k1_card_1(A),B))) ) ) )).

fof(fraenkel_a_2_0_card_5,axiom,(
    ! [A,B,C] :
      ( v1_card_1(C)
     => ( r2_hidden(A,a_2_0_card_5(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(B))
            & A = D
            & r2_hidden(k1_card_1(D),C) ) ) ) )).
%------------------------------------------------------------------------------