SET007 Axioms: SET007+171.ax


%------------------------------------------------------------------------------
% File     : SET007+171 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Mahlo and Inaccessible 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_lar [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   74 (   1 unit)
%            Number of atoms       :  501 (  19 equality)
%            Maximal formula depth :   15 (   8 average)
%            Number of connectives :  556 ( 129 ~  ;   1  |; 248  &)
%                                         (  18 <=>; 160 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   35 (   1 propositional; 0-3 arity)
%            Number of functors    :   27 (   2 constant; 0-3 arity)
%            Number of variables   :  180 (   0 singleton; 170 !;  10 ?)
%            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(cc1_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal1(A) ) ) )).

fof(cc2_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_ordinal1(A)
        & v4_ordinal1(A) )
     => ( ~ v1_xboole_0(A)
        & v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & ~ v1_finset_1(A) ) ) )).

fof(cc3_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v2_card_1(A) )
     => ( ~ v1_xboole_0(A)
        & v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) ) ) )).

fof(rc1_card_lar,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal1(A)
      & ~ v1_finset_1(A)
      & v1_card_1(A)
      & v1_card_5(A)
      & ~ v1_card_4(A) ) )).

fof(rc2_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ? [B] :
          ( m1_subset_1(B,A)
          & ~ v1_xboole_0(B)
          & v1_ordinal1(B)
          & v2_ordinal1(B)
          & v3_ordinal1(B)
          & v4_ordinal1(B)
          & ~ v1_finset_1(B)
          & v1_card_1(B) ) ) )).

fof(fc1_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_ordinal1(A) )
     => ( ~ v1_xboole_0(k4_classes1(A))
        & v1_ordinal1(k4_classes1(A)) ) ) )).

fof(d1_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( r1_card_lar(A,B)
        <=> ( r1_tarski(B,A)
            & k7_ordinal2(B) = A ) ) ) )).

fof(d2_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( r2_card_lar(A,B)
        <=> ( r1_tarski(B,A)
            & ! [C] :
                ( ( v3_ordinal1(C)
                  & v4_ordinal1(C)
                  & ~ v1_finset_1(C) )
               => ( ( r2_hidden(C,A)
                    & k7_ordinal2(k3_xboole_0(B,C)) = C )
                 => r2_hidden(C,B) ) ) ) ) ) )).

fof(d3_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( r3_card_lar(A,B)
        <=> ( r2_card_lar(A,B)
            & r1_card_lar(A,B) ) ) ) )).

fof(d4_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_card_lar(B,A)
          <=> k7_ordinal2(B) = A ) ) ) )).

fof(d5_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v2_card_lar(B,A)
          <=> ! [C] :
                ( ( v3_ordinal1(C)
                  & v4_ordinal1(C)
                  & ~ v1_finset_1(C) )
               => ( ( r2_hidden(C,A)
                    & k7_ordinal2(k3_xboole_0(B,C)) = C )
                 => r2_hidden(C,B) ) ) ) ) ) )).

fof(t1_card_lar,axiom,(
    $true )).

fof(t2_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( r3_card_lar(A,B)
          <=> ( v2_card_lar(B,A)
              & v1_card_lar(B,A) ) ) ) ) )).

fof(t3_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => r1_tarski(B,k7_ordinal2(B)) ) ) )).

fof(t4_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( ! [C] :
                ( v3_ordinal1(C)
               => ~ ( r2_hidden(C,B)
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ~ ( r2_hidden(D,B)
                            & r2_hidden(C,D) ) ) ) )
           => ( v1_xboole_0(B)
              | ( v3_ordinal1(k7_ordinal2(B))
                & v4_ordinal1(k7_ordinal2(B))
                & ~ v1_finset_1(k7_ordinal2(B)) ) ) ) ) ) )).

fof(t5_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( ~ ( ~ v1_card_lar(B,A)
                & ! [C] :
                    ( v3_ordinal1(C)
                   => ~ ( r2_hidden(C,A)
                        & r1_tarski(B,C) ) ) )
            & ~ ( ? [C] :
                    ( v3_ordinal1(C)
                    & r2_hidden(C,A)
                    & r1_tarski(B,C) )
                & v1_card_lar(B,A) ) ) ) ) )).

fof(t6_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( ( v3_ordinal1(B)
            & v4_ordinal1(B)
            & ~ v1_finset_1(B) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ~ ( k7_ordinal2(k3_xboole_0(C,B)) != B
                  & ! [D] :
                      ( v3_ordinal1(D)
                     => ~ ( r2_hidden(D,B)
                          & r1_tarski(k3_xboole_0(C,B),D) ) ) ) ) ) ) )).

fof(t7_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_card_lar(B,A)
          <=> ! [C] :
                ( v3_ordinal1(C)
               => ~ ( r2_hidden(C,A)
                    & ! [D] :
                        ( v3_ordinal1(D)
                       => ~ ( r2_hidden(D,B)
                            & r1_ordinal1(C,D) ) ) ) ) ) ) ) )).

fof(t8_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ~ ( v1_card_lar(B,A)
              & v1_xboole_0(B) ) ) ) )).

fof(t10_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ( ( v1_card_lar(C,A)
                  & r2_hidden(B,A) )
               => ( r2_hidden(k2_card_lar(A,C,B),C)
                  & r2_hidden(B,k2_card_lar(A,C,B)) ) ) ) ) ) )).

fof(t11_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ( v2_card_lar(k2_subset_1(A),A)
        & v1_card_lar(k2_subset_1(A),A) ) ) )).

fof(t12_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ( ( r2_hidden(B,A)
                  & v2_card_lar(C,A)
                  & v1_card_lar(C,A) )
               => ( v2_card_lar(k3_card_lar(A,C,B),A)
                  & v1_card_lar(k3_card_lar(A,C,B),A) ) ) ) ) ) )).

fof(t13_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(B,A)
           => ( v2_card_lar(k1_card_fil(A,B),A)
              & v1_card_lar(k1_card_fil(A,B),A) ) ) ) ) )).

fof(d7_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v3_card_lar(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(A))
               => ~ ( v2_card_lar(C,A)
                    & v1_card_lar(C,A)
                    & v1_xboole_0(k1_card_lar(A,B,C)) ) ) ) ) ) )).

fof(t14_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ( ( v3_card_lar(B,A)
                  & r1_tarski(B,C) )
               => v3_card_lar(C,A) ) ) ) ) )).

fof(d8_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( r4_card_lar(A,B)
        <=> ( r1_tarski(B,A)
            & ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(A))
               => ~ ( v2_card_lar(C,A)
                    & v1_card_lar(C,A)
                    & v1_xboole_0(k1_card_lar(A,B,C)) ) ) ) ) ) )).

fof(t15_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B,C] :
          ( ( r4_card_lar(A,B)
            & r1_tarski(B,C)
            & r1_tarski(C,A) )
         => r4_card_lar(A,C) ) ) )).

fof(t16_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v3_card_lar(B,A)
           => v1_card_lar(B,A) ) ) ) )).

fof(t17_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( v3_ordinal1(C)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ( r1_tarski(k3_xboole_0(D,B),C)
                   => r1_tarski(k1_card_lar(A,B,k4_card_lar(A,D)),k1_ordinal1(C)) ) ) ) ) ) )).

fof(t18_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ( r1_tarski(C,B)
               => r1_tarski(k4_card_lar(A,C),k1_ordinal1(B)) ) ) ) ) )).

fof(t19_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => v2_card_lar(k4_card_lar(A,B),A) ) ) )).

fof(t21_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( v1_card_lar(B,A)
           => r1_tarski(k1_card_5(A),k1_card_1(B)) ) ) ) )).

fof(t22_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
         => ( ! [C] :
                ( m1_card_lar(C,A,B)
               => v2_card_lar(C,A) )
           => v2_card_lar(k6_setfam_1(A,B),A) ) ) ) )).

fof(t23_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( r2_hidden(k3_card_1(np__0),k1_card_5(A))
           => ! [C] :
                ( ( v1_funct_1(C)
                  & v1_funct_2(C,k5_numbers,B)
                  & m2_relset_1(C,k5_numbers,B) )
               => r2_hidden(k7_ordinal2(k2_relat_1(C)),A) ) ) ) ) )).

fof(t24_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( r2_hidden(k3_card_1(np__0),k1_card_5(A))
       => ! [B] :
            ( ( ~ v1_xboole_0(B)
              & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
           => ( ( r2_hidden(k1_card_1(B),k1_card_5(A))
                & ! [C] :
                    ( m1_card_lar(C,A,B)
                   => ( v2_card_lar(C,A)
                      & v1_card_lar(C,A) ) ) )
             => ( v2_card_lar(k6_setfam_1(A,B),A)
                & v1_card_lar(k6_setfam_1(A,B),A) ) ) ) ) ) )).

fof(t25_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( ( r2_hidden(k3_card_1(np__0),k1_card_5(A))
              & v1_card_lar(B,A) )
           => ! [C] :
                ( v3_ordinal1(C)
               => ~ ( r2_hidden(C,A)
                    & ! [D] :
                        ( ( v3_ordinal1(D)
                          & v4_ordinal1(D)
                          & ~ v1_finset_1(D) )
                       => ~ ( r2_hidden(D,A)
                            & r2_hidden(C,D)
                            & r2_hidden(D,k4_card_lar(A,B)) ) ) ) ) ) ) ) )).

fof(t26_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( ( r2_hidden(k3_card_1(np__0),k1_card_5(A))
              & v1_card_lar(B,A) )
           => v1_card_lar(k4_card_lar(A,B),A) ) ) ) )).

fof(t27_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v5_card_lar(A)
       => v4_card_lar(A) ) ) )).

fof(t28_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v4_card_lar(A)
       => v1_card_5(A) ) ) )).

fof(t29_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v4_card_lar(A)
       => v2_card_1(A) ) ) )).

fof(t30_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v4_card_lar(A)
       => v4_card_fil(A) ) ) )).

fof(t31_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v5_card_lar(A)
       => v5_card_fil(A) ) ) )).

fof(t32_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v5_card_lar(A)
       => v6_card_fil(A) ) ) )).

fof(t33_card_lar,axiom,(
    ! [A] :
      ( ! [B] :
          ~ ( r2_hidden(B,A)
            & ! [C] :
                ~ ( r2_hidden(C,A)
                  & r1_tarski(B,C)
                  & v1_card_1(C) ) )
     => v1_card_1(k3_tarski(A)) ) )).

fof(t34_card_lar,axiom,(
    ! [A,B] :
      ( ( ~ v1_finset_1(B)
        & v1_card_1(B) )
     => ( ( r2_hidden(k1_card_1(A),k1_card_5(B))
          & ! [C] :
              ( r2_hidden(C,A)
             => r2_hidden(k1_card_1(C),B) ) )
       => r2_hidden(k1_card_1(k3_tarski(A)),B) ) ) )).

fof(t35_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ( ( v6_card_fil(A)
              & r2_hidden(B,A) )
           => r2_hidden(k1_card_1(k4_classes1(B)),A) ) ) ) )).

fof(t36_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v6_card_fil(A)
       => k1_card_1(k4_classes1(A)) = A ) ) )).

fof(t37_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v6_card_fil(A)
       => v2_classes1(k4_classes1(A)) ) ) )).

fof(t38_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v3_ordinal1(A) )
     => ~ v1_xboole_0(k4_classes1(A)) ) )).

fof(t39_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v6_card_fil(A)
       => ( ~ v1_xboole_0(k4_classes1(A))
          & v1_classes2(k4_classes1(A)) ) ) ) )).

fof(t40_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v6_card_fil(A)
       => v1_zf_model(k4_classes1(A)) ) ) )).

fof(dt_m1_card_lar,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_card_lar(C,A,B)
         => m1_subset_1(C,k1_zfmisc_1(A)) ) ) )).

fof(existence_m1_card_lar,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ? [C] : m1_card_lar(C,A,B) ) )).

fof(redefinition_m1_card_lar,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
     => ! [C] :
          ( m1_card_lar(C,A,B)
        <=> m1_subset_1(C,B) ) ) )).

fof(dt_k1_card_lar,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => m1_subset_1(k1_card_lar(A,B,C),k1_zfmisc_1(A)) ) )).

fof(commutativity_k1_card_lar,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => k1_card_lar(A,B,C) = k1_card_lar(A,C,B) ) )).

fof(idempotence_k1_card_lar,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => k1_card_lar(A,B,B) = B ) )).

fof(redefinition_k1_card_lar,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(C,k1_zfmisc_1(A))
     => k1_card_lar(A,B,C) = k3_xboole_0(B,C) ) )).

fof(dt_k2_card_lar,axiom,(
    ! [A,B,C] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A)
        & m1_subset_1(B,k1_zfmisc_1(A))
        & v3_ordinal1(C) )
     => m1_subset_1(k2_card_lar(A,B,C),B) ) )).

fof(dt_k3_card_lar,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => m1_subset_1(k3_card_lar(A,B,C),k1_zfmisc_1(A)) ) )).

fof(redefinition_k3_card_lar,axiom,(
    ! [A,B,C] :
      ( m1_subset_1(B,k1_zfmisc_1(A))
     => k3_card_lar(A,B,C) = k4_xboole_0(B,C) ) )).

fof(dt_k4_card_lar,axiom,(
    ! [A,B] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A)
        & m1_subset_1(B,k1_zfmisc_1(A)) )
     => m1_subset_1(k4_card_lar(A,B),k1_zfmisc_1(A)) ) )).

fof(t9_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( v3_ordinal1(B)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ~ ( v1_card_lar(C,A)
                  & r2_hidden(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,A)
                     => ~ r2_hidden(D,a_3_0_card_lar(A,B,C)) ) ) ) ) ) )).

fof(d6_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( v3_ordinal1(C)
             => ( ( v1_card_lar(B,A)
                  & r2_hidden(C,A) )
               => k2_card_lar(A,B,C) = k6_ordinal2(a_3_1_card_lar(A,B,C)) ) ) ) ) )).

fof(d9_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => k4_card_lar(A,B) = a_2_0_card_lar(A,B) ) ) )).

fof(t20_card_lar,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A)
        & ~ v1_finset_1(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ~ ( v1_card_lar(B,A)
              & ~ v1_card_lar(k4_card_lar(A,B),A)
              & ! [C] :
                  ( v3_ordinal1(C)
                 => ~ ( r2_hidden(C,A)
                      & r3_card_lar(A,a_3_2_card_lar(A,B,C)) ) ) ) ) ) )).

fof(d10_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v4_card_lar(A)
      <=> r4_card_lar(A,a_1_0_card_lar(A)) ) ) )).

fof(d11_card_lar,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v1_card_4(A) )
     => ( v5_card_lar(A)
      <=> r4_card_lar(A,a_1_1_card_lar(A)) ) ) )).

fof(fraenkel_a_3_0_card_lar,axiom,(
    ! [A,B,C,D] :
      ( ( v3_ordinal1(B)
        & v4_ordinal1(B)
        & ~ v1_finset_1(B)
        & v3_ordinal1(C)
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_0_card_lar(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & r2_hidden(E,D)
            & r2_hidden(C,E) ) ) ) )).

fof(fraenkel_a_3_1_card_lar,axiom,(
    ! [A,B,C,D] :
      ( ( v3_ordinal1(B)
        & v4_ordinal1(B)
        & ~ v1_finset_1(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & v3_ordinal1(D) )
     => ( r2_hidden(A,a_3_1_card_lar(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = E
            & r2_hidden(E,C)
            & r2_hidden(D,E) ) ) ) )).

fof(fraenkel_a_2_0_card_lar,axiom,(
    ! [A,B,C] :
      ( ( v3_ordinal1(B)
        & v4_ordinal1(B)
        & ~ v1_finset_1(B)
        & m1_subset_1(C,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_2_0_card_lar(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,B)
            & A = D
            & ~ v1_finset_1(D)
            & v4_ordinal1(D)
            & k7_ordinal2(k3_xboole_0(C,D)) = D ) ) ) )).

fof(fraenkel_a_3_2_card_lar,axiom,(
    ! [A,B,C,D] :
      ( ( v3_ordinal1(B)
        & v4_ordinal1(B)
        & ~ v1_finset_1(B)
        & m1_subset_1(C,k1_zfmisc_1(B))
        & v3_ordinal1(D) )
     => ( r2_hidden(A,a_3_2_card_lar(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = k1_ordinal1(E)
            & r2_hidden(E,C)
            & r2_hidden(D,k1_ordinal1(E)) ) ) ) )).

fof(fraenkel_a_1_0_card_lar,axiom,(
    ! [A,B] :
      ( ( ~ v1_finset_1(B)
        & v1_card_1(B)
        & ~ v1_card_4(B) )
     => ( r2_hidden(A,a_1_0_card_lar(B))
      <=> ? [C] :
            ( ~ v1_finset_1(C)
            & v1_card_1(C)
            & m1_subset_1(C,B)
            & A = C
            & v1_card_5(C) ) ) ) )).

fof(fraenkel_a_1_1_card_lar,axiom,(
    ! [A,B] :
      ( ( ~ v1_finset_1(B)
        & v1_card_1(B)
        & ~ v1_card_4(B) )
     => ( r2_hidden(A,a_1_1_card_lar(B))
      <=> ? [C] :
            ( ~ v1_finset_1(C)
            & v1_card_1(C)
            & m1_subset_1(C,B)
            & A = C
            & v6_card_fil(C) ) ) ) )).
%------------------------------------------------------------------------------