SET007 Axioms: SET007+148.ax


%------------------------------------------------------------------------------
% File     : SET007+148 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Countable Sets and Hessenberg's Theorem
% 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_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   97 (  10 unit)
%            Number of atoms       :  414 (  57 equality)
%            Maximal formula depth :   17 (   6 average)
%            Number of connectives :  390 (  73 ~  ;  16  |; 124  &)
%                                         (  20 <=>; 157 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   1 propositional; 0-3 arity)
%            Number of functors    :   42 (  11 constant; 0-4 arity)
%            Number of variables   :  169 (   0 singleton; 159 !;  10 ?)
%            Maximal term depth    :    4 (   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_4,axiom,
    ( ~ v1_xboole_0(k5_ordinal2)
    & v1_ordinal1(k5_ordinal2)
    & v2_ordinal1(k5_ordinal2)
    & v3_ordinal1(k5_ordinal2)
    & ~ v1_finset_1(k5_ordinal2)
    & v1_membered(k5_ordinal2)
    & v2_membered(k5_ordinal2)
    & v3_membered(k5_ordinal2)
    & v4_membered(k5_ordinal2)
    & v5_membered(k5_ordinal2) )).

fof(t1_card_4,axiom,(
    ! [A] :
      ( v1_finset_1(A)
    <=> v1_finset_1(k1_card_1(A)) ) )).

fof(t2_card_4,axiom,(
    ! [A] :
      ( v1_finset_1(A)
    <=> r2_hidden(k1_card_1(A),k3_card_1(np__0)) ) )).

fof(t3_card_4,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ( r2_hidden(k1_card_1(A),k3_card_1(np__0))
        & r2_hidden(k1_card_1(A),k5_ordinal2) ) ) )).

fof(t4_card_4,axiom,(
    ! [A] :
      ( v1_finset_1(A)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & k1_card_1(A) = k1_card_1(B) ) ) )).

fof(t5_card_4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k4_xboole_0(k1_ordinal1(A),k1_tarski(A)) = A ) )).

fof(t6_card_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_wellord2(B,A)
           => B = A ) ) ) )).

fof(t7_card_4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( v1_finset_1(A)
      <=> r2_hidden(A,k5_ordinal2) ) ) )).

fof(t8_card_4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( ~ v1_finset_1(A)
      <=> r1_ordinal1(k5_ordinal2,A) ) ) )).

fof(t9_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( v1_finset_1(A)
      <=> r2_hidden(A,k3_card_1(np__0)) ) ) )).

fof(t10_card_4,axiom,(
    $true )).

fof(t11_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( ~ v1_finset_1(A)
      <=> r1_ordinal1(k3_card_1(np__0),A) ) ) )).

fof(t12_card_4,axiom,(
    $true )).

fof(t13_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( v1_finset_1(A)
           => ( v1_finset_1(B)
              | ( r2_hidden(A,B)
                & r1_tarski(A,B) ) ) ) ) ) )).

fof(t14_card_4,axiom,(
    ! [A] :
      ( ~ ( ~ v1_finset_1(A)
          & ! [B] :
              ~ ( r1_tarski(B,A)
                & k1_card_1(B) = k3_card_1(np__0) ) )
      & ~ ( ? [B] :
              ( r1_tarski(B,A)
              & k1_card_1(B) = k3_card_1(np__0) )
          & v1_finset_1(A) ) ) )).

fof(t15_card_4,axiom,(
    ~ v1_finset_1(k5_numbers) )).

fof(t16_card_4,axiom,(
    $true )).

fof(t17_card_4,axiom,(
    ! [A] :
      ( A = k1_xboole_0
    <=> k1_card_1(A) = np__0 ) )).

fof(t18_card_4,axiom,(
    $true )).

fof(t19_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => r1_tarski(np__0,A) ) )).

fof(t20_card_4,axiom,(
    ! [A,B] :
      ( k1_card_1(A) = k1_card_1(B)
    <=> k2_card_1(A) = k2_card_1(B) ) )).

fof(t21_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( A = B
          <=> k2_card_1(B) = k2_card_1(A) ) ) ) )).

fof(t22_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( r2_hidden(A,B)
          <=> r1_tarski(k2_card_1(A),B) ) ) ) )).

fof(t23_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( r2_hidden(A,k2_card_1(B))
          <=> r1_tarski(A,B) ) ) ) )).

fof(t24_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( r2_hidden(np__0,A)
      <=> r1_tarski(np__1,A) ) ) )).

fof(t25_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( r2_hidden(np__1,A)
      <=> r1_tarski(np__2,A) ) ) )).

fof(t26_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( v1_finset_1(A)
           => ( ( ~ r1_tarski(B,A)
                & ~ r2_hidden(B,A) )
              | v1_finset_1(B) ) ) ) ) )).

fof(t27_card_4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( v4_ordinal1(A)
      <=> ! [B] :
            ( v3_ordinal1(B)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r2_hidden(B,A)
                 => r2_hidden(k14_ordinal2(B,C),A) ) ) ) ) ) )).

fof(t28_card_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( k14_ordinal2(B,k1_ordinal1(A)) = k14_ordinal2(k1_ordinal1(B),A)
            & k14_ordinal2(B,k1_nat_1(A,np__1)) = k14_ordinal2(k1_ordinal1(B),A) ) ) ) )).

fof(t29_card_4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & k15_ordinal2(A,k1_ordinal1(k4_ordinal2)) = k14_ordinal2(A,B) ) ) )).

fof(t30_card_4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( v4_ordinal1(A)
       => k15_ordinal2(A,k1_ordinal1(k4_ordinal2)) = A ) ) )).

fof(t31_card_4,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( r1_ordinal1(k5_ordinal2,A)
       => k14_ordinal2(k4_ordinal2,A) = A ) ) )).

fof(t32_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( ~ v1_finset_1(A)
       => v4_ordinal1(A) ) ) )).

fof(t33_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( ~ v1_finset_1(A)
       => k1_card_2(A,A) = A ) ) )).

fof(t34_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ~ ( ~ v1_finset_1(A)
              & ( r1_tarski(B,A)
                | r2_hidden(B,A) )
              & ~ ( k1_card_2(A,B) = A
                  & k1_card_2(B,A) = A ) ) ) ) )).

fof(t35_card_4,axiom,(
    ! [A,B] :
      ~ ( ~ v1_finset_1(A)
        & ( r2_wellord2(A,B)
          | r2_wellord2(B,A) )
        & ~ ( r2_wellord2(k2_xboole_0(A,B),A)
            & k1_card_1(k2_xboole_0(A,B)) = k1_card_1(A) ) ) )).

fof(t36_card_4,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ( v1_finset_1(A)
        | ( r2_wellord2(k2_xboole_0(A,B),A)
          & k1_card_1(k2_xboole_0(A,B)) = k1_card_1(A) ) ) ) )).

fof(t37_card_4,axiom,(
    ! [A,B] :
      ~ ( ~ v1_finset_1(A)
        & ( r2_hidden(k1_card_1(B),k1_card_1(A))
          | r1_tarski(k1_card_1(B),k1_card_1(A)) )
        & ~ ( r2_wellord2(k2_xboole_0(A,B),A)
            & k1_card_1(k2_xboole_0(A,B)) = k1_card_1(A) ) ) )).

fof(t38_card_4,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & v1_card_1(A) )
     => ! [B] :
          ( ( v1_finset_1(B)
            & v1_card_1(B) )
         => v1_finset_1(k1_card_2(A,B)) ) ) )).

fof(t39_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( ~ v1_finset_1(A)
           => ( ~ v1_finset_1(k1_card_2(A,B))
              & ~ v1_finset_1(k1_card_2(B,A)) ) ) ) ) )).

fof(t40_card_4,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & v1_card_1(A) )
     => ! [B] :
          ( ( v1_finset_1(B)
            & v1_card_1(B) )
         => v1_finset_1(k2_card_2(A,B)) ) ) )).

fof(t41_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => ! [D] :
                  ( v1_card_1(D)
                 => ( ~ ( ~ ( r2_hidden(A,B)
                            & r2_hidden(C,D) )
                        & ~ ( r1_tarski(A,B)
                            & r2_hidden(C,D) )
                        & ~ ( r2_hidden(A,B)
                            & r1_tarski(C,D) )
                        & ~ ( r1_tarski(A,B)
                            & r1_tarski(C,D) ) )
                   => ( r1_tarski(k1_card_2(A,C),k1_card_2(B,D))
                      & r1_tarski(k1_card_2(C,A),k1_card_2(B,D)) ) ) ) ) ) ) )).

fof(t42_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => ( ( r2_hidden(A,B)
                  | r1_tarski(A,B) )
               => ( r1_tarski(k1_card_2(C,A),k1_card_2(C,B))
                  & r1_tarski(k1_card_2(C,A),k1_card_2(B,C))
                  & r1_tarski(k1_card_2(A,C),k1_card_2(C,B))
                  & r1_tarski(k1_card_2(A,C),k1_card_2(B,C)) ) ) ) ) ) )).

fof(d1_card_4,axiom,(
    ! [A] :
      ( v1_card_4(A)
    <=> r1_tarski(k1_card_1(A),k3_card_1(np__0)) ) )).

fof(t43_card_4,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => v1_card_4(A) ) )).

fof(t44_card_4,axiom,
    ( v1_card_4(k5_ordinal2)
    & v1_card_4(k5_numbers) )).

fof(t45_card_4,axiom,(
    ! [A] :
      ( v1_card_4(A)
    <=> ? [B] :
          ( v1_relat_1(B)
          & v1_funct_1(B)
          & k1_relat_1(B) = k5_numbers
          & r1_tarski(A,k2_relat_1(B)) ) ) )).

fof(t46_card_4,axiom,(
    ! [A,B] :
      ( ( r1_tarski(A,B)
        & v1_card_4(B) )
     => v1_card_4(A) ) )).

fof(t47_card_4,axiom,(
    ! [A,B] :
      ( ( v1_card_4(A)
        & v1_card_4(B) )
     => v1_card_4(k2_xboole_0(A,B)) ) )).

fof(t48_card_4,axiom,(
    ! [A,B] :
      ( v1_card_4(A)
     => ( v1_card_4(k3_xboole_0(A,B))
        & v1_card_4(k3_xboole_0(B,A)) ) ) )).

fof(t49_card_4,axiom,(
    ! [A,B] :
      ( v1_card_4(A)
     => v1_card_4(k4_xboole_0(A,B)) ) )).

fof(t50_card_4,axiom,(
    ! [A,B] :
      ( ( v1_card_4(A)
        & v1_card_4(B) )
     => v1_card_4(k5_xboole_0(A,B)) ) )).

fof(t51_card_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ ( ~ ( B = np__0
                    & A != np__0 )
                & k3_newton(B,A) = np__0 )
            & ~ ( k3_newton(B,A) != np__0
                & B = np__0
                & A != np__0 ) ) ) ) )).

fof(t52_card_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( k2_nat_1(k1_card_4(np__2,A),k1_nat_1(k2_nat_1(np__2,B),np__1)) = k2_nat_1(k1_card_4(np__2,C),k1_nat_1(k2_nat_1(np__2,D),np__1))
                   => ( A = C
                      & B = D ) ) ) ) ) ) )).

fof(t53_card_4,axiom,
    ( r2_wellord2(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),k5_numbers)
    & k1_card_1(k5_numbers) = k1_card_1(k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers)) )).

fof(t54_card_4,axiom,(
    k2_card_2(k3_card_1(np__0),k3_card_1(np__0)) = k3_card_1(np__0) )).

fof(t55_card_4,axiom,(
    ! [A,B] :
      ( ( v1_card_4(A)
        & v1_card_4(B) )
     => v1_card_4(k2_zfmisc_1(A,B)) ) )).

fof(t56_card_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( r2_wellord2(k4_finseq_2(np__1,A),A)
        & k1_card_1(k4_finseq_2(np__1,A)) = k1_card_1(A) ) ) )).

fof(t57_card_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_wellord2(k2_zfmisc_1(k4_finseq_2(B,A),k4_finseq_2(C,A)),k4_finseq_2(k1_nat_1(B,C),A))
                & k1_card_1(k2_zfmisc_1(k4_finseq_2(B,A),k4_finseq_2(C,A))) = k1_card_1(k4_finseq_2(k1_nat_1(B,C),A)) ) ) ) ) )).

fof(t58_card_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( v1_card_4(A)
           => v1_card_4(k4_finseq_2(B,A)) ) ) ) )).

fof(t59_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( ( r1_tarski(k1_card_1(k1_relat_1(C)),A)
                  & ! [D] :
                      ( r2_hidden(D,k1_relat_1(C))
                     => r1_tarski(k1_card_1(k1_funct_1(C,D)),B) ) )
               => r1_tarski(k1_card_1(k3_card_3(C)),k2_card_2(A,B)) ) ) ) ) )).

fof(t60_card_4,axiom,(
    ! [A,B] :
      ( v1_card_1(B)
     => ! [C] :
          ( v1_card_1(C)
         => ( ( r1_tarski(k1_card_1(A),B)
              & ! [D] :
                  ( r2_hidden(D,A)
                 => r1_tarski(k1_card_1(D),C) ) )
           => r1_tarski(k1_card_1(k3_tarski(A)),k2_card_2(B,C)) ) ) ) )).

fof(t61_card_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ( v1_card_4(k1_relat_1(A))
          & ! [B] :
              ( r2_hidden(B,k1_relat_1(A))
             => v1_card_4(k1_funct_1(A,B)) ) )
       => v1_card_4(k3_card_3(A)) ) ) )).

fof(t62_card_4,axiom,(
    ! [A] :
      ( ( v1_card_4(A)
        & ! [B] :
            ( r2_hidden(B,A)
           => v1_card_4(B) ) )
     => v1_card_4(k3_tarski(A)) ) )).

fof(t63_card_4,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( ( v1_finset_1(k1_relat_1(A))
          & ! [B] :
              ( r2_hidden(B,k1_relat_1(A))
             => v1_finset_1(k1_funct_1(A,B)) ) )
       => v1_finset_1(k3_card_3(A)) ) ) )).

fof(t64_card_4,axiom,(
    $true )).

fof(t65_card_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( v1_card_4(A)
       => v1_card_4(k3_finseq_2(A)) ) ) )).

fof(t66_card_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => r1_tarski(k3_card_1(np__0),k1_card_1(k3_finseq_2(A))) ) )).

fof(t67_card_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_tarski(k2_card_2(k3_card_1(np__0),k1_card_1(A)),k3_card_1(np__0))
        & r1_tarski(k2_card_2(k1_card_1(A),k3_card_1(np__0)),k3_card_1(np__0)) ) ) )).

fof(t68_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => ! [D] :
                  ( v1_card_1(D)
                 => ( ~ ( ~ ( r2_hidden(A,B)
                            & r2_hidden(C,D) )
                        & ~ ( r1_tarski(A,B)
                            & r2_hidden(C,D) )
                        & ~ ( r2_hidden(A,B)
                            & r1_tarski(C,D) )
                        & ~ ( r1_tarski(A,B)
                            & r1_tarski(C,D) ) )
                   => ( r1_tarski(k2_card_2(A,C),k2_card_2(B,D))
                      & r1_tarski(k2_card_2(C,A),k2_card_2(B,D)) ) ) ) ) ) ) )).

fof(t69_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => ( ( r2_hidden(A,B)
                  | r1_tarski(A,B) )
               => ( r1_tarski(k2_card_2(C,A),k2_card_2(C,B))
                  & r1_tarski(k2_card_2(C,A),k2_card_2(B,C))
                  & r1_tarski(k2_card_2(A,C),k2_card_2(C,B))
                  & r1_tarski(k2_card_2(A,C),k2_card_2(B,C)) ) ) ) ) ) )).

fof(t70_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => ! [D] :
                  ( v1_card_1(D)
                 => ~ ( ~ ( ~ ( r2_hidden(A,B)
                              & r2_hidden(C,D) )
                          & ~ ( r1_tarski(A,B)
                              & r2_hidden(C,D) )
                          & ~ ( r2_hidden(A,B)
                              & r1_tarski(C,D) )
                          & ~ ( r1_tarski(A,B)
                              & r1_tarski(C,D) ) )
                      & A != np__0
                      & ~ r1_tarski(k3_card_2(A,C),k3_card_2(B,D)) ) ) ) ) ) )).

fof(t71_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => ~ ( ( r2_hidden(A,B)
                    | r1_tarski(A,B) )
                  & C != np__0
                  & ~ ( r1_tarski(k3_card_2(C,A),k3_card_2(C,B))
                      & r1_tarski(k3_card_2(A,C),k3_card_2(B,C)) ) ) ) ) ) )).

fof(t72_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( r1_tarski(A,k1_card_2(A,B))
            & r1_tarski(B,k1_card_2(A,B)) ) ) ) )).

fof(t73_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( A != np__0
           => ( r1_tarski(B,k2_card_2(B,A))
              & r1_tarski(B,k2_card_2(A,B)) ) ) ) ) )).

fof(t74_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => ! [D] :
                  ( v1_card_1(D)
                 => ( ( r2_hidden(A,B)
                      & r2_hidden(C,D) )
                   => ( r2_hidden(k1_card_2(A,C),k1_card_2(B,D))
                      & r2_hidden(k1_card_2(C,A),k1_card_2(B,D)) ) ) ) ) ) ) )).

fof(t75_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => ( r2_hidden(k1_card_2(A,B),k1_card_2(A,C))
               => r2_hidden(B,C) ) ) ) ) )).

fof(t76_card_4,axiom,(
    ! [A,B] :
      ( ( k1_card_2(k1_card_1(A),k1_card_1(B)) = k1_card_1(A)
        & r2_hidden(k1_card_1(B),k1_card_1(A)) )
     => k1_card_1(k4_xboole_0(A,B)) = k1_card_1(A) ) )).

fof(t77_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( ~ v1_finset_1(A)
       => k2_card_2(A,A) = A ) ) )).

fof(t78_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( r2_hidden(np__0,B)
           => ( v1_finset_1(A)
              | ( ~ r1_tarski(B,A)
                & ~ r2_hidden(B,A) )
              | ( k2_card_2(A,B) = A
                & k2_card_2(B,A) = A ) ) ) ) ) )).

fof(t79_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ~ ( ~ v1_finset_1(A)
              & ( r1_tarski(B,A)
                | r2_hidden(B,A) )
              & ~ ( r1_tarski(k2_card_2(A,B),A)
                  & r1_tarski(k2_card_2(B,A),A) ) ) ) ) )).

fof(t80_card_4,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ( r2_wellord2(k2_zfmisc_1(A,A),A)
        & k1_card_1(k2_zfmisc_1(A,A)) = k1_card_1(A) ) ) )).

fof(t81_card_4,axiom,(
    ! [A,B] :
      ( v1_finset_1(B)
     => ( v1_finset_1(A)
        | B = k1_xboole_0
        | ( r2_wellord2(k2_zfmisc_1(A,B),A)
          & k1_card_1(k2_zfmisc_1(A,B)) = k1_card_1(A) ) ) ) )).

fof(t82_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => ! [D] :
                  ( v1_card_1(D)
                 => ( ( r2_hidden(A,B)
                      & r2_hidden(C,D) )
                   => ( r2_hidden(k2_card_2(A,C),k2_card_2(B,D))
                      & r2_hidden(k2_card_2(C,A),k2_card_2(B,D)) ) ) ) ) ) ) )).

fof(t83_card_4,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( v1_card_1(C)
             => ( r2_hidden(k2_card_2(A,B),k2_card_2(A,C))
               => r2_hidden(B,C) ) ) ) ) )).

fof(t84_card_4,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => k1_card_1(A) = k2_card_2(k3_card_1(np__0),k1_card_1(A)) ) )).

fof(t85_card_4,axiom,(
    ! [A,B] :
      ( v1_finset_1(A)
     => ( A = k1_xboole_0
        | v1_finset_1(B)
        | k2_card_2(k1_card_1(B),k1_card_1(A)) = k1_card_1(B) ) ) )).

fof(t86_card_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ v1_finset_1(A)
              & B != np__0
              & ~ ( r2_wellord2(k4_finseq_2(B,A),A)
                  & k1_card_1(k4_finseq_2(B,A)) = k1_card_1(A) ) ) ) ) )).

fof(t87_card_4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v1_finset_1(A)
       => k1_card_1(A) = k1_card_1(k3_finseq_2(A)) ) ) )).

fof(dt_k1_card_4,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k1_card_4(A,B),k1_numbers,k5_numbers) ) )).

fof(redefinition_k1_card_4,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k1_card_4(A,B) = k2_newton(A,B) ) )).

fof(s1_card_4,axiom,(
    v1_card_4(a_0_0_card_4) )).

fof(s2_card_4,axiom,(
    v1_card_4(a_0_1_card_4) )).

fof(s3_card_4,axiom,(
    v1_card_4(a_0_2_card_4) )).

fof(fraenkel_a_0_0_card_4,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_card_4)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = f1_s1_card_4(B)
          & p1_s1_card_4(B) ) ) )).

fof(fraenkel_a_0_1_card_4,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_card_4)
    <=> ? [B,C] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & m2_subset_1(C,k1_numbers,k5_numbers)
          & A = f1_s2_card_4(B,C)
          & p1_s2_card_4(B,C) ) ) )).

fof(fraenkel_a_0_2_card_4,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_2_card_4)
    <=> ? [B,C,D] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & m2_subset_1(C,k1_numbers,k5_numbers)
          & m2_subset_1(D,k1_numbers,k5_numbers)
          & A = f1_s3_card_4(B,C,D)
          & p1_s3_card_4(B,C,D) ) ) )).
%------------------------------------------------------------------------------