SET007 Axioms: SET007+170.ax


%------------------------------------------------------------------------------
% File     : SET007+170 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Basic Facts about Inaccessible and Measurable 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_fil [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   95 (   5 unit)
%            Number of atoms       :  552 (  29 equality)
%            Maximal formula depth :   17 (   8 average)
%            Number of connectives :  612 ( 155 ~  ;   3  |; 227  &)
%                                         (  30 <=>; 197 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   1 propositional; 0-3 arity)
%            Number of functors    :   37 (   6 constant; 0-6 arity)
%            Number of variables   :  217 (   0 singleton; 198 !;  19 ?)
%            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_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ( v1_ordinal1(k1_card_1(A))
        & v2_ordinal1(k1_card_1(A))
        & v3_ordinal1(k1_card_1(A))
        & ~ v1_xboole_0(k1_card_1(A))
        & ~ v1_finset_1(k1_card_1(A))
        & v1_card_1(k1_card_1(A)) ) ) )).

fof(fc2_card_fil,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ~ v1_xboole_0(k7_setfam_1(A,B)) ) )).

fof(rc1_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ? [B] :
          ( m1_card_fil(B,A)
          & ~ v1_xboole_0(B)
          & ~ v2_card_fil(B,A)
          & v3_card_fil(B,A) ) ) )).

fof(cc1_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ( ( v1_card_fil(B,A)
              & v3_card_fil(B,A) )
           => ~ v2_card_fil(B,A) ) ) ) )).

fof(cc2_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & v4_card_fil(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & ~ v1_xboole_0(A)
        & ~ v1_finset_1(A)
        & v1_card_1(A)
        & v2_card_1(A)
        & v1_card_5(A) ) ) )).

fof(cc3_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & v5_card_fil(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & ~ v1_xboole_0(A)
        & ~ v1_finset_1(A)
        & v1_card_1(A)
        & v2_card_1(A) ) ) )).

fof(cc4_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & v6_card_fil(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & ~ v1_xboole_0(A)
        & ~ v1_finset_1(A)
        & v1_card_1(A)
        & v2_card_1(A)
        & v1_card_5(A)
        & v5_card_fil(A) ) ) )).

fof(cc5_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & v6_card_fil(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & ~ v1_xboole_0(A)
        & ~ v1_finset_1(A)
        & v1_card_1(A)
        & v2_card_1(A)
        & v1_card_5(A)
        & v4_card_fil(A) ) ) )).

fof(fc3_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v1_ordinal1(k2_card_1(A))
        & v2_ordinal1(k2_card_1(A))
        & v3_ordinal1(k2_card_1(A))
        & ~ v1_xboole_0(k2_card_1(A))
        & ~ v1_finset_1(k2_card_1(A))
        & v1_card_1(k2_card_1(A))
        & ~ v2_card_1(k2_card_1(A)) ) ) )).

fof(rc2_card_fil,axiom,(
    ? [A] :
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & ~ v1_xboole_0(A)
      & ~ v1_finset_1(A)
      & v1_card_1(A)
      & ~ v2_card_1(A) ) )).

fof(cc6_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v2_card_1(A) )
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & ~ v1_xboole_0(A)
        & ~ v1_finset_1(A)
        & v1_card_1(A)
        & v1_card_5(A) ) ) )).

fof(fc4_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v2_card_1(A) )
     => ( v1_ordinal1(k7_card_fil(A))
        & v2_ordinal1(k7_card_fil(A))
        & v3_ordinal1(k7_card_fil(A))
        & ~ v1_xboole_0(k7_card_fil(A))
        & ~ v1_finset_1(k7_card_fil(A))
        & v1_card_1(k7_card_fil(A)) ) ) )).

fof(t1_card_fil,axiom,(
    ! [A,B] :
      ( ~ v1_finset_1(B)
     => r2_hidden(k1_card_1(k1_tarski(A)),k1_card_1(B)) ) )).

fof(t2_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v1_xboole_0(k1_tarski(A))
        & m1_subset_1(k1_tarski(A),k1_zfmisc_1(k1_zfmisc_1(A)))
        & ~ r2_hidden(k1_xboole_0,k1_tarski(A))
        & ! [B] :
            ( m1_subset_1(B,k1_zfmisc_1(A))
           => ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(A))
               => ( ( ( r2_hidden(B,k1_tarski(A))
                      & r2_hidden(C,k1_tarski(A)) )
                   => r2_hidden(k5_subset_1(A,B,C),k1_tarski(A)) )
                  & ( ( r2_hidden(B,k1_tarski(A))
                      & r1_tarski(B,C) )
                   => r2_hidden(C,k1_tarski(A)) ) ) ) ) ) ) )).

fof(d1_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( m1_card_fil(B,A)
          <=> ( ~ r2_hidden(k1_xboole_0,B)
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(A))
                 => ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(A))
                     => ( ( ( r2_hidden(C,B)
                            & r2_hidden(D,B) )
                         => r2_hidden(k5_subset_1(A,C,D),B) )
                        & ( ( r2_hidden(C,B)
                            & r1_tarski(C,D) )
                         => r2_hidden(D,B) ) ) ) ) ) ) ) ) )).

fof(t3_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_card_fil(B,A)
        <=> ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
            & ~ r2_hidden(k1_xboole_0,B)
            & ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(A))
               => ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(A))
                   => ( ( ( r2_hidden(C,B)
                          & r2_hidden(D,B) )
                       => r2_hidden(k5_subset_1(A,C,D),B) )
                      & ( ( r2_hidden(C,B)
                          & r1_tarski(C,D) )
                       => r2_hidden(D,B) ) ) ) ) ) ) ) )).

fof(t4_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => m1_card_fil(k1_tarski(A),A) ) )).

fof(t5_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => r2_hidden(A,B) ) ) )).

fof(t6_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_card_fil(C,A)
             => ~ ( r2_hidden(B,C)
                  & r2_hidden(k1_card_fil(A,B),C) ) ) ) ) )).

fof(t7_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
             => ( ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(A))
                   => ( r2_hidden(D,C)
                    <=> r2_hidden(k3_subset_1(A,D),B) ) )
               => ( ~ r2_hidden(A,C)
                  & ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(A))
                     => ! [E] :
                          ( m1_subset_1(E,k1_zfmisc_1(A))
                         => ( ( ( r2_hidden(D,C)
                                & r2_hidden(E,C) )
                             => r2_hidden(k4_subset_1(A,D,E),C) )
                            & ( ( r2_hidden(D,C)
                                & r1_tarski(E,D) )
                             => r2_hidden(E,C) ) ) ) ) ) ) ) ) ) )).

fof(d2_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( m2_card_fil(B,A)
          <=> ( ~ r2_hidden(A,B)
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(A))
                 => ! [D] :
                      ( m1_subset_1(D,k1_zfmisc_1(A))
                     => ( ( ( r2_hidden(C,B)
                            & r2_hidden(D,B) )
                         => r2_hidden(k4_subset_1(A,C,D),B) )
                        & ( ( r2_hidden(C,B)
                            & r1_tarski(D,C) )
                         => r2_hidden(D,B) ) ) ) ) ) ) ) ) )).

fof(t10_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ! [C] :
              ( m2_card_fil(C,A)
             => ( ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(A))
                   => ~ ( r2_hidden(D,B)
                        & r2_hidden(D,k2_card_fil(A,B)) ) )
                & ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(A))
                   => ~ ( r2_hidden(D,C)
                        & r2_hidden(D,k7_setfam_1(A,C)) ) ) ) ) ) ) )).

fof(t11_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_card_fil(B,A)
         => r2_hidden(k1_xboole_0,B) ) ) )).

fof(d3_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
             => ( r1_card_fil(A,B,C)
              <=> ! [D] :
                    ( ~ v1_xboole_0(D)
                   => ( ( r1_tarski(D,C)
                        & r2_hidden(k1_card_1(D),B) )
                     => r2_hidden(k1_setfam_1(D),C) ) ) ) ) ) ) )).

fof(d4_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( v1_card_1(B)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
             => ( r2_card_fil(A,B,C)
              <=> ! [D] :
                    ( ~ v1_xboole_0(D)
                   => ( ( r1_tarski(D,C)
                        & r2_hidden(k1_card_1(D),B) )
                     => r2_hidden(k3_tarski(D),C) ) ) ) ) ) ) )).

fof(t12_card_fil,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B))) )
             => ( r1_card_fil(B,A,C)
               => r2_card_fil(B,A,k7_setfam_1(B,C)) ) ) ) ) )).

fof(d5_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ( v1_card_fil(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(A))
               => ( r2_hidden(C,B)
                 => k1_card_1(C) = k1_card_1(A) ) ) ) ) ) )).

fof(d6_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ( v2_card_fil(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,k1_zfmisc_1(A))
                & r2_hidden(C,B)
                & ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(A))
                   => ( r2_hidden(D,B)
                     => r1_tarski(C,D) ) ) ) ) ) ) )).

fof(d7_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ( v3_card_fil(B,A)
          <=> ! [C] :
                ( m1_subset_1(C,k1_zfmisc_1(A))
               => ( r2_hidden(C,B)
                  | r2_hidden(k1_card_fil(A,C),B) ) ) ) ) ) )).

fof(t13_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_card_fil(C,A)
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ( r2_hidden(D,k3_card_fil(A,C,B))
                  <=> ? [E] :
                        ( m1_subset_1(E,k1_zfmisc_1(A))
                        & r2_hidden(E,C)
                        & r1_tarski(k5_subset_1(A,E,B),D) ) ) ) ) ) ) )).

fof(t14_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ! [C] :
              ( m1_card_fil(C,A)
             => ( ! [D] :
                    ( m1_subset_1(D,k1_zfmisc_1(A))
                   => ~ ( r2_hidden(D,C)
                        & r1_xboole_0(D,B) ) )
               => ( r2_hidden(B,k3_card_fil(A,C,B))
                  & m1_card_fil(k3_card_fil(A,C,B),A)
                  & r1_tarski(C,k3_card_fil(A,C,B)) ) ) ) ) ) )).

fof(t15_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( r2_hidden(B,k4_card_fil(A))
        <=> m1_card_fil(B,A) ) ) )).

fof(t16_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k4_card_fil(A))) )
         => ( v6_ordinal1(B)
           => m1_card_fil(k3_tarski(B),A) ) ) ) )).

fof(t17_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ? [C] :
              ( m1_card_fil(C,A)
              & r1_tarski(B,C)
              & v3_card_fil(C,A) ) ) ) )).

fof(d11_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => k6_card_fil(A) = k2_card_fil(A,k5_card_fil(A)) ) )).

fof(t18_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( r2_hidden(B,k5_card_fil(A))
          <=> r2_hidden(k1_card_1(k1_card_fil(A,B)),k1_card_1(A)) ) ) ) )).

fof(t19_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(A))
         => ( r2_hidden(B,k6_card_fil(A))
          <=> r2_hidden(k1_card_1(B),k1_card_1(A)) ) ) ) )).

fof(t20_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ( r1_tarski(k5_card_fil(A),B)
           => v1_card_fil(B,A) ) ) ) )).

fof(t21_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ( ( v1_card_fil(B,A)
              & v3_card_fil(B,A) )
           => r1_tarski(k5_card_fil(A),B) ) ) ) )).

fof(t22_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B] :
          ( ( v3_card_fil(B,A)
            & m1_card_fil(B,A) )
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ( r2_hidden(C,B)
              <=> ~ r2_hidden(C,k2_card_fil(A,B)) ) ) ) ) )).

fof(t23_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ( ( v3_card_fil(B,A)
              & r1_card_fil(A,k1_card_1(A),B) )
           => ( v2_card_fil(B,A)
              | v1_card_fil(B,A) ) ) ) ) )).

fof(t24_card_fil,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => r1_tarski(k2_card_1(A),k3_card_2(np__2,A)) ) )).

fof(d12_card_fil,axiom,
    ( r3_card_fil
  <=> ! [A] :
        ( ( ~ v1_finset_1(A)
          & v1_card_1(A) )
       => k2_card_1(A) = k3_card_2(np__2,A) ) )).

fof(d13_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v4_card_fil(A)
      <=> ( v1_card_5(A)
          & v2_card_1(A) ) ) ) )).

fof(t25_card_fil,axiom,(
    v4_card_fil(k3_card_1(np__0)) )).

fof(d14_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v5_card_fil(A)
      <=> ! [B] :
            ( v1_card_1(B)
           => ( r2_hidden(B,A)
             => r2_hidden(k3_card_2(np__2,B),A) ) ) ) ) )).

fof(t26_card_fil,axiom,(
    v5_card_fil(k3_card_1(np__0)) )).

fof(t27_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v5_card_fil(A)
       => v2_card_1(A) ) ) )).

fof(t28_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( ( r3_card_fil
          & v2_card_1(A) )
       => v5_card_fil(A) ) ) )).

fof(d15_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v6_card_fil(A)
      <=> ( v1_card_5(A)
          & v5_card_fil(A) ) ) ) )).

fof(t29_card_fil,axiom,(
    v6_card_fil(k3_card_1(np__0)) )).

fof(t30_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v6_card_fil(A)
       => v4_card_fil(A) ) ) )).

fof(t31_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( ( r3_card_fil
          & v4_card_fil(A) )
       => v6_card_fil(A) ) ) )).

fof(d16_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v7_card_fil(A)
      <=> ? [B] :
            ( m1_card_fil(B,A)
            & r1_card_fil(A,A,B)
            & ~ v2_card_fil(B,A)
            & v3_card_fil(B,A) ) ) ) )).

fof(t32_card_fil,axiom,(
    ! [A] :
      ( ( v3_ordinal1(A)
        & v4_ordinal1(A) )
     => ! [B] :
          ( ( r1_tarski(B,A)
            & k7_ordinal2(B) = A )
         => k3_tarski(B) = A ) ) )).

fof(t33_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v7_card_fil(A)
       => v1_card_5(A) ) ) )).

fof(d17_card_fil,axiom,(
    ! [A] :
      ( ( v1_card_1(A)
        & ~ v2_card_1(A) )
     => ! [B] :
          ( v1_card_1(B)
         => ( B = k7_card_fil(A)
          <=> A = k2_card_1(B) ) ) ) )).

fof(t34_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v2_card_1(A) )
     => ? [B] :
          ( v1_funct_1(B)
          & v1_funct_2(B,k2_zfmisc_1(k7_card_fil(A),A),k1_zfmisc_1(A))
          & m2_relset_1(B,k2_zfmisc_1(k7_card_fil(A),A),k1_zfmisc_1(A))
          & r4_card_fil(A,B) ) ) )).

fof(t35_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v2_card_1(A) )
     => ! [B] :
          ( m2_card_fil(B,A)
         => ~ ( r2_card_fil(A,A,B)
              & r1_tarski(k6_card_fil(A),B)
              & ! [C] :
                  ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A)))
                 => ~ ( k1_card_1(C) = A
                      & ! [D] :
                          ~ ( r2_hidden(D,C)
                            & r2_hidden(D,B) )
                      & ! [D,E] :
                          ( ( r2_hidden(D,C)
                            & r2_hidden(E,C) )
                         => ( D = E
                            | r1_xboole_0(D,E) ) ) ) ) ) ) ) )).

fof(t36_card_fil,axiom,(
    ! [A,B] :
      ( v1_card_1(B)
     => ~ ( r1_tarski(B,k1_card_1(A))
          & ! [C] :
              ~ ( r1_tarski(C,A)
                & k1_card_1(C) = B ) ) ) )).

fof(t37_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v2_card_1(A) )
     => ! [B] :
          ( m1_card_fil(B,A)
         => ~ ( v1_card_fil(B,A)
              & v3_card_fil(B,A)
              & r1_card_fil(A,A,B) ) ) ) )).

fof(t38_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v7_card_fil(A)
       => v2_card_1(A) ) ) )).

fof(t39_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v7_card_fil(A)
       => v4_card_fil(A) ) ) )).

fof(t40_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v7_card_fil(A)
       => v5_card_fil(A) ) ) )).

fof(t41_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A) )
     => ( v7_card_fil(A)
       => v6_card_fil(A) ) ) )).

fof(dt_m1_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ) )).

fof(existence_m1_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] : m1_card_fil(B,A) ) )).

fof(dt_m2_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m2_card_fil(B,A)
         => ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ) )).

fof(existence_m2_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ? [B] : m2_card_fil(B,A) ) )).

fof(dt_k1_card_fil,axiom,(
    ! [A,B] : m1_subset_1(k1_card_fil(A,B),k1_zfmisc_1(A)) )).

fof(redefinition_k1_card_fil,axiom,(
    ! [A,B] : k1_card_fil(A,B) = k4_xboole_0(A,B) )).

fof(dt_k2_card_fil,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_card_fil(B,A) )
     => m2_card_fil(k2_card_fil(A,B),A) ) )).

fof(involutiveness_k2_card_fil,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_card_fil(B,A) )
     => k2_card_fil(A,k2_card_fil(A,B)) = B ) )).

fof(redefinition_k2_card_fil,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_card_fil(B,A) )
     => k2_card_fil(A,B) = k7_setfam_1(A,B) ) )).

fof(dt_k3_card_fil,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_card_fil(B,A)
        & m1_subset_1(C,k1_zfmisc_1(A)) )
     => ( ~ v1_xboole_0(k3_card_fil(A,B,C))
        & m1_subset_1(k3_card_fil(A,B,C),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) )).

fof(dt_k4_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ( ~ v1_xboole_0(k4_card_fil(A))
        & m1_subset_1(k4_card_fil(A),k1_zfmisc_1(k1_zfmisc_1(k1_zfmisc_1(A)))) ) ) )).

fof(dt_k5_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => m1_card_fil(k5_card_fil(A),A) ) )).

fof(dt_k6_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => m2_card_fil(k6_card_fil(A),A) ) )).

fof(dt_k7_card_fil,axiom,(
    ! [A] :
      ( ( v1_card_1(A)
        & ~ v2_card_1(A) )
     => v1_card_1(k7_card_fil(A)) ) )).

fof(t8_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => k7_setfam_1(A,B) = a_2_0_card_fil(A,B) ) ) )).

fof(t9_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => k7_setfam_1(A,B) = a_2_1_card_fil(A,B) ) ) )).

fof(d8_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_card_fil(B,A)
         => ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => k3_card_fil(A,B,C) = a_3_0_card_fil(A,B,C) ) ) ) )).

fof(d9_card_fil,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => k4_card_fil(A) = a_1_0_card_fil(A) ) )).

fof(d10_card_fil,axiom,(
    ! [A] :
      ( ~ v1_finset_1(A)
     => k5_card_fil(A) = a_1_1_card_fil(A) ) )).

fof(d18_card_fil,axiom,(
    ! [A] :
      ( ( ~ v1_finset_1(A)
        & v1_card_1(A)
        & ~ v2_card_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k2_zfmisc_1(k7_card_fil(A),A),k1_zfmisc_1(A))
            & m2_relset_1(B,k2_zfmisc_1(k7_card_fil(A),A),k1_zfmisc_1(A)) )
         => ( r4_card_fil(A,B)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k7_card_fil(A))
                 => ! [D] :
                      ( m1_subset_1(D,A)
                     => ! [E] :
                          ( m1_subset_1(E,A)
                         => ( D != E
                           => v1_xboole_0(k5_subset_1(A,k2_binop_1(k7_card_fil(A),A,k1_zfmisc_1(A),B,C,D),k2_binop_1(k7_card_fil(A),A,k1_zfmisc_1(A),B,C,E))) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,A)
                 => ! [D] :
                      ( m1_subset_1(D,k7_card_fil(A))
                     => ! [E] :
                          ( m1_subset_1(E,k7_card_fil(A))
                         => ( D != E
                           => v1_xboole_0(k5_subset_1(A,k2_binop_1(k7_card_fil(A),A,k1_zfmisc_1(A),B,D,C),k2_binop_1(k7_card_fil(A),A,k1_zfmisc_1(A),B,E,C))) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,k7_card_fil(A))
                 => r1_tarski(k1_card_1(k1_card_fil(A,k3_tarski(a_3_2_card_fil(A,B,C)))),k7_card_fil(A)) )
              & ! [C] :
                  ( m1_subset_1(C,A)
                 => r1_tarski(k1_card_1(k1_card_fil(A,k3_tarski(a_3_3_card_fil(A,B,C)))),k7_card_fil(A)) ) ) ) ) ) )).

fof(s1_card_fil,axiom,
    ( r2_hidden(f2_s1_card_fil,a_0_0_card_fil)
   => p1_s1_card_fil(f2_s1_card_fil) )).

fof(fraenkel_a_2_0_card_fil,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B))) )
     => ( r2_hidden(A,a_2_0_card_fil(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(B))
            & A = D
            & r2_hidden(k3_subset_1(B,D),C) ) ) ) )).

fof(fraenkel_a_2_1_card_fil,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B))) )
     => ( r2_hidden(A,a_2_1_card_fil(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,k1_zfmisc_1(B))
            & A = k3_subset_1(B,D)
            & r2_hidden(D,C) ) ) ) )).

fof(fraenkel_a_3_0_card_fil,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_card_fil(C,B)
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_0_card_fil(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_zfmisc_1(B))
            & A = E
            & ? [F] :
                ( m1_subset_1(F,k1_zfmisc_1(B))
                & r2_hidden(F,a_3_1_card_fil(B,C,D))
                & r1_tarski(F,E) ) ) ) ) )).

fof(fraenkel_a_3_1_card_fil,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_card_fil(C,B)
        & m1_subset_1(D,k1_zfmisc_1(B)) )
     => ( r2_hidden(A,a_3_1_card_fil(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k1_zfmisc_1(B))
            & A = k5_subset_1(B,E,D)
            & r2_hidden(E,C) ) ) ) )).

fof(fraenkel_a_1_0_card_fil,axiom,(
    ! [A,B] :
      ( ~ v1_xboole_0(B)
     => ( r2_hidden(A,a_1_0_card_fil(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(B)))
            & A = C
            & m1_card_fil(C,B) ) ) ) )).

fof(fraenkel_a_1_1_card_fil,axiom,(
    ! [A,B] :
      ( ~ v1_finset_1(B)
     => ( r2_hidden(A,a_1_1_card_fil(B))
      <=> ? [C] :
            ( m1_subset_1(C,k1_zfmisc_1(B))
            & A = C
            & r2_hidden(k1_card_1(k1_card_fil(B,C)),k1_card_1(B)) ) ) ) )).

fof(fraenkel_a_3_2_card_fil,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_finset_1(B)
        & v1_card_1(B)
        & ~ v2_card_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(k7_card_fil(B),B),k1_zfmisc_1(B))
        & m2_relset_1(C,k2_zfmisc_1(k7_card_fil(B),B),k1_zfmisc_1(B))
        & m1_subset_1(D,k7_card_fil(B)) )
     => ( r2_hidden(A,a_3_2_card_fil(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,B)
            & A = k2_binop_1(k7_card_fil(B),B,k1_zfmisc_1(B),C,D,E)
            & r2_hidden(E,B) ) ) ) )).

fof(fraenkel_a_3_3_card_fil,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_finset_1(B)
        & v1_card_1(B)
        & ~ v2_card_1(B)
        & v1_funct_1(C)
        & v1_funct_2(C,k2_zfmisc_1(k7_card_fil(B),B),k1_zfmisc_1(B))
        & m2_relset_1(C,k2_zfmisc_1(k7_card_fil(B),B),k1_zfmisc_1(B))
        & m1_subset_1(D,B) )
     => ( r2_hidden(A,a_3_3_card_fil(B,C,D))
      <=> ? [E] :
            ( m1_subset_1(E,k7_card_fil(B))
            & A = k2_binop_1(k7_card_fil(B),B,k1_zfmisc_1(B),C,E,D)
            & r2_hidden(E,k7_card_fil(B)) ) ) ) )).

fof(fraenkel_a_0_0_card_fil,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_card_fil)
    <=> ? [B] :
          ( m1_subset_1(B,f1_s1_card_fil)
          & A = B
          & p1_s1_card_fil(B) ) ) )).
%------------------------------------------------------------------------------