SET007 Axioms: SET007+54.ax


%------------------------------------------------------------------------------
% File     : SET007+54 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Cardinal Numbers
% 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_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  111 (  49 unit)
%            Number of atoms       :  307 (  50 equality)
%            Maximal formula depth :   17 (   4 average)
%            Number of connectives :  206 (  10 ~  ;   3  |;  71  &)
%                                         (  20 <=>; 102 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   28 (   1 propositional; 0-3 arity)
%            Number of functors    :   27 (   7 constant; 0-2 arity)
%            Number of variables   :  131 (   0 singleton; 121 !;  10 ?)
%            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(rc1_card_1,axiom,(
    ? [A] : v1_card_1(A) )).

fof(cc1_card_1,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A) ) ) )).

fof(fc1_card_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( v1_ordinal1(k3_card_1(A))
        & v2_ordinal1(k3_card_1(A))
        & v3_ordinal1(k3_card_1(A))
        & v1_card_1(k3_card_1(A)) ) ) )).

fof(cc2_card_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_card_1(A) ) ) )).

fof(cc3_card_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( v1_ordinal1(A)
        & v2_ordinal1(A)
        & v3_ordinal1(A)
        & v4_ordinal2(A)
        & v1_xcmplx_0(A)
        & v1_finset_1(A)
        & v1_xreal_0(A)
        & ~ v3_xreal_0(A)
        & v1_card_1(A) ) ) )).

fof(rc2_card_1,axiom,(
    ? [A] :
      ( v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v1_finset_1(A)
      & v1_card_1(A) ) )).

fof(fc2_card_1,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_finset_1(k1_card_1(A))
        & v1_card_1(k1_card_1(A)) ) ) )).

fof(d1_card_1,axiom,(
    ! [A] :
      ( v1_card_1(A)
    <=> ? [B] :
          ( v3_ordinal1(B)
          & A = B
          & ! [C] :
              ( v3_ordinal1(C)
             => ( r2_wellord2(C,B)
               => r1_ordinal1(B,C) ) ) ) ) )).

fof(t1_card_1,axiom,(
    $true )).

fof(t2_card_1,axiom,(
    $true )).

fof(t3_card_1,axiom,(
    $true )).

fof(t4_card_1,axiom,(
    ! [A] :
    ? [B] :
      ( v3_ordinal1(B)
      & r2_wellord2(A,B) ) )).

fof(t5_card_1,axiom,(
    $true )).

fof(t6_card_1,axiom,(
    $true )).

fof(t7_card_1,axiom,(
    $true )).

fof(t8_card_1,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( A = B
          <=> r2_wellord2(A,B) ) ) ) )).

fof(t9_card_1,axiom,(
    $true )).

fof(t10_card_1,axiom,(
    $true )).

fof(t11_card_1,axiom,(
    $true )).

fof(t12_card_1,axiom,(
    $true )).

fof(t13_card_1,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( r2_hidden(A,B)
          <=> ( r1_tarski(A,B)
              & A != B ) ) ) ) )).

fof(t14_card_1,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ! [B] :
          ( v1_card_1(B)
         => ( r2_hidden(A,B)
          <=> ~ r1_tarski(B,A) ) ) ) )).

fof(d2_card_1,axiom,(
    $true )).

fof(d3_card_1,axiom,(
    $true )).

fof(d4_card_1,axiom,(
    $true )).

fof(d5_card_1,axiom,(
    ! [A,B] :
      ( v1_card_1(B)
     => ( B = k1_card_1(A)
      <=> r2_wellord2(A,B) ) ) )).

fof(t15_card_1,axiom,(
    $true )).

fof(t16_card_1,axiom,(
    $true )).

fof(t17_card_1,axiom,(
    $true )).

fof(t18_card_1,axiom,(
    $true )).

fof(t19_card_1,axiom,(
    $true )).

fof(t20_card_1,axiom,(
    $true )).

fof(t21_card_1,axiom,(
    ! [A,B] :
      ( r2_wellord2(A,B)
    <=> k1_card_1(A) = k1_card_1(B) ) )).

fof(t22_card_1,axiom,(
    ! [A] :
      ( v1_relat_1(A)
     => ( v2_wellord1(A)
       => r2_wellord2(k3_relat_1(A),k2_wellord2(A)) ) ) )).

fof(t23_card_1,axiom,(
    ! [A,B] :
      ( v1_card_1(B)
     => ( r1_tarski(A,B)
       => r1_tarski(k1_card_1(A),B) ) ) )).

fof(t24_card_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => r1_ordinal1(k1_card_1(A),A) ) )).

fof(t25_card_1,axiom,(
    ! [A,B] :
      ( v1_card_1(B)
     => ( r2_hidden(A,B)
       => r2_hidden(k1_card_1(A),B) ) ) )).

fof(t26_card_1,axiom,(
    ! [A,B] :
      ( r1_tarski(k1_card_1(A),k1_card_1(B))
    <=> ? [C] :
          ( v1_relat_1(C)
          & v1_funct_1(C)
          & v2_funct_1(C)
          & k1_relat_1(C) = A
          & r1_tarski(k2_relat_1(C),B) ) ) )).

fof(t27_card_1,axiom,(
    ! [A,B] :
      ( r1_tarski(A,B)
     => r1_tarski(k1_card_1(A),k1_card_1(B)) ) )).

fof(t28_card_1,axiom,(
    ! [A,B] :
      ( r1_tarski(k1_card_1(A),k1_card_1(B))
    <=> ? [C] :
          ( v1_relat_1(C)
          & v1_funct_1(C)
          & k1_relat_1(C) = B
          & r1_tarski(A,k2_relat_1(C)) ) ) )).

fof(t29_card_1,axiom,(
    ! [A] : ~ r2_wellord2(A,k1_zfmisc_1(A)) )).

fof(t30_card_1,axiom,(
    ! [A] : r2_hidden(k1_card_1(A),k1_card_1(k1_zfmisc_1(A))) )).

fof(d6_card_1,axiom,(
    ! [A,B] :
      ( v1_card_1(B)
     => ( B = k2_card_1(A)
      <=> ( r2_hidden(k1_card_1(A),B)
          & ! [C] :
              ( v1_card_1(C)
             => ( r2_hidden(k1_card_1(A),C)
               => r1_tarski(B,C) ) ) ) ) ) )).

fof(t31_card_1,axiom,(
    $true )).

fof(t32_card_1,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => r2_hidden(A,k2_card_1(A)) ) )).

fof(t33_card_1,axiom,(
    ! [A] : r2_hidden(k1_card_1(k1_xboole_0),k2_card_1(A)) )).

fof(t34_card_1,axiom,(
    ! [A,B] :
      ( k1_card_1(A) = k1_card_1(B)
     => k2_card_1(A) = k2_card_1(B) ) )).

fof(t35_card_1,axiom,(
    ! [A,B] :
      ( r2_wellord2(A,B)
     => k2_card_1(A) = k2_card_1(B) ) )).

fof(t36_card_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => r2_hidden(A,k2_card_1(A)) ) )).

fof(d7_card_1,axiom,(
    ! [A] :
      ( v1_card_1(A)
     => ( v2_card_1(A)
      <=> ! [B] :
            ( v1_card_1(B)
           => A != k2_card_1(B) ) ) ) )).

fof(d8_card_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( B = k3_card_1(A)
        <=> ? [C] :
              ( v1_relat_1(C)
              & v1_funct_1(C)
              & v5_ordinal1(C)
              & B = k1_ordinal2(C)
              & k1_relat_1(C) = k1_ordinal1(A)
              & k1_funct_1(C,k1_xboole_0) = k1_card_1(k5_numbers)
              & ! [D] :
                  ( v3_ordinal1(D)
                 => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                   => k1_funct_1(C,k1_ordinal1(D)) = k2_card_1(k3_tarski(k1_tarski(k1_funct_1(C,D)))) ) )
              & ! [D] :
                  ( v3_ordinal1(D)
                 => ( ( r2_hidden(D,k1_ordinal1(A))
                      & v4_ordinal1(D) )
                   => ( D = k1_xboole_0
                      | k1_funct_1(C,D) = k1_card_1(k8_ordinal2(k2_ordinal1(C,D))) ) ) ) ) ) ) )).

fof(t37_card_1,axiom,(
    $true )).

fof(t38_card_1,axiom,(
    k3_card_1(np__0) = k1_card_1(k5_numbers) )).

fof(t39_card_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => k3_card_1(k1_ordinal1(A)) = k2_card_1(k3_card_1(A)) ) )).

fof(t40_card_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ( v4_ordinal1(A)
       => ( A = k1_xboole_0
          | ! [B] :
              ( ( v1_relat_1(B)
                & v1_funct_1(B)
                & v5_ordinal1(B) )
             => ( ( k1_relat_1(B) = A
                  & ! [C] :
                      ( v3_ordinal1(C)
                     => ( r2_hidden(C,A)
                       => k1_funct_1(B,C) = k3_card_1(C) ) ) )
               => k3_card_1(A) = k1_card_1(k8_ordinal2(B)) ) ) ) ) ) )).

fof(t41_card_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r2_hidden(k3_card_1(A),k3_card_1(B)) ) ) ) )).

fof(t42_card_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( k3_card_1(A) = k3_card_1(B)
           => A = B ) ) ) )).

fof(t43_card_1,axiom,(
    ! [A] :
      ( v3_ordinal1(A)
     => ! [B] :
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,B)
          <=> r1_tarski(k3_card_1(A),k3_card_1(B)) ) ) ) )).

fof(t44_card_1,axiom,(
    ! [A,B,C] :
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C)
        & r2_wellord2(A,C) )
     => ( r2_wellord2(A,B)
        & r2_wellord2(B,C) ) ) )).

fof(t45_card_1,axiom,(
    ! [A,B] :
      ( r1_tarski(k1_zfmisc_1(A),B)
     => ( r2_hidden(k1_card_1(A),k1_card_1(B))
        & ~ r2_wellord2(A,B) ) ) )).

fof(t46_card_1,axiom,(
    ! [A] :
      ( r2_wellord2(A,k1_xboole_0)
    <=> A = k1_xboole_0 ) )).

fof(t47_card_1,axiom,(
    k1_card_1(k1_xboole_0) = k1_xboole_0 )).

fof(t48_card_1,axiom,(
    ! [A,B] :
      ( r2_wellord2(A,k1_tarski(B))
    <=> ? [C] : A = k1_tarski(C) ) )).

fof(t49_card_1,axiom,(
    ! [A,B] :
      ( k1_card_1(A) = k1_card_1(k1_tarski(B))
    <=> ? [C] : A = k1_tarski(C) ) )).

fof(t50_card_1,axiom,(
    ! [A] : k1_card_1(k1_tarski(A)) = k4_ordinal2 )).

fof(t51_card_1,axiom,(
    np__0 = k1_xboole_0 )).

fof(t52_card_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_ordinal1(A) = k1_nat_1(A,np__1) ) )).

fof(t53_card_1,axiom,(
    $true )).

fof(t54_card_1,axiom,(
    $true )).

fof(t55_card_1,axiom,(
    $true )).

fof(t56_card_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,B)
          <=> r1_ordinal1(A,B) ) ) ) )).

fof(t57_card_1,axiom,(
    $true )).

fof(t58_card_1,axiom,(
    ! [A,B,C,D] :
      ( ( r1_xboole_0(A,B)
        & r1_xboole_0(C,D)
        & r2_wellord2(A,C)
        & r2_wellord2(B,D) )
     => r2_wellord2(k2_xboole_0(A,B),k2_xboole_0(C,D)) ) )).

fof(t59_card_1,axiom,(
    ! [A,B,C] :
      ( ( r2_hidden(A,B)
        & r2_hidden(C,B) )
     => r2_wellord2(k4_xboole_0(B,k1_tarski(A)),k4_xboole_0(B,k1_tarski(C))) ) )).

fof(t60_card_1,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( ( r1_tarski(A,k1_relat_1(B))
          & v2_funct_1(B) )
       => r2_wellord2(A,k9_relat_1(B,A)) ) ) )).

fof(t61_card_1,axiom,(
    ! [A,B,C,D] :
      ( ( r2_wellord2(A,B)
        & r2_hidden(C,A)
        & r2_hidden(D,B) )
     => r2_wellord2(k4_xboole_0(A,k1_tarski(C)),k4_xboole_0(B,k1_tarski(D))) ) )).

fof(t62_card_1,axiom,(
    $true )).

fof(t63_card_1,axiom,(
    $true )).

fof(t64_card_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_wellord2(A,B)
           => A = B ) ) ) )).

fof(t65_card_1,axiom,(
    ! [A] :
      ( r2_hidden(A,k5_ordinal2)
     => v1_card_1(A) ) )).

fof(t66_card_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => A = k1_card_1(A) ) )).

fof(t67_card_1,axiom,(
    $true )).

fof(t68_card_1,axiom,(
    ! [A,B] :
      ( ( r2_wellord2(A,B)
        & v1_finset_1(A) )
     => v1_finset_1(B) ) )).

fof(t69_card_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( v1_finset_1(A)
        & v1_finset_1(k1_card_1(A)) ) ) )).

fof(t70_card_1,axiom,(
    $true )).

fof(t71_card_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( k1_card_1(A) = k1_card_1(B)
           => A = B ) ) ) )).

fof(t72_card_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_tarski(k1_card_1(A),k1_card_1(B))
          <=> r1_xreal_0(A,B) ) ) ) )).

fof(t73_card_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_hidden(k1_card_1(A),k1_card_1(B))
          <=> ~ r1_xreal_0(B,A) ) ) ) )).

fof(t74_card_1,axiom,(
    ! [A] :
      ~ ( v1_finset_1(A)
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ r2_wellord2(A,B) ) ) )).

fof(t75_card_1,axiom,(
    $true )).

fof(t76_card_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_card_1(k1_card_1(A)) = k1_card_1(k1_nat_1(A,np__1)) ) )).

fof(d9_card_1,axiom,(
    $true )).

fof(d10_card_1,axiom,(
    $true )).

fof(d11_card_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k4_card_1(A)
          <=> k1_card_1(B) = k1_card_1(A) ) ) ) )).

fof(t77_card_1,axiom,(
    $true )).

fof(t78_card_1,axiom,(
    k4_card_1(k1_xboole_0) = np__0 )).

fof(t79_card_1,axiom,(
    ! [A] : k4_card_1(k1_tarski(A)) = np__1 )).

fof(t80_card_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ( r1_tarski(A,B)
           => r1_xreal_0(k4_card_1(A),k4_card_1(B)) ) ) ) )).

fof(t81_card_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => ! [B] :
          ( v1_finset_1(B)
         => ( r2_wellord2(A,B)
           => k4_card_1(A) = k4_card_1(B) ) ) ) )).

fof(t82_card_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => v1_finset_1(k2_card_1(A)) ) )).

fof(t83_card_1,axiom,(
    k3_card_1(np__0) = k5_ordinal2 )).

fof(t84_card_1,axiom,(
    k1_card_1(k5_ordinal2) = k5_ordinal2 )).

fof(t85_card_1,axiom,(
    v2_card_1(k1_card_1(k5_ordinal2)) )).

fof(t86_card_1,axiom,(
    ! [A] :
      ( ( v1_finset_1(A)
        & v1_card_1(A) )
     => ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = k1_card_1(B) ) ) )).

fof(s1_card_1,axiom,
    ( ( p1_s1_card_1(k1_xboole_0)
      & ! [A] :
          ( v1_card_1(A)
         => ( p1_s1_card_1(A)
           => p1_s1_card_1(k2_card_1(A)) ) )
      & ! [A] :
          ( v1_card_1(A)
         => ( ( v2_card_1(A)
              & ! [B] :
                  ( v1_card_1(B)
                 => ( r2_hidden(B,A)
                   => p1_s1_card_1(B) ) ) )
           => ( A = k1_xboole_0
              | p1_s1_card_1(A) ) ) ) )
   => ! [A] :
        ( v1_card_1(A)
       => p1_s1_card_1(A) ) )).

fof(s2_card_1,axiom,
    ( ! [A] :
        ( v1_card_1(A)
       => ( ! [B] :
              ( v1_card_1(B)
             => ( r2_hidden(B,A)
               => p1_s2_card_1(B) ) )
         => p1_s2_card_1(A) ) )
   => ! [A] :
        ( v1_card_1(A)
       => p1_s2_card_1(A) ) )).

fof(dt_k1_card_1,axiom,(
    ! [A] : v1_card_1(k1_card_1(A)) )).

fof(dt_k2_card_1,axiom,(
    ! [A] : v1_card_1(k2_card_1(A)) )).

fof(dt_k3_card_1,axiom,(
    $true )).

fof(dt_k4_card_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => m2_subset_1(k4_card_1(A),k1_numbers,k5_numbers) ) )).

fof(redefinition_k4_card_1,axiom,(
    ! [A] :
      ( v1_finset_1(A)
     => k4_card_1(A) = k1_card_1(A) ) )).
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