fof(t85_card_1,theorem,(
    v2_card_1(k1_card_1(k5_ordinal2)) ),
    inference(mizar_proof,[status(thm)],[d7_card_1,l46_card_1,t32_card_1,t83_card_1,t41_ordinal1,t19_ordinal2,d21_ordinal2,t52_card_1,t66_card_1,t76_card_1,l46_card_1,t83_card_1]),
    [file(card_1,t85_card_1),interesting(1.00)]).

fof(t43_card_1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,B)
          <=> r1_tarski(k3_card_1(A),k3_card_1(B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,t41_card_1,t42_card_1,t21_ordinal1,d2_ordinal1]),
    [file(card_1,t43_card_1),interesting(0.90)]).

fof(t42_card_1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( k3_card_1(A) = k3_card_1(B)
           => A = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_card_1,t41_card_1,t24_ordinal1]),
    [file(card_1,t42_card_1),interesting(0.82)]).

fof(t83_card_1,theorem,(
    k3_card_1(0) = k5_ordinal2 ),
    inference(mizar_proof,[status(thm)],[l46_card_1,t24_card_1,d10_xboole_0,d3_tarski,t23_ordinal1,t52_card_1,t33_ordinal1,d5_card_1,t27_card_1,t10_ordinal1,l46_card_1]),
    [file(card_1,t83_card_1),interesting(0.82)]).

fof(t36_card_1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => r2_hidden(A,k2_card_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[d5_card_1,t26_ordinal1,d5_card_1,t27_card_1,t32_card_1,t35_card_1,t7_ordinal1]),
    [file(card_1,t36_card_1),interesting(0.79)]).

fof(t65_card_1,theorem,(
    ! [A] : 
      ( r2_hidden(A,k5_ordinal2)
     => v1_card_1(A) ) ),
    inference(mizar_proof,[status(thm)],[d1_card_1,t23_ordinal1,t26_ordinal1,t19_ordinal1,t64_card_1]),
    [file(card_1,t65_card_1),interesting(0.76)]).

fof(t30_card_1,theorem,(
    ! [A] : r2_hidden(k1_card_1(A),k1_card_1(k1_zfmisc_1(A))) ),
    inference(mizar_proof,[status(thm)],[s3_funct_1,d8_funct_1,t6_zfmisc_1,d3_tarski,d5_funct_1,d3_tarski,d1_tarski,t26_card_1,t29_card_1,t21_card_1,d8_xboole_0,t21_ordinal1]),
    [file(card_1,t30_card_1),interesting(0.75)]).

fof(t41_card_1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r2_hidden(k3_card_1(A),k3_card_1(B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t39_card_1,d8_xboole_0,t32_card_1,t19_ordinal1,t32_card_1,t21_ordinal1,t34_ordinal1,s2_ordinal2,l46_card_1,t41_ordinal1,d5_funct_1,t27_ordinal2,t35_ordinal2,d5_card_1,d2_ordinal1,t27_card_1,t32_card_1,t39_card_1,s1_ordinal2,t24_ordinal1]),
    [file(card_1,t41_card_1),interesting(0.74)]).

fof(t84_card_1,theorem,(
    k1_card_1(k5_ordinal2) = k5_ordinal2 ),
    inference(mizar_proof,[status(thm)],[l46_card_1,t83_card_1]),
    [file(card_1,t84_card_1),interesting(0.73)]).

fof(t33_card_1,theorem,(
    ! [A] : r2_hidden(k1_card_1(k1_xboole_0),k2_card_1(A)) ),
    inference(mizar_proof,[status(thm)],[t2_xboole_1,d6_card_1,t27_card_1,t22_ordinal1]),
    [file(card_1,t33_card_1),interesting(0.72)]).

fof(t45_card_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(k1_zfmisc_1(A),B)
     => ( r2_hidden(k1_card_1(A),k1_card_1(B))
        & ~ r2_wellord2(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t27_card_1,t30_card_1,t21_card_1]),
    [file(card_1,t45_card_1),interesting(0.70)]).

fof(t27_card_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => r1_tarski(k1_card_1(A),k1_card_1(B)) ) ),
    inference(mizar_proof,[status(thm)],[t52_funct_1,t71_relat_1,t26_card_1]),
    [file(card_1,t27_card_1),interesting(0.69)]).

fof(t48_card_1,theorem,(
    ! [A,B] : 
      ( r2_wellord2(A,k1_tarski(B))
    <=> ? [C] : A = k1_tarski(C) ) ),
    inference(mizar_proof,[status(thm)],[d4_wellord2,t14_funct_1,d4_wellord2,t13_funcop_1,t19_funcop_1,d8_funct_1,d1_tarski,t19_funcop_1,d10_xboole_0,d3_tarski,d1_tarski,d5_funct_1]),
    [file(card_1,t48_card_1),interesting(0.69)]).

fof(t82_card_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => v1_finset_1(k2_card_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[d5_card_1,t35_card_1,t76_card_1,t69_card_1]),
    [file(card_1,t82_card_1),interesting(0.69)]).

fof(t23_card_1,theorem,(
    ! [A,B] : 
      ( v1_card_1(B)
     => ( r1_tarski(A,B)
       => r1_tarski(k1_card_1(A),B) ) ) ),
    inference(mizar_proof,[status(thm)],[t9_wellord2,t17_wellord2,d1_wellord2,t22_card_1,s1_ordinal1,d1_card_1,t22_wellord2,t1_xboole_1,d5_card_1]),
    [file(card_1,t23_card_1),interesting(0.67)]).

fof(t39_card_1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k3_card_1(k1_ordinal1(A)) = k2_card_1(k3_card_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[l46_card_1,t31_zfmisc_1]),
    [file(card_1,t39_card_1),interesting(0.67)]).

fof(t28_card_1,theorem,(
    ! [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)) ) ) ),
    inference(mizar_proof,[status(thm)],[t26_card_1,s2_funct_1,d3_tarski,t56_funct_1,d5_funct_1,d5_funct_1,s3_funct_1,t2_xboole_1,t27_card_1,t65_relat_1,d5_funct_1,d5_funct_1,d1_tarski,d13_funct_1,t28_wellord2,t46_relat_1,d8_funct_1,t23_funct_1,d5_funct_1,d13_funct_1,d1_tarski,d3_tarski,d5_funct_1,t23_funct_1,d5_funct_1,d13_funct_1,t26_card_1]),
    [file(card_1,t28_card_1),interesting(0.66)]).

fof(t29_card_1,theorem,(
    ! [A] : ~ r2_wellord2(A,k1_zfmisc_1(A)) ),
    inference(mizar_proof,[status(thm)],[d4_wellord2,s1_xboole_0,d3_tarski,d5_funct_1]),
    [file(card_1,t29_card_1),interesting(0.66)]).

fof(t64_card_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_wellord2(A,B)
           => A = B ) ) ) ),
    inference(mizar_proof,[status(thm)],[s5_nat_1,d4_wellord2,t65_relat_1,t22_nat_1,d1_ordinal1,t52_card_1,t40_xboole_1,t65_zfmisc_1,t65_relat_1,t22_nat_1,d1_ordinal1,t52_card_1,t40_xboole_1,t65_zfmisc_1,d1_tarski,d2_xboole_0,t61_card_1,t31_xreal_1]),
    [file(card_1,t64_card_1),interesting(0.65)]).

fof(t38_card_1,theorem,(
    k3_card_1(0) = k1_card_1(k5_numbers) ),
    inference(mizar_proof,[status(thm)],[l46_card_1]),
    [file(card_1,t38_card_1),interesting(0.64)]).

fof(t50_card_1,theorem,(
    ! [A] : k1_card_1(k1_tarski(A)) = k4_ordinal2 ),
    inference(mizar_proof,[status(thm)],[d1_ordinal1,d4_ordinal2,d1_card_1,t48_card_1,d1_tarski,t23_ordinal1,d2_ordinal1,t39_zfmisc_1,t48_card_1,d5_card_1]),
    [file(card_1,t50_card_1),interesting(0.64)]).

fof(t61_card_1,theorem,(
    ! [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))) ) ),
    inference(mizar_proof,[status(thm)],[d4_wellord2,t36_xboole_1,t60_card_1,t123_funct_1,t146_relat_1,t117_funct_1,d5_funct_1,t59_card_1,t22_wellord2]),
    [file(card_1,t61_card_1),interesting(0.64)]).

fof(t74_card_1,theorem,(
    ! [A] : ~ ( v1_finset_1(A)
      & ! [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ r2_wellord2(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t51_card_1,t37_zfmisc_1,t12_xboole_1,t52_card_1,d1_ordinal1,t48_card_1,t3_xboole_0,d1_tarski,d3_xboole_0,d1_tarski,d7_xboole_0,t58_card_1,s2_finset_1]),
    [file(card_1,t74_card_1),interesting(0.64)]).

fof(t49_card_1,theorem,(
    ! [A,B] : 
      ( k1_card_1(A) = k1_card_1(k1_tarski(B))
    <=> ? [C] : A = k1_tarski(C) ) ),
    inference(mizar_proof,[status(thm)],[t21_card_1,t48_card_1]),
    [file(card_1,t49_card_1),interesting(0.64)]).

fof(t32_card_1,theorem,(
    ! [A] : 
      ( v1_card_1(A)
     => r2_hidden(A,k2_card_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[d5_card_1,d6_card_1]),
    [file(card_1,t32_card_1),interesting(0.63)]).

fof(t35_card_1,theorem,(
    ! [A,B] : 
      ( r2_wellord2(A,B)
     => k2_card_1(A) = k2_card_1(B) ) ),
    inference(mizar_proof,[status(thm)],[t21_card_1,t34_card_1]),
    [file(card_1,t35_card_1),interesting(0.63)]).

fof(t4_card_1,theorem,(
    ! [A] : 
    ? [B] : 
      ( v3_ordinal1(B)
      & r2_wellord2(A,B) ) ),
    inference(mizar_proof,[status(thm)],[t26_wellord2,t25_wellord2,d2_wellord2,d8_wellord1,d7_wellord1,d4_wellord2,t25_wellord2,d1_wellord2]),
    [file(card_1,t4_card_1),interesting(0.63)]).

fof(t79_card_1,theorem,(
    ! [A] : k4_card_1(k1_tarski(A)) = 1 ),
    inference(mizar_proof,[status(thm)],[d3_tarski,t38_nat_1,t19_nat_1,d1_tarski,d10_xboole_0,d3_tarski,d1_tarski,t30_axioms,t49_card_1,d5_card_1]),
    [file(card_1,t79_card_1),interesting(0.63)]).

fof(t24_card_1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => r1_ordinal1(k1_card_1(A),A) ) ),
    inference(mizar_proof,[status(thm)],[d1_card_1,d5_card_1]),
    [file(card_1,t24_card_1),interesting(0.62)]).

fof(t22_card_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( v2_wellord1(A)
       => r2_wellord2(k3_relat_1(A),k2_wellord2(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_wellord2,d8_wellord1,d4_wellord2,d1_wellord2,d7_wellord1]),
    [file(card_1,t22_card_1),interesting(0.62)]).

fof(t80_card_1,theorem,(
    ! [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)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_card_1,t27_card_1,t56_card_1]),
    [file(card_1,t80_card_1),interesting(0.62)]).

fof(t21_card_1,theorem,(
    ! [A,B] : 
      ( r2_wellord2(A,B)
    <=> k1_card_1(A) = k1_card_1(B) ) ),
    inference(mizar_proof,[status(thm)],[d5_card_1,t22_wellord2,d5_card_1,t22_wellord2]),
    [file(card_1,t21_card_1),interesting(0.60)]).

fof(t44_card_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C)
        & r2_wellord2(A,C) )
     => ( r2_wellord2(A,B)
        & r2_wellord2(B,C) ) ) ),
    inference(mizar_proof,[status(thm)],[t21_card_1,t27_card_1,d10_xboole_0,t21_card_1]),
    [file(card_1,t44_card_1),interesting(0.60)]).

fof(t59_card_1,theorem,(
    ! [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))) ) ),
    inference(mizar_proof,[status(thm)],[s2_funct_1,d4_wellord2,d8_funct_1,d4_xboole_0,d1_tarski,d3_tarski,d5_funct_1,d4_xboole_0,d1_tarski,d1_tarski,d4_xboole_0,d1_tarski,d4_xboole_0,d4_xboole_0,d10_xboole_0,d3_tarski,d4_xboole_0,d1_tarski,d1_tarski,d4_xboole_0,d5_funct_1,d1_tarski,d4_xboole_0,d5_funct_1]),
    [file(card_1,t59_card_1),interesting(0.60)]).

fof(t76_card_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_card_1(k1_card_1(A)) = k1_card_1(k1_nat_1(A,1)) ) ),
    inference(mizar_proof,[status(thm)],[d5_card_1,t38_nat_1,t73_card_1,t66_card_1,d5_card_1,t52_card_1,t33_ordinal1,d6_card_1]),
    [file(card_1,t76_card_1),interesting(0.60)]).

fof(t8_card_1,theorem,(
    ! [A] : 
      ( v1_card_1(A)
     => ! [B] : 
          ( v1_card_1(B)
         => ( A = B
          <=> r2_wellord2(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_card_1,d1_card_1,d10_xboole_0]),
    [file(card_1,t8_card_1),interesting(0.59)]).

fof(t58_card_1,theorem,(
    ! [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)) ) ),
    inference(mizar_proof,[status(thm)],[d7_xboole_0,d4_wellord2,d4_wellord2,d2_xboole_0,d3_xboole_0,s2_funct_1,d4_wellord2,d8_funct_1,d5_funct_1,d3_xboole_0,d3_xboole_0,d5_funct_1,d3_xboole_0,d3_xboole_0,d8_funct_1,d3_tarski,d5_funct_1,d5_funct_1,d3_xboole_0,d2_xboole_0,d5_funct_1,d3_xboole_0,d2_xboole_0,d2_xboole_0,d10_xboole_0,d3_tarski,d5_funct_1,d2_xboole_0,d5_funct_1,d3_xboole_0,d5_funct_1,d2_xboole_0,d5_funct_1,d3_xboole_0,d2_xboole_0]),
    [file(card_1,t58_card_1),interesting(0.59)]).

fof(t26_card_1,theorem,(
    ! [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) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_card_1,d4_wellord2,d4_wellord2,t62_funct_1,t46_funct_1,t55_funct_1,t45_relat_1,t46_relat_1,d5_card_1,d4_wellord2,d4_wellord2,t46_funct_1,t46_relat_1,t45_relat_1,t23_card_1,d5_card_1,t22_wellord2,t22_wellord2,t8_card_1]),
    [file(card_1,t26_card_1),interesting(0.58)]).

fof(l1_card_1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,B)
           => r1_ordinal1(k1_ordinal1(A),k1_ordinal1(B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,t13_ordinal1,t10_ordinal1,t22_ordinal1]),
    [file(card_1,l1_card_1),interesting(0.58)]).

fof(t56_card_1,theorem,(
    ! [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) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t18_nat_1,t2_xboole_1,t28_nat_1,t29_nat_1,l1_card_1,t52_card_1,t52_card_1,t33_ordinal1,d2_ordinal1,d5_real_1,t38_nat_1,s1_nat_1]),
    [file(card_1,t56_card_1),interesting(0.58)]).

fof(t73_card_1,theorem,(
    ! [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) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,t71_card_1,t72_card_1,t21_ordinal1,d2_ordinal1,d5_real_1]),
    [file(card_1,t73_card_1),interesting(0.57)]).

fof(t34_card_1,theorem,(
    ! [A,B] : 
      ( k1_card_1(A) = k1_card_1(B)
     => k2_card_1(A) = k2_card_1(B) ) ),
    inference(mizar_proof,[status(thm)],[d6_card_1,d6_card_1]),
    [file(card_1,t34_card_1),interesting(0.56)]).

fof(t68_card_1,theorem,(
    ! [A,B] : 
      ( ( r2_wellord2(A,B)
        & v1_finset_1(A) )
     => v1_finset_1(B) ) ),
    inference(mizar_proof,[status(thm)],[d4_wellord2,d1_finset_1,d1_finset_1,t47_relat_1,t46_relat_1]),
    [file(card_1,t68_card_1),interesting(0.56)]).

fof(t72_card_1,theorem,(
    ! [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) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t56_card_1,t66_card_1]),
    [file(card_1,t72_card_1),interesting(0.55)]).

fof(t69_card_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( v1_finset_1(A)
        & v1_finset_1(k1_card_1(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[t71_relat_1,d1_finset_1,t66_card_1]),
    [file(card_1,t69_card_1),interesting(0.55)]).

fof(t52_card_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_ordinal1(A) = k1_nat_1(A,1) ) ),
    inference(mizar_proof,[status(thm)],[t30_axioms,d3_tarski,t38_nat_1,t13_ordinal1,t38_nat_1,d10_xboole_0,d3_tarski,t38_nat_1,d5_real_1,t13_ordinal1]),
    [file(card_1,t52_card_1),interesting(0.54)]).

fof(t60_card_1,theorem,(
    ! [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)) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_wellord2,t84_funct_1,t91_relat_1,t148_relat_1]),
    [file(card_1,t60_card_1),interesting(0.54)]).

fof(t66_card_1,theorem,(
    ! [A] : 
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => A = k1_card_1(A) ) ),
    inference(mizar_proof,[status(thm)],[d5_card_1]),
    [file(card_1,t66_card_1),interesting(0.54)]).

fof(t25_card_1,theorem,(
    ! [A,B] : 
      ( v1_card_1(B)
     => ( r2_hidden(A,B)
       => r2_hidden(k1_card_1(A),B) ) ) ),
    inference(mizar_proof,[status(thm)],[t23_ordinal1,t24_card_1,t22_ordinal1]),
    [file(card_1,t25_card_1),interesting(0.53)]).

fof(t46_card_1,theorem,(
    ! [A] : 
      ( r2_wellord2(A,k1_xboole_0)
    <=> A = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[d4_wellord2,t65_relat_1]),
    [file(card_1,t46_card_1),interesting(0.51)]).

fof(t14_card_1,theorem,(
    ! [A] : 
      ( v1_card_1(A)
     => ! [B] : 
          ( v1_card_1(B)
         => ( r2_hidden(A,B)
          <=> ~ r1_tarski(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t7_ordinal1,t26_ordinal1]),
    [file(card_1,t14_card_1),interesting(0.50)]).

fof(t47_card_1,theorem,(
    k1_card_1(k1_xboole_0) = k1_xboole_0 ),
    inference(mizar_proof,[status(thm)],[d1_card_1,t2_xboole_1,d5_card_1]),
    [file(card_1,t47_card_1),interesting(0.50)]).

fof(t71_card_1,theorem,(
    ! [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 ) ) ) ),
    inference(mizar_proof,[status(thm)],[t66_card_1]),
    [file(card_1,t71_card_1),interesting(0.49)]).

fof(t13_card_1,theorem,(
    ! [A] : 
      ( v1_card_1(A)
     => ! [B] : 
          ( v1_card_1(B)
         => ( r2_hidden(A,B)
          <=> ( r1_tarski(A,B)
              & A != B ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,t21_ordinal1,d2_ordinal1]),
    [file(card_1,t13_card_1),interesting(0.48)]).

fof(t78_card_1,theorem,(
    k4_card_1(k1_xboole_0) = 0 ),
    inference(mizar_proof,[status(thm)],[d5_card_1]),
    [file(card_1,t78_card_1),interesting(0.47)]).

fof(s2_card_1,theorem,
    ( ! [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) ) ),
    inference(mizar_proof,[status(thm)],[s2_ordinal1]),
    [file(card_1,s2_card_1),interesting(0.46)]).

fof(t81_card_1,theorem,(
    ! [A] : 
      ( v1_finset_1(A)
     => ! [B] : 
          ( v1_finset_1(B)
         => ( r2_wellord2(A,B)
           => k4_card_1(A) = k4_card_1(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t21_card_1]),
    [file(card_1,t81_card_1),interesting(0.46)]).

fof(t86_card_1,theorem,(
    ! [A] : 
      ( ( v1_finset_1(A)
        & v1_card_1(A) )
     => ? [B] : 
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = k1_card_1(B) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_card_1]),
    [file(card_1,t86_card_1),interesting(0.37)]).

fof(s1_card_1,theorem,
    ( ( 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) ) ),
    inference(mizar_proof,[status(thm)],[d7_card_1,t32_card_1,s2_ordinal1]),
    [file(card_1,s1_card_1),interesting(0.29)]).

fof(t40_card_1,theorem,(
    ! [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)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[l46_card_1]),
    [file(card_1,t40_card_1),interesting(0.21)]).

fof(d7_card_1,definition,(
    ! [A] : 
      ( v1_card_1(A)
     => ( v2_card_1(A)
      <=> ! [B] : 
            ( v1_card_1(B)
           => A != k2_card_1(B) ) ) ) ),
    file(card_1,d7_card_1),
    [interesting(0.00)]).

fof(d5_card_1,definition,(
    ! [A,B] : 
      ( v1_card_1(B)
     => ( B = k1_card_1(A)
      <=> r2_wellord2(A,B) ) ) ),
    file(card_1,d5_card_1),
    [interesting(0.00)]).

fof(d6_card_1,definition,(
    ! [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) ) ) ) ) ) ),
    file(card_1,d6_card_1),
    [interesting(0.00)]).

fof(s2_ordinal1,theorem,
    ( ! [A] : 
        ( v3_ordinal1(A)
       => ( ! [B] : 
              ( v3_ordinal1(B)
             => ( r2_hidden(B,A)
               => p1_s2_ordinal1(B) ) )
         => p1_s2_ordinal1(A) ) )
   => ! [A] : 
        ( v3_ordinal1(A)
       => p1_s2_ordinal1(A) ) ),
    file(ordinal1,s2_ordinal1),
    [interesting(0.00)]).

fof(t10_card_1,theorem,(
    $true ),
    file(card_1,t10_card_1),
    [interesting(0.00)]).

fof(t11_card_1,theorem,(
    $true ),
    file(card_1,t11_card_1),
    [interesting(0.00)]).

fof(t12_card_1,theorem,(
    $true ),
    file(card_1,t12_card_1),
    [interesting(0.00)]).

fof(d8_xboole_0,definition,(
    ! [A,B] : 
      ( r2_xboole_0(A,B)
    <=> ( r1_tarski(A,B)
        & A != B ) ) ),
    file(xboole_0,d8_xboole_0),
    [interesting(0.00)]).

fof(t21_ordinal1,theorem,(
    ! [A] : 
      ( v1_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_xboole_0(A,B)
           => r2_hidden(A,B) ) ) ) ),
    file(ordinal1,t21_ordinal1),
    [interesting(0.00)]).

fof(d2_ordinal1,definition,(
    ! [A] : 
      ( v1_ordinal1(A)
    <=> ! [B] : 
          ( r2_hidden(B,A)
         => r1_tarski(B,A) ) ) ),
    file(ordinal1,d2_ordinal1),
    [interesting(0.00)]).

fof(t7_ordinal1,theorem,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & r1_tarski(B,A) ) ),
    file(ordinal1,t7_ordinal1),
    [interesting(0.00)]).

fof(t26_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,B)
            | r2_hidden(B,A) ) ) ) ),
    file(ordinal1,t26_ordinal1),
    [interesting(0.00)]).

fof(t15_card_1,theorem,(
    $true ),
    file(card_1,t15_card_1),
    [interesting(0.00)]).

fof(t16_card_1,theorem,(
    $true ),
    file(card_1,t16_card_1),
    [interesting(0.00)]).

fof(t17_card_1,theorem,(
    $true ),
    file(card_1,t17_card_1),
    [interesting(0.00)]).

fof(t18_card_1,theorem,(
    $true ),
    file(card_1,t18_card_1),
    [interesting(0.00)]).

fof(t19_card_1,theorem,(
    $true ),
    file(card_1,t19_card_1),
    [interesting(0.00)]).

fof(t1_card_1,theorem,(
    $true ),
    file(card_1,t1_card_1),
    [interesting(0.00)]).

fof(t20_card_1,theorem,(
    $true ),
    file(card_1,t20_card_1),
    [interesting(0.00)]).

fof(t23_ordinal1,theorem,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ( r2_hidden(A,B)
       => v3_ordinal1(A) ) ) ),
    file(ordinal1,t23_ordinal1),
    [interesting(0.00)]).

fof(d1_card_1,definition,(
    ! [A] : 
      ( v1_card_1(A)
    <=> ? [B] : 
          ( v3_ordinal1(B)
          & A = B
          & ! [C] : 
              ( v3_ordinal1(C)
             => ( r2_wellord2(C,B)
               => r1_ordinal1(B,C) ) ) ) ) ),
    file(card_1,d1_card_1),
    [interesting(0.00)]).

fof(t22_ordinal1,theorem,(
    ! [A] : 
      ( v1_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( ( r1_tarski(A,B)
                  & r2_hidden(B,C) )
               => r2_hidden(A,C) ) ) ) ) ),
    file(ordinal1,t22_ordinal1),
    [interesting(0.00)]).

fof(d4_wellord2,definition,(
    ! [A,B] : 
      ( r2_wellord2(A,B)
    <=> ? [C] : 
          ( v1_relat_1(C)
          & v1_funct_1(C)
          & v2_funct_1(C)
          & k1_relat_1(C) = A
          & k2_relat_1(C) = B ) ) ),
    file(wellord2,d4_wellord2),
    [interesting(0.00)]).

fof(t62_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => v2_funct_1(k2_funct_1(A)) ) ) ),
    file(funct_1,t62_funct_1),
    [interesting(0.00)]).

fof(t46_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( v2_funct_1(A)
              & v2_funct_1(B) )
           => v2_funct_1(k5_relat_1(A,B)) ) ) ) ),
    file(funct_1,t46_funct_1),
    [interesting(0.00)]).

fof(t55_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
       => ( k2_relat_1(A) = k1_relat_1(k2_funct_1(A))
          & k1_relat_1(A) = k2_relat_1(k2_funct_1(A)) ) ) ) ),
    file(funct_1,t55_funct_1),
    [interesting(0.00)]).

fof(t45_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => r1_tarski(k2_relat_1(k5_relat_1(A,B)),k2_relat_1(B)) ) ) ),
    file(relat_1,t45_relat_1),
    [interesting(0.00)]).

fof(t46_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r1_tarski(k2_relat_1(A),k1_relat_1(B))
           => k1_relat_1(k5_relat_1(A,B)) = k1_relat_1(A) ) ) ) ),
    file(relat_1,t46_relat_1),
    [interesting(0.00)]).

fof(t9_wellord2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( r1_tarski(B,A)
         => v2_wellord1(k1_wellord2(B)) ) ) ),
    file(wellord2,t9_wellord2),
    [interesting(0.00)]).

fof(t17_wellord2,theorem,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ( r1_tarski(A,B)
       => r1_ordinal1(k2_wellord2(k1_wellord2(A)),B) ) ) ),
    file(wellord2,t17_wellord2),
    [interesting(0.00)]).

fof(d1_wellord2,definition,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( B = k1_wellord2(A)
      <=> ( k3_relat_1(B) = A
          & ! [C,D] : 
              ( ( r2_hidden(C,A)
                & r2_hidden(D,A) )
             => ( r2_hidden(k4_tarski(C,D),B)
              <=> r1_tarski(C,D) ) ) ) ) ) ),
    file(wellord2,d1_wellord2),
    [interesting(0.00)]).

fof(d2_wellord2,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( v2_wellord1(A)
       => ! [B] : 
            ( v3_ordinal1(B)
           => ( B = k2_wellord2(A)
            <=> r4_wellord1(A,k1_wellord2(B)) ) ) ) ) ),
    file(wellord2,d2_wellord2),
    [interesting(0.00)]).

fof(d8_wellord1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r4_wellord1(A,B)
          <=> ? [C] : 
                ( v1_relat_1(C)
                & v1_funct_1(C)
                & r3_wellord1(A,B,C) ) ) ) ) ),
    file(wellord1,d8_wellord1),
    [interesting(0.00)]).

fof(d7_wellord1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C) )
             => ( r3_wellord1(A,B,C)
              <=> ( k1_relat_1(C) = k3_relat_1(A)
                  & k2_relat_1(C) = k3_relat_1(B)
                  & v2_funct_1(C)
                  & ! [D,E] : 
                      ( r2_hidden(k4_tarski(D,E),A)
                    <=> ( r2_hidden(D,k3_relat_1(A))
                        & r2_hidden(E,k3_relat_1(A))
                        & r2_hidden(k4_tarski(k1_funct_1(C,D),k1_funct_1(C,E)),B) ) ) ) ) ) ) ) ),
    file(wellord1,d7_wellord1),
    [interesting(0.00)]).

fof(s1_ordinal1,theorem,
    ( ? [A] : 
        ( v3_ordinal1(A)
        & p1_s1_ordinal1(A) )
   => ? [A] : 
        ( v3_ordinal1(A)
        & p1_s1_ordinal1(A)
        & ! [B] : 
            ( v3_ordinal1(B)
           => ( p1_s1_ordinal1(B)
             => r1_ordinal1(A,B) ) ) ) ),
    file(ordinal1,s1_ordinal1),
    [interesting(0.00)]).

fof(t22_wellord2,theorem,(
    ! [A,B,C] : 
      ( ( r2_wellord2(A,B)
        & r2_wellord2(B,C) )
     => r2_wellord2(A,C) ) ),
    file(wellord2,t22_wellord2),
    [interesting(0.00)]).

fof(t1_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(B,C) )
     => r1_tarski(A,C) ) ),
    file(xboole_1,t1_xboole_1),
    [interesting(0.00)]).

fof(d10_xboole_0,definition,(
    ! [A,B] : 
      ( A = B
    <=> ( r1_tarski(A,B)
        & r1_tarski(B,A) ) ) ),
    file(xboole_0,d10_xboole_0),
    [interesting(0.00)]).

fof(s2_funct_1,theorem,
    ( ( ! [A,B,C] : 
          ( ( r2_hidden(A,f1_s2_funct_1)
            & p1_s2_funct_1(A,B)
            & p1_s2_funct_1(A,C) )
         => B = C )
      & ! [A] : ~ ( r2_hidden(A,f1_s2_funct_1)
          & ! [B] : ~ p1_s2_funct_1(A,B) ) )
   => ? [A] : 
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & k1_relat_1(A) = f1_s2_funct_1
        & ! [B] : 
            ( r2_hidden(B,f1_s2_funct_1)
           => p1_s2_funct_1(B,k1_funct_1(A,B)) ) ) ),
    file(funct_1,s2_funct_1),
    [interesting(0.00)]).

fof(d3_tarski,definition,(
    ! [A,B] : 
      ( r1_tarski(A,B)
    <=> ! [C] : 
          ( r2_hidden(C,A)
         => r2_hidden(C,B) ) ) ),
    file(tarski,d3_tarski),
    [interesting(0.00)]).

fof(t56_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( ( v2_funct_1(B)
          & r2_hidden(A,k1_relat_1(B)) )
       => ( A = k1_funct_1(k2_funct_1(B),k1_funct_1(B,A))
          & A = k1_funct_1(k5_relat_1(B,k2_funct_1(B)),A) ) ) ) ),
    file(funct_1,t56_funct_1),
    [interesting(0.00)]).

fof(d5_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( B = k2_relat_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : 
                  ( r2_hidden(D,k1_relat_1(A))
                  & C = k1_funct_1(A,D) ) ) ) ) ),
    file(funct_1,d5_funct_1),
    [interesting(0.00)]).

fof(s3_funct_1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & k1_relat_1(A) = f1_s3_funct_1
      & ! [B] : 
          ( r2_hidden(B,f1_s3_funct_1)
         => k1_funct_1(A,B) = f2_s3_funct_1(B) ) ) ),
    file(funct_1,s3_funct_1),
    [interesting(0.00)]).

fof(t2_xboole_1,theorem,(
    ! [A] : r1_tarski(k1_xboole_0,A) ),
    file(xboole_1,t2_xboole_1),
    [interesting(0.00)]).

fof(t52_funct_1,theorem,(
    ! [A] : v2_funct_1(k6_relat_1(A)) ),
    file(funct_1,t52_funct_1),
    [interesting(0.00)]).

fof(t71_relat_1,theorem,(
    ! [A] : 
      ( k1_relat_1(k6_relat_1(A)) = A
      & k2_relat_1(k6_relat_1(A)) = A ) ),
    file(relat_1,t71_relat_1),
    [interesting(0.00)]).

fof(t65_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ( k1_relat_1(A) = k1_xboole_0
      <=> k2_relat_1(A) = k1_xboole_0 ) ) ),
    file(relat_1,t65_relat_1),
    [interesting(0.00)]).

fof(d1_tarski,definition,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    [interesting(0.00)]).

fof(d13_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B,C] : 
          ( C = k10_relat_1(A,B)
        <=> ! [D] : 
              ( r2_hidden(D,C)
            <=> ( r2_hidden(D,k1_relat_1(A))
                & r2_hidden(k1_funct_1(A,D),B) ) ) ) ) ),
    file(funct_1,d13_funct_1),
    [interesting(0.00)]).

fof(t28_wellord2,theorem,(
    ! [A] : 
      ( ~ v1_xboole_0(A)
     => ~ ( ! [B] : ~ ( r2_hidden(B,A)
              & B = k1_xboole_0 )
          & ! [B] : 
              ( ( v1_relat_1(B)
                & v1_funct_1(B) )
             => ~ ( k1_relat_1(B) = A
                  & ! [C] : 
                      ( r2_hidden(C,A)
                     => r2_hidden(k1_funct_1(B,C),C) ) ) ) ) ) ),
    file(wellord2,t28_wellord2),
    [interesting(0.00)]).

fof(d8_funct_1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v2_funct_1(A)
      <=> ! [B,C] : 
            ( ( r2_hidden(B,k1_relat_1(A))
              & r2_hidden(C,k1_relat_1(A))
              & k1_funct_1(A,B) = k1_funct_1(A,C) )
           => B = C ) ) ) ),
    file(funct_1,d8_funct_1),
    [interesting(0.00)]).

fof(t23_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ! [C] : 
          ( ( v1_relat_1(C)
            & v1_funct_1(C) )
         => ( r2_hidden(A,k1_relat_1(B))
           => k1_funct_1(k5_relat_1(B,C),A) = k1_funct_1(C,k1_funct_1(B,A)) ) ) ) ),
    file(funct_1,t23_funct_1),
    [interesting(0.00)]).

fof(t2_card_1,theorem,(
    $true ),
    file(card_1,t2_card_1),
    [interesting(0.00)]).

fof(t31_card_1,theorem,(
    $true ),
    file(card_1,t31_card_1),
    [interesting(0.00)]).

fof(t37_card_1,theorem,(
    $true ),
    file(card_1,t37_card_1),
    [interesting(0.00)]).

fof(d8_card_1,definition,(
    ! [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))) ) ) ) ) ) ) ),
    file(card_1,d8_card_1),
    [interesting(0.00)]).

fof(s8_ordinal2,theorem,
    ( ! [A] : 
        ( v3_ordinal1(A)
       => ! [B] : 
            ( B = f1_s8_ordinal2(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) = f2_s8_ordinal2
                & ! [D] : 
                    ( v3_ordinal1(D)
                   => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                     => k1_funct_1(C,k1_ordinal1(D)) = f3_s8_ordinal2(D,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) = f4_s8_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
   => f1_s8_ordinal2(k1_xboole_0) = f2_s8_ordinal2 ),
    file(ordinal2,s8_ordinal2),
    [interesting(0.00)]).

fof(s9_ordinal2,theorem,
    ( ! [A] : 
        ( v3_ordinal1(A)
       => ! [B] : 
            ( B = f4_s9_ordinal2(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) = f1_s9_ordinal2
                & ! [D] : 
                    ( v3_ordinal1(D)
                   => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                     => k1_funct_1(C,k1_ordinal1(D)) = f2_s9_ordinal2(D,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) = f3_s9_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
   => ! [A] : 
        ( v3_ordinal1(A)
       => f4_s9_ordinal2(k1_ordinal1(A)) = f2_s9_ordinal2(A,f4_s9_ordinal2(A)) ) ),
    file(ordinal2,s9_ordinal2),
    [interesting(0.00)]).

fof(s10_ordinal2,theorem,
    ( ( ! [A] : 
          ( v3_ordinal1(A)
         => ! [B] : 
              ( B = f3_s10_ordinal2(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) = f4_s10_ordinal2
                  & ! [D] : 
                      ( v3_ordinal1(D)
                     => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                       => k1_funct_1(C,k1_ordinal1(D)) = f5_s10_ordinal2(D,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) = f6_s10_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
      & f2_s10_ordinal2 != k1_xboole_0
      & v4_ordinal1(f2_s10_ordinal2)
      & k1_relat_1(f1_s10_ordinal2) = f2_s10_ordinal2
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( r2_hidden(A,f2_s10_ordinal2)
           => k1_funct_1(f1_s10_ordinal2,A) = f3_s10_ordinal2(A) ) ) )
   => f3_s10_ordinal2(f2_s10_ordinal2) = f6_s10_ordinal2(f2_s10_ordinal2,f1_s10_ordinal2) ),
    file(ordinal2,s10_ordinal2),
    [interesting(0.00)]).

fof(l46_card_1,theorem,
    ( k3_card_1(0) = k1_card_1(k5_numbers)
    & ! [A] : 
        ( v3_ordinal1(A)
       => k3_card_1(k1_ordinal1(A)) = k2_card_1(k3_tarski(k1_tarski(k3_card_1(A)))) )
    & ! [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)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_card_1,s8_ordinal2,s9_ordinal2,s10_ordinal2]),
    [file(card_1,l46_card_1),interesting(0.00)]).

fof(t3_card_1,theorem,(
    $true ),
    file(card_1,t3_card_1),
    [interesting(0.00)]).

fof(t31_zfmisc_1,theorem,(
    ! [A] : k3_tarski(k1_tarski(A)) = A ),
    file(zfmisc_1,t31_zfmisc_1),
    [interesting(0.00)]).

fof(t19_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v1_ordinal1(C)
             => ( ( r2_hidden(C,A)
                  & r2_hidden(A,B) )
               => r2_hidden(C,B) ) ) ) ) ),
    file(ordinal1,t19_ordinal1),
    [interesting(0.00)]).

fof(t34_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,k1_ordinal1(B))
          <=> r1_ordinal1(A,B) ) ) ) ),
    file(ordinal1,t34_ordinal1),
    [interesting(0.00)]).

fof(s2_ordinal2,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & k1_relat_1(A) = f1_s2_ordinal2
      & ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(B,f1_s2_ordinal2)
           => k1_funct_1(A,B) = f2_s2_ordinal2(B) ) ) ) ),
    file(ordinal2,s2_ordinal2),
    [interesting(0.00)]).

fof(t41_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( v4_ordinal1(A)
      <=> ! [B] : 
            ( v3_ordinal1(B)
           => ( r2_hidden(B,A)
             => r2_hidden(k1_ordinal1(B),A) ) ) ) ) ),
    file(ordinal1,t41_ordinal1),
    [interesting(0.00)]).

fof(t27_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( r2_hidden(A,B)
         => r2_hidden(A,k7_ordinal2(B)) ) ) ),
    file(ordinal2,t27_ordinal2),
    [interesting(0.00)]).

fof(t35_ordinal2,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A) )
     => ( k8_ordinal2(A) = k7_ordinal2(k2_relat_1(A))
        & k9_ordinal2(A) = k6_ordinal2(k2_relat_1(A)) ) ) ),
    file(ordinal2,t35_ordinal2),
    [interesting(0.00)]).

fof(s1_ordinal2,theorem,
    ( ( p1_s1_ordinal2(k1_xboole_0)
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( p1_s1_ordinal2(A)
           => p1_s1_ordinal2(k1_ordinal1(A)) ) )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( ( v4_ordinal1(A)
              & ! [B] : 
                  ( v3_ordinal1(B)
                 => ( r2_hidden(B,A)
                   => p1_s1_ordinal2(B) ) ) )
           => ( A = k1_xboole_0
              | p1_s1_ordinal2(A) ) ) ) )
   => ! [A] : 
        ( v3_ordinal1(A)
       => p1_s1_ordinal2(A) ) ),
    file(ordinal2,s1_ordinal2),
    [interesting(0.00)]).

fof(t24_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ~ ( ~ r2_hidden(A,B)
              & A != B
              & ~ r2_hidden(B,A) ) ) ) ),
    file(ordinal1,t24_ordinal1),
    [interesting(0.00)]).

fof(t6_zfmisc_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(k1_tarski(A),k1_tarski(B))
     => A = B ) ),
    file(zfmisc_1,t6_zfmisc_1),
    [interesting(0.00)]).

fof(s1_xboole_0,theorem,(
    ? [A] : 
    ! [B] : 
      ( r2_hidden(B,A)
    <=> ( r2_hidden(B,f1_s1_xboole_0)
        & p1_s1_xboole_0(B) ) ) ),
    file(xboole_0,s1_xboole_0),
    [interesting(0.00)]).

fof(t26_wellord2,theorem,(
    ! [A] : 
    ? [B] : 
      ( v1_relat_1(B)
      & r2_wellord1(B,A) ) ),
    file(wellord2,t26_wellord2),
    [interesting(0.00)]).

fof(t25_wellord2,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( r2_wellord1(B,A)
       => ( k3_relat_1(k2_wellord1(B,A)) = A
          & v2_wellord1(k2_wellord1(B,A)) ) ) ) ),
    file(wellord2,t25_wellord2),
    [interesting(0.00)]).

fof(d1_ordinal1,definition,(
    ! [A] : k1_ordinal1(A) = k2_xboole_0(A,k1_tarski(A)) ),
    file(ordinal1,d1_ordinal1),
    [interesting(0.00)]).

fof(d4_ordinal2,definition,(
    k4_ordinal2 = k1_ordinal1(k1_xboole_0) ),
    file(ordinal2,d4_ordinal2),
    [interesting(0.00)]).

fof(t14_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( k1_relat_1(B) = k1_tarski(A)
       => k2_relat_1(B) = k1_tarski(k1_funct_1(B,A)) ) ) ),
    file(funct_1,t14_funct_1),
    [interesting(0.00)]).

fof(t13_funcop_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(B,A)
     => k1_funct_1(k2_funcop_1(A,C),B) = C ) ),
    file(funcop_1,t13_funcop_1),
    [interesting(0.00)]).

fof(t19_funcop_1,theorem,(
    ! [A,B] : 
      ( k1_relat_1(k2_funcop_1(A,B)) = A
      & r1_tarski(k2_relat_1(k2_funcop_1(A,B)),k1_tarski(B)) ) ),
    file(funcop_1,t19_funcop_1),
    [interesting(0.00)]).

fof(t39_zfmisc_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,k1_tarski(B))
    <=> ( A = k1_xboole_0
        | A = k1_tarski(B) ) ) ),
    file(zfmisc_1,t39_zfmisc_1),
    [interesting(0.00)]).

fof(t53_card_1,theorem,(
    $true ),
    file(card_1,t53_card_1),
    [interesting(0.00)]).

fof(t54_card_1,theorem,(
    $true ),
    file(card_1,t54_card_1),
    [interesting(0.00)]).

fof(t55_card_1,theorem,(
    $true ),
    file(card_1,t55_card_1),
    [interesting(0.00)]).

fof(t57_card_1,theorem,(
    $true ),
    file(card_1,t57_card_1),
    [interesting(0.00)]).

fof(t5_card_1,theorem,(
    $true ),
    file(card_1,t5_card_1),
    [interesting(0.00)]).

fof(t62_card_1,theorem,(
    $true ),
    file(card_1,t62_card_1),
    [interesting(0.00)]).

fof(t63_card_1,theorem,(
    $true ),
    file(card_1,t63_card_1),
    [interesting(0.00)]).

fof(s5_nat_1,theorem,
    ( ? [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & p1_s5_nat_1(A) )
   => ? [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & p1_s5_nat_1(A)
        & ! [B] : 
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( p1_s5_nat_1(B)
             => r1_xreal_0(A,B) ) ) ) ),
    file(nat_1,s5_nat_1),
    [interesting(0.00)]).

fof(t22_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ~ ( A != 0
          & ! [B] : 
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => A != k1_nat_1(B,1) ) ) ) ),
    file(nat_1,t22_nat_1),
    [interesting(0.00)]).

fof(t30_axioms,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => A = a_1_0_axioms(A) ) ),
    file(axioms,t30_axioms),
    [interesting(0.00)]).

fof(t38_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(k2_xcmplx_0(B,1),A)
          <=> r1_xreal_0(A,B) ) ) ) ),
    file(nat_1,t38_nat_1),
    [interesting(0.00)]).

fof(t13_ordinal1,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,k1_ordinal1(B))
    <=> ( r2_hidden(A,B)
        | A = B ) ) ),
    file(ordinal1,t13_ordinal1),
    [interesting(0.00)]).

fof(d5_real_1,definition,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
          <=> ~ ( r1_xreal_0(B,A)
                & B != A ) ) ) ) ),
    file(real_1,d5_real_1),
    [interesting(0.00)]).

fof(t40_xboole_1,theorem,(
    ! [A,B] : k4_xboole_0(k2_xboole_0(A,B),B) = k4_xboole_0(A,B) ),
    file(xboole_1,t40_xboole_1),
    [interesting(0.00)]).

fof(t65_zfmisc_1,theorem,(
    ! [A,B] : 
      ( k4_xboole_0(A,k1_tarski(B)) = A
    <=> ~ r2_hidden(B,A) ) ),
    file(zfmisc_1,t65_zfmisc_1),
    [interesting(0.00)]).

fof(d2_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k2_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            | r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d2_xboole_0),
    [interesting(0.00)]).

fof(t36_xboole_1,theorem,(
    ! [A,B] : r1_tarski(k4_xboole_0(A,B),A) ),
    file(xboole_1,t36_xboole_1),
    [interesting(0.00)]).

fof(t84_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( v2_funct_1(B)
       => v2_funct_1(k7_relat_1(B,A)) ) ) ),
    file(funct_1,t84_funct_1),
    [interesting(0.00)]).

fof(t91_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => ( r1_tarski(A,k1_relat_1(B))
       => k1_relat_1(k7_relat_1(B,A)) = A ) ) ),
    file(relat_1,t91_relat_1),
    [interesting(0.00)]).

fof(t148_relat_1,theorem,(
    ! [A,B] : 
      ( v1_relat_1(B)
     => k2_relat_1(k7_relat_1(B,A)) = k9_relat_1(B,A) ) ),
    file(relat_1,t148_relat_1),
    [interesting(0.00)]).

fof(t123_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( v2_funct_1(C)
       => k9_relat_1(C,k4_xboole_0(A,B)) = k4_xboole_0(k9_relat_1(C,A),k9_relat_1(C,B)) ) ) ),
    file(funct_1,t123_funct_1),
    [interesting(0.00)]).

fof(t146_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => k9_relat_1(A,k1_relat_1(A)) = k2_relat_1(A) ) ),
    file(relat_1,t146_relat_1),
    [interesting(0.00)]).

fof(t117_funct_1,theorem,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B) )
     => ( r2_hidden(A,k1_relat_1(B))
       => k9_relat_1(B,k1_tarski(A)) = k1_tarski(k1_funct_1(B,A)) ) ) ),
    file(funct_1,t117_funct_1),
    [interesting(0.00)]).

fof(d4_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k4_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & ~ r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d4_xboole_0),
    [interesting(0.00)]).

fof(t31_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ~ ( ~ r1_xreal_0(A,0)
              & r1_xreal_0(k2_xcmplx_0(B,A),B) ) ) ) ),
    file(xreal_1,t31_xreal_1),
    [interesting(0.00)]).

fof(t67_card_1,theorem,(
    $true ),
    file(card_1,t67_card_1),
    [interesting(0.00)]).

fof(d1_finset_1,definition,(
    ! [A] : 
      ( v1_finset_1(A)
    <=> ? [B] : 
          ( v1_relat_1(B)
          & v1_funct_1(B)
          & k2_relat_1(B) = A
          & r2_hidden(k1_relat_1(B),k5_ordinal2) ) ) ),
    file(finset_1,d1_finset_1),
    [interesting(0.00)]).

fof(t47_relat_1,theorem,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( v1_relat_1(B)
         => ( r1_tarski(k1_relat_1(A),k2_relat_1(B))
           => k2_relat_1(k5_relat_1(B,A)) = k2_relat_1(A) ) ) ) ),
    file(relat_1,t47_relat_1),
    [interesting(0.00)]).

fof(t6_card_1,theorem,(
    $true ),
    file(card_1,t6_card_1),
    [interesting(0.00)]).

fof(t70_card_1,theorem,(
    $true ),
    file(card_1,t70_card_1),
    [interesting(0.00)]).

fof(t51_card_1,theorem,(
    0 = k1_xboole_0 ),
    file(card_1,t51_card_1),
    [interesting(0.00)]).

fof(t37_zfmisc_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(k1_tarski(A),B)
    <=> r2_hidden(A,B) ) ),
    file(zfmisc_1,t37_zfmisc_1),
    [interesting(0.00)]).

fof(t12_xboole_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => k2_xboole_0(A,B) = B ) ),
    file(xboole_1,t12_xboole_1),
    [interesting(0.00)]).

fof(t3_xboole_0,theorem,(
    ! [A,B] : 
      ( ~ ( ~ r1_xboole_0(A,B)
          & ! [C] : ~ ( r2_hidden(C,A)
              & r2_hidden(C,B) ) )
      & ~ ( ? [C] : 
              ( r2_hidden(C,A)
              & r2_hidden(C,B) )
          & r1_xboole_0(A,B) ) ) ),
    file(xboole_0,t3_xboole_0),
    [interesting(0.00)]).

fof(d3_xboole_0,definition,(
    ! [A,B,C] : 
      ( C = k3_xboole_0(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( r2_hidden(D,A)
            & r2_hidden(D,B) ) ) ) ),
    file(xboole_0,d3_xboole_0),
    [interesting(0.00)]).

fof(d7_xboole_0,definition,(
    ! [A,B] : 
      ( r1_xboole_0(A,B)
    <=> k3_xboole_0(A,B) = k1_xboole_0 ) ),
    file(xboole_0,d7_xboole_0),
    [interesting(0.00)]).

fof(s2_finset_1,theorem,
    ( ( v1_finset_1(f1_s2_finset_1)
      & p1_s2_finset_1(k1_xboole_0)
      & ! [A,B] : 
          ( ( r2_hidden(A,f1_s2_finset_1)
            & r1_tarski(B,f1_s2_finset_1)
            & p1_s2_finset_1(B) )
         => p1_s2_finset_1(k2_xboole_0(B,k1_tarski(A))) ) )
   => p1_s2_finset_1(f1_s2_finset_1) ),
    file(finset_1,s2_finset_1),
    [interesting(0.00)]).

fof(t75_card_1,theorem,(
    $true ),
    file(card_1,t75_card_1),
    [interesting(0.00)]).

fof(t77_card_1,theorem,(
    $true ),
    file(card_1,t77_card_1),
    [interesting(0.00)]).

fof(t19_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ~ ( 0 != A
          & r1_xreal_0(A,0) ) ) ),
    file(nat_1,t19_nat_1),
    [interesting(0.00)]).

fof(t7_card_1,theorem,(
    $true ),
    file(card_1,t7_card_1),
    [interesting(0.00)]).

fof(t18_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => r1_xreal_0(0,A) ) ),
    file(nat_1,t18_nat_1),
    [interesting(0.00)]).

fof(t28_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => ~ ( r1_xreal_0(A,B)
              & ! [C] : 
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => B != k2_xcmplx_0(A,C) ) ) ) ) ),
    file(nat_1,t28_nat_1),
    [interesting(0.00)]).

fof(t29_nat_1,theorem,(
    ! [A] : 
      ( v4_ordinal2(A)
     => ! [B] : 
          ( v4_ordinal2(B)
         => r1_xreal_0(A,k2_xcmplx_0(A,B)) ) ) ),
    file(nat_1,t29_nat_1),
    [interesting(0.00)]).

fof(t10_ordinal1,theorem,(
    ! [A] : r2_hidden(A,k1_ordinal1(A)) ),
    file(ordinal1,t10_ordinal1),
    [interesting(0.00)]).

fof(t33_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r1_ordinal1(k1_ordinal1(A),B) ) ) ) ),
    file(ordinal1,t33_ordinal1),
    [interesting(0.00)]).

fof(s1_nat_1,theorem,
    ( ( p1_s1_nat_1(0)
      & ! [A] : 
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( p1_s1_nat_1(A)
           => p1_s1_nat_1(k1_nat_1(A,1)) ) ) )
   => ! [A] : 
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => p1_s1_nat_1(A) ) ),
    file(nat_1,s1_nat_1),
    [interesting(0.00)]).

fof(t19_ordinal2,theorem,
    ( r2_hidden(k1_xboole_0,k5_ordinal2)
    & v4_ordinal1(k5_ordinal2)
    & ! [A] : 
        ( v3_ordinal1(A)
       => ( ( r2_hidden(k1_xboole_0,A)
            & v4_ordinal1(A) )
         => r1_ordinal1(k5_ordinal2,A) ) ) ),
    file(ordinal2,t19_ordinal2),
    [interesting(0.00)]).

fof(d21_ordinal2,definition,(
    ! [A] : 
      ( v4_ordinal2(A)
    <=> r2_hidden(A,k5_ordinal2) ) ),
    file(ordinal2,d21_ordinal2),
    [interesting(0.00)]).

fof(t9_card_1,theorem,(
    $true ),
    file(card_1,t9_card_1),
    [interesting(0.00)]).