fof(t37_ordinal1,theorem,(
    ! [A] : ~ ! [B] : 
        ( r2_hidden(B,A)
      <=> v3_ordinal1(B) ) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal1,d3_tarski,t23_ordinal1,d3_ordinal1,t24_ordinal1,d4_ordinal1]),
    [file(ordinal1,t37_ordinal1),interesting(1.00)]).

fof(t38_ordinal1,theorem,(
    ! [A] : ~ ! [B] : 
        ( v3_ordinal1(B)
       => r2_hidden(B,A) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0,t37_ordinal1]),
    [file(ordinal1,t38_ordinal1),interesting(0.98)]).

fof(t36_ordinal1,theorem,(
    ! [A] : ~ ( ! [B] : 
          ( r2_hidden(B,A)
         => v3_ordinal1(B) )
      & ! [B] : 
          ( v3_ordinal1(B)
         => ~ r1_tarski(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[t35_ordinal1,t29_ordinal1,d3_tarski,d4_tarski,d3_tarski,t34_ordinal1]),
    [file(ordinal1,t36_ordinal1),interesting(0.98)]).

fof(t30_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => v3_ordinal1(k3_tarski(A)) ) ),
    inference(mizar_proof,[status(thm)],[d4_ordinal1,d4_ordinal1,d4_tarski,t23_ordinal1,d2_ordinal1,t92_zfmisc_1,d2_ordinal1,d4_ordinal1,d3_ordinal1,d4_tarski,d4_tarski,t23_ordinal1,t23_ordinal1,t24_ordinal1]),
    [file(ordinal1,t30_ordinal1),interesting(0.89)]).

fof(t35_ordinal1,theorem,(
    ! [A] : 
      ( ! [B] : 
          ( r2_hidden(B,A)
         => v3_ordinal1(B) )
     => v3_ordinal1(k3_tarski(A)) ) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal1,d4_tarski,d2_ordinal1,d3_tarski,d4_tarski,d4_ordinal1,d3_ordinal1,d4_tarski,d4_tarski,t23_ordinal1,t24_ordinal1]),
    [file(ordinal1,t35_ordinal1),interesting(0.89)]).

fof(t34_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,k1_ordinal1(B))
          <=> r1_ordinal1(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_xboole_0,d2_ordinal1,d1_tarski,t24_ordinal1,t10_ordinal1,d2_ordinal1,d2_ordinal1,d10_xboole_0,t14_ordinal1]),
    [file(ordinal1,t34_ordinal1),interesting(0.89)]).

fof(t21_ordinal1,theorem,(
    ! [A] : 
      ( v1_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_xboole_0(A,B)
           => r2_hidden(A,B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,t37_xboole_1,t60_xboole_1,t7_tarski,d4_xboole_0,d2_ordinal1,d4_xboole_0,d3_tarski,d2_ordinal1,d4_xboole_0,d3_ordinal1,d3_tarski,d10_xboole_0]),
    [file(ordinal1,t21_ordinal1),interesting(0.88)]).

fof(t39_ordinal1,theorem,(
    ! [A] : 
    ? [B] : 
      ( v3_ordinal1(B)
      & ~ r2_hidden(B,A)
      & ! [C] : 
          ( v3_ordinal1(C)
         => ( ~ r2_hidden(C,A)
           => r1_ordinal1(B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t38_ordinal1,s1_xboole_0,d3_tarski,t10_ordinal1,t32_ordinal1,d2_ordinal1,t24_ordinal1,d2_ordinal1]),
    [file(ordinal1,t39_ordinal1),interesting(0.88)]).

fof(t26_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r1_ordinal1(A,B)
            | r2_hidden(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t24_ordinal1,d2_ordinal1]),
    [file(ordinal1,t26_ordinal1),interesting(0.88)]).

fof(t23_ordinal1,theorem,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ( r2_hidden(A,B)
       => v3_ordinal1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal1,d3_ordinal1,d3_ordinal1,d3_tarski,d4_xboole_0,t7_tarski,d4_xboole_0,d2_ordinal1,d3_ordinal1,t3_ordinal1,d2_ordinal1,d4_ordinal1]),
    [file(ordinal1,t23_ordinal1),interesting(0.85)]).

fof(t14_ordinal1,theorem,(
    ! [A] : A != k1_ordinal1(A) ),
    inference(mizar_proof,[status(thm)],[t10_ordinal1]),
    [file(ordinal1,t14_ordinal1),interesting(0.84)]).

fof(t24_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ~ ( ~ r2_hidden(A,B)
              & A != B
              & ~ r2_hidden(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t21_ordinal1,d8_xboole_0,d3_tarski,d4_xboole_0,t7_tarski,d4_xboole_0,d2_ordinal1,t23_ordinal1,d4_xboole_0,d3_tarski,d8_xboole_0,t21_ordinal1]),
    [file(ordinal1,t24_ordinal1),interesting(0.83)]).

fof(t10_ordinal1,theorem,(
    ! [A] : r2_hidden(A,k1_ordinal1(A)) ),
    inference(mizar_proof,[status(thm)],[d1_tarski,d2_xboole_0]),
    [file(ordinal1,t10_ordinal1),interesting(0.83)]).

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) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d6_ordinal1,d4_tarski,d1_tarski,d3_tarski,t23_ordinal1,d2_ordinal1,t8_xboole_1,d8_xboole_0,t21_ordinal1,d2_ordinal1,t23_ordinal1,t10_ordinal1,d4_tarski,d3_tarski,d4_tarski,d2_ordinal1,d3_tarski,d10_xboole_0,d6_ordinal1]),
    [file(ordinal1,t41_ordinal1),interesting(0.83)]).

fof(t29_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => v3_ordinal1(k1_ordinal1(A)) ) ),
    inference(mizar_proof,[status(thm)],[d1_tarski,d2_xboole_0,d2_ordinal1,t7_xboole_1,t1_xboole_1,t7_xboole_1,d2_ordinal1,d3_ordinal1,d3_ordinal1,d4_ordinal1]),
    [file(ordinal1,t29_ordinal1),interesting(0.83)]).

fof(t49_ordinal1,theorem,(
    ! [A] : 
      ( ( ! [B] : 
            ( r2_hidden(B,A)
           => ( v1_relat_1(B)
              & v1_funct_1(B)
              & v5_ordinal1(B) ) )
        & v6_ordinal1(A) )
     => ( v1_relat_1(k3_tarski(A))
        & v1_funct_1(k3_tarski(A))
        & v5_ordinal1(k3_tarski(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_tarski,d1_relat_1,d1_relat_1,d1_funct_1,d4_tarski,d4_tarski,d9_ordinal1,d9_xboole_0,d1_funct_1,s1_funct_1,d5_funct_1,t8_funct_1,t36_ordinal1,d2_ordinal1,s1_ordinal1,d4_relat_1,d4_tarski,d4_relat_1,t8_funct_1,d5_funct_1,d3_tarski,d10_xboole_0,d3_tarski,t23_ordinal1,t26_ordinal1,d5_funct_1,t8_funct_1,d2_ordinal1,d10_xboole_0,t92_zfmisc_1,d4_relat_1,d4_relat_1,d7_ordinal1]),
    [file(ordinal1,t49_ordinal1),interesting(0.83)]).

fof(t31_ordinal1,theorem,(
    ! [A] : 
      ( ! [B] : 
          ( r2_hidden(B,A)
         => ( v3_ordinal1(B)
            & r1_tarski(B,A) ) )
     => v3_ordinal1(A) ) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal1,d4_ordinal1,d3_ordinal1,t24_ordinal1]),
    [file(ordinal1,t31_ordinal1),interesting(0.81)]).

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) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0,t10_ordinal1,d3_tarski,t32_ordinal1,d8_xboole_0,t21_ordinal1,d2_ordinal1]),
    [file(ordinal1,s1_ordinal1),interesting(0.81)]).

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) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0,t36_xboole_1,t32_ordinal1,d4_xboole_0,d2_ordinal1,d4_xboole_0,d8_xboole_0,t21_ordinal1,d4_xboole_0,t10_ordinal1]),
    [file(ordinal1,s2_ordinal1),interesting(0.80)]).

fof(t33_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(A,B)
          <=> r1_ordinal1(k1_ordinal1(A),B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal1,d1_tarski,d3_tarski,t8_xboole_1,t10_ordinal1]),
    [file(ordinal1,t33_ordinal1),interesting(0.79)]).

fof(t3_ordinal1,theorem,(
    ! [A,B,C] : ~ ( r2_hidden(A,B)
      & r2_hidden(B,C)
      & r2_hidden(C,A) ) ),
    inference(mizar_proof,[status(thm)],[d1_enumset1,t7_tarski,d1_enumset1]),
    [file(ordinal1,t3_ordinal1),interesting(0.78)]).

fof(t45_ordinal1,theorem,(
    ! [A] : m1_ordinal1(k1_xboole_0,A) ),
    inference(mizar_proof,[status(thm)],[d4_relat_1,d7_ordinal1,t65_relat_1,t2_xboole_1,d8_ordinal1]),
    [file(ordinal1,t45_ordinal1),interesting(0.75)]).

fof(t47_ordinal1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => ! [C] : 
          ( m1_ordinal1(C,A)
         => m1_ordinal1(C,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_ordinal1,t1_xboole_1,d8_ordinal1]),
    [file(ordinal1,t47_ordinal1),interesting(0.75)]).

fof(t27_ordinal1,theorem,(
    v3_ordinal1(k1_xboole_0) ),
    inference(mizar_proof,[status(thm)],[d4_ordinal1,l18_ordinal1]),
    [file(ordinal1,t27_ordinal1),interesting(0.74)]).

fof(t42_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( ~ ( ~ v4_ordinal1(A)
            & ! [B] : 
                ( v3_ordinal1(B)
               => A != k1_ordinal1(B) ) )
        & ~ ( ? [B] : 
                ( v3_ordinal1(B)
                & A = k1_ordinal1(B) )
            & v4_ordinal1(A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t41_ordinal1,t33_ordinal1,d8_xboole_0,t21_ordinal1,t10_ordinal1,t41_ordinal1]),
    [file(ordinal1,t42_ordinal1),interesting(0.74)]).

fof(t32_ordinal1,theorem,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ~ ( r1_tarski(A,B)
          & A != k1_xboole_0
          & ! [C] : 
              ( v3_ordinal1(C)
             => ~ ( r2_hidden(C,A)
                  & ! [D] : 
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,A)
                       => r1_ordinal1(C,D) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t7_tarski,t23_ordinal1,t24_ordinal1,d2_ordinal1]),
    [file(ordinal1,t32_ordinal1),interesting(0.73)]).

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) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal1,t7_ordinal1,t1_xboole_1,d8_xboole_0,t21_ordinal1]),
    [file(ordinal1,t22_ordinal1),interesting(0.73)]).

fof(l18_ordinal1,theorem,
    ( v1_ordinal1(k1_xboole_0)
    & v2_ordinal1(k1_xboole_0) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal1,d3_ordinal1]),
    [file(ordinal1,l18_ordinal1),interesting(0.73)]).

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) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal1]),
    [file(ordinal1,t19_ordinal1),interesting(0.67)]).

fof(t12_ordinal1,theorem,(
    ! [A,B] : 
      ( k1_ordinal1(A) = k1_ordinal1(B)
     => A = B ) ),
    inference(mizar_proof,[status(thm)],[t10_ordinal1,d2_xboole_0,t10_ordinal1,d2_xboole_0,d1_tarski]),
    [file(ordinal1,t12_ordinal1),interesting(0.67)]).

fof(t13_ordinal1,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,k1_ordinal1(B))
    <=> ( r2_hidden(A,B)
        | A = B ) ) ),
    inference(mizar_proof,[status(thm)],[d2_xboole_0,d1_tarski,d1_tarski,d2_xboole_0]),
    [file(ordinal1,t13_ordinal1),interesting(0.67)]).

fof(t4_ordinal1,theorem,(
    ! [A,B,C,D] : ~ ( r2_hidden(A,B)
      & r2_hidden(B,C)
      & r2_hidden(C,D)
      & r2_hidden(D,A) ) ),
    inference(mizar_proof,[status(thm)],[d2_enumset1,t7_tarski,d2_enumset1]),
    [file(ordinal1,t4_ordinal1),interesting(0.67)]).

fof(t25_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => r3_xboole_0(A,B) ) ) ),
    inference(mizar_proof,[status(thm)],[d9_xboole_0]),
    [file(ordinal1,t25_ordinal1),interesting(0.65)]).

fof(t5_ordinal1,theorem,(
    ! [A,B,C,D,E] : ~ ( r2_hidden(A,B)
      & r2_hidden(B,C)
      & r2_hidden(C,D)
      & r2_hidden(D,E)
      & r2_hidden(E,A) ) ),
    inference(mizar_proof,[status(thm)],[d3_enumset1,d3_enumset1,d3_enumset1,d3_enumset1,d3_enumset1,d3_enumset1,d3_enumset1,t7_tarski]),
    [file(ordinal1,t5_ordinal1),interesting(0.65)]).

fof(t6_ordinal1,theorem,(
    ! [A,B,C,D,E,F] : ~ ( r2_hidden(A,B)
      & r2_hidden(B,C)
      & r2_hidden(C,D)
      & r2_hidden(D,E)
      & r2_hidden(E,F)
      & r2_hidden(F,A) ) ),
    inference(mizar_proof,[status(thm)],[d4_enumset1,d4_enumset1,d4_enumset1,d4_enumset1,d4_enumset1,d4_enumset1,d4_enumset1,d4_enumset1,t7_tarski]),
    [file(ordinal1,t6_ordinal1),interesting(0.64)]).

fof(t48_ordinal1,theorem,(
    ! [A,B] : 
      ( m1_ordinal1(B,A)
     => ! [C] : 
          ( v3_ordinal1(C)
         => m1_ordinal1(k2_ordinal1(B,C),A) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_ordinal1,t47_ordinal1]),
    [file(ordinal1,t48_ordinal1),interesting(0.63)]).

fof(t50_ordinal1,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ~ ( ~ r2_xboole_0(A,B)
              & A != B
              & ~ r2_xboole_0(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0]),
    [file(ordinal1,t50_ordinal1),interesting(0.57)]).

fof(s4_ordinal1,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & k1_relat_1(A) = f1_s4_ordinal1
      & ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v5_ordinal1(C) )
             => ( ( r2_hidden(B,f1_s4_ordinal1)
                  & C = k2_ordinal1(A,B) )
               => k1_funct_1(A,B) = f2_s4_ordinal1(C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s3_ordinal1,s1_funct_1,d4_relat_1,d3_tarski,d10_xboole_0,d3_tarski,t23_ordinal1,d4_relat_1,d4_relat_1,t8_funct_1,s2_funct_1,d7_ordinal1,d2_ordinal1,t91_relat_1,t8_funct_1,t19_ordinal1,d2_ordinal1,t91_relat_1,d2_ordinal1,t82_funct_1,t70_funct_1,t8_funct_1,t23_ordinal1,t19_ordinal1,t72_funct_1,t9_funct_1,s2_ordinal1]),
    [file(ordinal1,s4_ordinal1),interesting(0.55)]).

fof(t46_ordinal1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v3_ordinal1(k1_relat_1(A))
       => m1_ordinal1(A,k2_relat_1(A)) ) ) ),
    inference(mizar_proof,[status(thm)],[d7_ordinal1,d8_ordinal1]),
    [file(ordinal1,t46_ordinal1),interesting(0.50)]).

fof(s3_ordinal1,theorem,
    ( ( k1_relat_1(f3_s3_ordinal1) = f1_s3_ordinal1
      & ! [A] : 
          ( v3_ordinal1(A)
         => ! [B] : 
              ( ( v1_relat_1(B)
                & v1_funct_1(B)
                & v5_ordinal1(B) )
             => ( ( r2_hidden(A,f1_s3_ordinal1)
                  & B = k2_ordinal1(f3_s3_ordinal1,A) )
               => k1_funct_1(f3_s3_ordinal1,A) = f2_s3_ordinal1(B) ) ) )
      & k1_relat_1(f4_s3_ordinal1) = f1_s3_ordinal1
      & ! [A] : 
          ( v3_ordinal1(A)
         => ! [B] : 
              ( ( v1_relat_1(B)
                & v1_funct_1(B)
                & v5_ordinal1(B) )
             => ( ( r2_hidden(A,f1_s3_ordinal1)
                  & B = k2_ordinal1(f4_s3_ordinal1,A) )
               => k1_funct_1(f4_s3_ordinal1,A) = f2_s3_ordinal1(B) ) ) ) )
   => f3_s3_ordinal1 = f4_s3_ordinal1 ),
    inference(mizar_proof,[status(thm)],[t9_funct_1,s1_xboole_0,d3_tarski,t32_ordinal1,d2_ordinal1,t7_ordinal1,t91_relat_1,t23_ordinal1,t70_funct_1,t9_funct_1]),
    [file(ordinal1,s3_ordinal1),interesting(0.17)]).

fof(t9_funct_1,theorem,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B) )
         => ( ( k1_relat_1(A) = k1_relat_1(B)
              & ! [C] : 
                  ( r2_hidden(C,k1_relat_1(A))
                 => k1_funct_1(A,C) = k1_funct_1(B,C) ) )
           => A = B ) ) ) ),
    file(funct_1,t9_funct_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(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(t7_tarski,theorem,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & ! [C] : ~ ( r2_hidden(C,B)
          & ! [D] : ~ ( r2_hidden(D,B)
              & r2_hidden(D,C) ) ) ) ),
    file(tarski,t7_tarski),
    [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(d3_ordinal1,definition,(
    ! [A] : 
      ( v2_ordinal1(A)
    <=> ! [B,C] : ~ ( r2_hidden(B,A)
          & r2_hidden(C,A)
          & ~ r2_hidden(B,C)
          & B != C
          & ~ r2_hidden(C,B) ) ) ),
    file(ordinal1,d3_ordinal1),
    [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(d1_enumset1,definition,(
    ! [A,B,C,D] : 
      ( D = k1_enumset1(A,B,C)
    <=> ! [E] : 
          ( r2_hidden(E,D)
        <=> ~ ( E != A
              & E != B
              & E != C ) ) ) ),
    file(enumset1,d1_enumset1),
    [interesting(0.00)]).

fof(d4_ordinal1,definition,(
    ! [A] : 
      ( v3_ordinal1(A)
    <=> ( v1_ordinal1(A)
        & v2_ordinal1(A) ) ) ),
    file(ordinal1,d4_ordinal1),
    [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(t37_xboole_1,theorem,(
    ! [A,B] : 
      ( k4_xboole_0(A,B) = k1_xboole_0
    <=> r1_tarski(A,B) ) ),
    file(xboole_1,t37_xboole_1),
    [interesting(0.00)]).

fof(t60_xboole_1,theorem,(
    ! [A,B] : ~ ( r1_tarski(A,B)
      & r2_xboole_0(B,A) ) ),
    file(xboole_1,t60_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(t7_ordinal1,theorem,(
    ! [A,B] : ~ ( r2_hidden(A,B)
      & r1_tarski(B,A) ) ),
    file(ordinal1,t7_ordinal1),
    [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(t70_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,k1_relat_1(k7_relat_1(C,A)))
       => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ),
    file(funct_1,t70_funct_1),
    [interesting(0.00)]).

fof(s1_funct_1,theorem,
    ( ! [A,B,C] : 
        ( ( p1_s1_funct_1(A,B)
          & p1_s1_funct_1(A,C) )
       => B = C )
   => ? [A] : 
        ( v1_relat_1(A)
        & v1_funct_1(A)
        & ! [B,C] : 
            ( r2_hidden(k4_tarski(B,C),A)
          <=> ( r2_hidden(B,f1_s1_funct_1)
              & p1_s1_funct_1(B,C) ) ) ) ),
    file(funct_1,s1_funct_1),
    [interesting(0.00)]).

fof(d4_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
     => ! [B] : 
          ( B = k1_relat_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ? [D] : r2_hidden(k4_tarski(C,D),A) ) ) ) ),
    file(relat_1,d4_relat_1),
    [interesting(0.00)]).

fof(t8_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(k4_tarski(A,B),C)
      <=> ( r2_hidden(A,k1_relat_1(C))
          & B = k1_funct_1(C,A) ) ) ) ),
    file(funct_1,t8_funct_1),
    [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(d7_ordinal1,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v5_ordinal1(A)
      <=> v3_ordinal1(k1_relat_1(A)) ) ) ),
    file(ordinal1,d7_ordinal1),
    [interesting(0.00)]).

fof(t82_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r1_tarski(A,B)
       => ( k7_relat_1(k7_relat_1(C,A),B) = k7_relat_1(C,A)
          & k7_relat_1(k7_relat_1(C,B),A) = k7_relat_1(C,A) ) ) ) ),
    file(funct_1,t82_funct_1),
    [interesting(0.00)]).

fof(t72_funct_1,theorem,(
    ! [A,B,C] : 
      ( ( v1_relat_1(C)
        & v1_funct_1(C) )
     => ( r2_hidden(B,A)
       => k1_funct_1(k7_relat_1(C,A),B) = k1_funct_1(C,B) ) ) ),
    file(funct_1,t72_funct_1),
    [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(d1_tarski,definition,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    [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(s5_ordinal1,theorem,
    ( ( ! [A] : 
          ( v3_ordinal1(A)
         => ! [B] : 
              ( B = f2_s5_ordinal1(A)
            <=> ? [C] : 
                  ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v5_ordinal1(C)
                  & B = f3_s5_ordinal1(C)
                  & k1_relat_1(C) = A
                  & ! [D] : 
                      ( v3_ordinal1(D)
                     => ( r2_hidden(D,A)
                       => k1_funct_1(C,D) = f3_s5_ordinal1(k2_ordinal1(C,D)) ) ) ) ) )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( r2_hidden(A,k1_relat_1(f1_s5_ordinal1))
           => k1_funct_1(f1_s5_ordinal1,A) = f2_s5_ordinal1(A) ) ) )
   => ! [A] : 
        ( v3_ordinal1(A)
       => ( r2_hidden(A,k1_relat_1(f1_s5_ordinal1))
         => k1_funct_1(f1_s5_ordinal1,A) = f3_s5_ordinal1(k2_ordinal1(f1_s5_ordinal1,A)) ) ) ),
    inference(mizar_proof,[status(thm)],[t23_ordinal1,d2_ordinal1,t91_relat_1,d2_ordinal1,t72_funct_1,t82_funct_1,t9_funct_1]),
    [file(ordinal1,s5_ordinal1),interesting(0.00)]).

fof(t11_ordinal1,theorem,(
    $true ),
    file(ordinal1,t11_ordinal1),
    [interesting(0.00)]).

fof(t15_ordinal1,theorem,(
    $true ),
    file(ordinal1,t15_ordinal1),
    [interesting(0.00)]).

fof(t16_ordinal1,theorem,(
    $true ),
    file(ordinal1,t16_ordinal1),
    [interesting(0.00)]).

fof(t17_ordinal1,theorem,(
    $true ),
    file(ordinal1,t17_ordinal1),
    [interesting(0.00)]).

fof(t18_ordinal1,theorem,(
    $true ),
    file(ordinal1,t18_ordinal1),
    [interesting(0.00)]).

fof(t1_ordinal1,theorem,(
    $true ),
    file(ordinal1,t1_ordinal1),
    [interesting(0.00)]).

fof(t20_ordinal1,theorem,(
    $true ),
    file(ordinal1,t20_ordinal1),
    [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(d9_xboole_0,definition,(
    ! [A,B] : 
      ( r3_xboole_0(A,B)
    <=> ( r1_tarski(A,B)
        | r1_tarski(B,A) ) ) ),
    file(xboole_0,d9_xboole_0),
    [interesting(0.00)]).

fof(t28_ordinal1,theorem,(
    $true ),
    file(ordinal1,t28_ordinal1),
    [interesting(0.00)]).

fof(t2_ordinal1,theorem,(
    $true ),
    file(ordinal1,t2_ordinal1),
    [interesting(0.00)]).

fof(d4_tarski,definition,(
    ! [A,B] : 
      ( B = k3_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ? [D] : 
              ( r2_hidden(C,D)
              & r2_hidden(D,A) ) ) ) ),
    file(tarski,d4_tarski),
    [interesting(0.00)]).

fof(t92_zfmisc_1,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => r1_tarski(A,k3_tarski(B)) ) ),
    file(zfmisc_1,t92_zfmisc_1),
    [interesting(0.00)]).

fof(t40_ordinal1,theorem,(
    $true ),
    file(ordinal1,t40_ordinal1),
    [interesting(0.00)]).

fof(d6_ordinal1,definition,(
    ! [A] : 
      ( v4_ordinal1(A)
    <=> A = k3_tarski(A) ) ),
    file(ordinal1,d6_ordinal1),
    [interesting(0.00)]).

fof(t8_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,B) )
     => r1_tarski(k2_xboole_0(A,C),B) ) ),
    file(xboole_1,t8_xboole_1),
    [interesting(0.00)]).

fof(t43_ordinal1,theorem,(
    $true ),
    file(ordinal1,t43_ordinal1),
    [interesting(0.00)]).

fof(t44_ordinal1,theorem,(
    $true ),
    file(ordinal1,t44_ordinal1),
    [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(t2_xboole_1,theorem,(
    ! [A] : r1_tarski(k1_xboole_0,A) ),
    file(xboole_1,t2_xboole_1),
    [interesting(0.00)]).

fof(d8_ordinal1,definition,(
    ! [A,B] : 
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B) )
     => ( m1_ordinal1(B,A)
      <=> r1_tarski(k2_relat_1(B),A) ) ) ),
    file(ordinal1,d8_ordinal1),
    [interesting(0.00)]).

fof(d1_relat_1,definition,(
    ! [A] : 
      ( v1_relat_1(A)
    <=> ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D] : B != k4_tarski(C,D) ) ) ),
    file(relat_1,d1_relat_1),
    [interesting(0.00)]).

fof(d1_funct_1,definition,(
    ! [A] : 
      ( v1_funct_1(A)
    <=> ! [B,C,D] : 
          ( ( r2_hidden(k4_tarski(B,C),A)
            & r2_hidden(k4_tarski(B,D),A) )
         => C = D ) ) ),
    file(funct_1,d1_funct_1),
    [interesting(0.00)]).

fof(d9_ordinal1,definition,(
    ! [A] : 
      ( v6_ordinal1(A)
    <=> ! [B,C] : 
          ( ( r2_hidden(B,A)
            & r2_hidden(C,A) )
         => r3_xboole_0(B,C) ) ) ),
    file(ordinal1,d9_ordinal1),
    [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(t7_xboole_1,theorem,(
    ! [A,B] : r1_tarski(A,k2_xboole_0(A,B)) ),
    file(xboole_1,t7_xboole_1),
    [interesting(0.00)]).

fof(d2_enumset1,definition,(
    ! [A,B,C,D,E] : 
      ( E = k2_enumset1(A,B,C,D)
    <=> ! [F] : 
          ( r2_hidden(F,E)
        <=> ~ ( F != A
              & F != B
              & F != C
              & F != D ) ) ) ),
    file(enumset1,d2_enumset1),
    [interesting(0.00)]).

fof(d3_enumset1,definition,(
    ! [A,B,C,D,E,F] : 
      ( F = k3_enumset1(A,B,C,D,E)
    <=> ! [G] : 
          ( r2_hidden(G,F)
        <=> ~ ( G != A
              & G != B
              & G != C
              & G != D
              & G != E ) ) ) ),
    file(enumset1,d3_enumset1),
    [interesting(0.00)]).

fof(d4_enumset1,definition,(
    ! [A,B,C,D,E,F,G] : 
      ( G = k4_enumset1(A,B,C,D,E,F)
    <=> ! [H] : 
          ( r2_hidden(H,G)
        <=> ~ ( H != A
              & H != B
              & H != C
              & H != D
              & H != E
              & H != F ) ) ) ),
    file(enumset1,d4_enumset1),
    [interesting(0.00)]).

fof(t8_ordinal1,theorem,(
    $true ),
    file(ordinal1,t8_ordinal1),
    [interesting(0.00)]).

fof(t9_ordinal1,theorem,(
    $true ),
    file(ordinal1,t9_ordinal1),
    [interesting(0.00)]).