fof(t59_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k15_ordinal2(C,A),k15_ordinal2(C,B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t52_ordinal2,t10_ordinal1,t50_ordinal2,t51_ordinal2,t53_ordinal2,t1_xboole_1,t42_ordinal1,s3_ordinal2,t54_ordinal2,d5_funct_1,t23_ordinal1,t57_ordinal2,t22_ordinal1,t5_ordinal2,t28_ordinal2,t1_xboole_1,s2_ordinal1,t3_xboole_1]),
    [file(ordinal2,t59_ordinal2),interesting(1.00)]).

fof(t63_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( k16_ordinal2(A,k4_ordinal2) = A
        & k16_ordinal2(k4_ordinal2,A) = k4_ordinal2 ) ) ),
    inference(mizar_proof,[status(thm)],[t61_ordinal2,t60_ordinal2,t56_ordinal2,t60_ordinal2,t61_ordinal2,t56_ordinal2,s3_ordinal2,t62_ordinal2,d3_tarski,d5_funct_1,t23_ordinal1,d1_tarski,t23_ordinal1,d5_funct_1,d1_tarski,d13_ordinal2,d14_ordinal2,s1_ordinal2]),
    [file(ordinal2,t63_ordinal2),interesting(0.99)]).

fof(t47_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k14_ordinal2(k1_xboole_0,A) = A ) ),
    inference(mizar_proof,[status(thm)],[t44_ordinal2,t45_ordinal2,s3_ordinal2,t46_ordinal2,d5_funct_1,t23_ordinal1,d3_tarski,d10_xboole_0,d3_tarski,t23_ordinal1,d5_funct_1,t26_ordinal2,s1_ordinal2]),
    [file(ordinal2,t47_ordinal2),interesting(0.95)]).

fof(t55_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k15_ordinal2(A,k1_xboole_0) = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[t52_ordinal2,t53_ordinal2,t44_ordinal2,s3_ordinal2,t54_ordinal2,t2_xboole_1,d8_xboole_0,t21_ordinal1,d5_funct_1,t23_ordinal1,d1_tarski,d3_tarski,d10_xboole_0,d3_tarski,t52_ordinal2,d1_tarski,d5_funct_1,t26_ordinal2,t2_ordinal2,s1_ordinal2]),
    [file(ordinal2,t55_ordinal2),interesting(0.94)]).

fof(t57_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( r2_hidden(B,C)
               => ( A = k1_xboole_0
                  | r2_hidden(k15_ordinal2(B,A),k15_ordinal2(C,A)) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t2_xboole_1,d8_xboole_0,t21_ordinal1,t10_ordinal1,d8_xboole_0,t44_ordinal2,t49_ordinal2,t53_ordinal2,t21_ordinal1,t34_ordinal1,t44_ordinal2,t49_ordinal2,t53_ordinal2,t19_ordinal1,t42_ordinal1,s3_ordinal2,t54_ordinal2,t41_ordinal1,d5_funct_1,t27_ordinal2,t92_zfmisc_1,t44_ordinal2,t49_ordinal2,t53_ordinal2,s2_ordinal1]),
    [file(ordinal2,t57_ordinal2),interesting(0.93)]).

fof(t58_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k15_ordinal2(A,C),k15_ordinal2(B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t55_ordinal2,d8_xboole_0,t21_ordinal1,t57_ordinal2,d2_ordinal1]),
    [file(ordinal2,t58_ordinal2),interesting(0.93)]).

fof(t44_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k14_ordinal2(A,k1_xboole_0) = A ) ),
    inference(mizar_proof,[status(thm)],[d18_ordinal2,s14_ordinal2]),
    [file(ordinal2,t44_ordinal2),interesting(0.92)]).

fof(t56_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ( k15_ordinal2(k4_ordinal2,A) = A
        & k15_ordinal2(A,k4_ordinal2) = A ) ) ),
    inference(mizar_proof,[status(thm)],[t53_ordinal2,t52_ordinal2,t47_ordinal2,t10_ordinal1,t53_ordinal2,t45_ordinal2,t44_ordinal2,t42_ordinal1,s3_ordinal2,t54_ordinal2,d3_tarski,t23_ordinal1,d5_funct_1,d10_xboole_0,d3_tarski,d5_funct_1,t23_ordinal1,t26_ordinal2,d6_ordinal1,t52_ordinal2,s2_ordinal1]),
    [file(ordinal2,t56_ordinal2),interesting(0.91)]).

fof(t60_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k16_ordinal2(A,k1_xboole_0) = k4_ordinal2 ) ),
    inference(mizar_proof,[status(thm)],[d20_ordinal2,s14_ordinal2]),
    [file(ordinal2,t60_ordinal2),interesting(0.89)]).

fof(t48_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k14_ordinal2(A,k4_ordinal2) = k1_ordinal1(A) ) ),
    inference(mizar_proof,[status(thm)],[t45_ordinal2,t44_ordinal2]),
    [file(ordinal2,t48_ordinal2),interesting(0.87)]).

fof(t52_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k15_ordinal2(k1_xboole_0,A) = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[d19_ordinal2,s14_ordinal2]),
    [file(ordinal2,t52_ordinal2),interesting(0.87)]).

fof(t51_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k14_ordinal2(A,C),k14_ordinal2(B,C)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t44_ordinal2,t10_ordinal1,t45_ordinal2,t1_ordinal2,t42_ordinal1,s3_ordinal2,t46_ordinal2,d5_funct_1,t23_ordinal1,t49_ordinal2,t22_ordinal1,t28_ordinal2,s2_ordinal1]),
    [file(ordinal2,t51_ordinal2),interesting(0.85)]).

fof(t53_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => k15_ordinal2(k1_ordinal1(A),B) = k14_ordinal2(k15_ordinal2(A,B),B) ) ) ),
    inference(mizar_proof,[status(thm)],[d19_ordinal2,s15_ordinal2]),
    [file(ordinal2,t53_ordinal2),interesting(0.84)]).

fof(t45_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => k14_ordinal2(A,k1_ordinal1(B)) = k1_ordinal1(k14_ordinal2(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d18_ordinal2,s15_ordinal2]),
    [file(ordinal2,t45_ordinal2),interesting(0.83)]).

fof(t61_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => k16_ordinal2(A,k1_ordinal1(B)) = k15_ordinal2(A,k16_ordinal2(A,B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d20_ordinal2,s15_ordinal2]),
    [file(ordinal2,t61_ordinal2),interesting(0.83)]).

fof(t49_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( r2_hidden(A,B)
               => r2_hidden(k14_ordinal2(C,A),k14_ordinal2(C,B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t10_ordinal1,d8_xboole_0,t45_ordinal2,t21_ordinal1,t34_ordinal1,t45_ordinal2,t10_ordinal1,t19_ordinal1,t42_ordinal1,s3_ordinal2,t46_ordinal2,d5_funct_1,t27_ordinal2,s2_ordinal1]),
    [file(ordinal2,t49_ordinal2),interesting(0.82)]).

fof(t50_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( r1_ordinal1(A,B)
               => r1_ordinal1(k14_ordinal2(C,A),k14_ordinal2(C,B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_xboole_0,t21_ordinal1,t49_ordinal2,d2_ordinal1]),
    [file(ordinal2,t50_ordinal2),interesting(0.82)]).

fof(t2_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k3_tarski(k1_ordinal1(A)) = A ) ),
    inference(mizar_proof,[status(thm)],[t10_ordinal1,d3_tarski,d4_tarski,t23_ordinal1,t34_ordinal1,d10_xboole_0,t92_zfmisc_1]),
    [file(ordinal2,t2_ordinal2),interesting(0.80)]).

fof(t26_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k7_ordinal2(A) = A ) ),
    inference(mizar_proof,[status(thm)],[t10_ordinal2,t10_ordinal2,d7_ordinal2]),
    [file(ordinal2,t26_ordinal2),interesting(0.79)]).

fof(t65_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ? [B] : 
          ( v3_ordinal1(B)
          & ? [C] : 
              ( v3_ordinal1(C)
              & v4_ordinal1(B)
              & v4_ordinal2(C)
              & A = k14_ordinal2(B,C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t42_ordinal1,d5_ordinal2,d21_ordinal2,t44_ordinal2,t10_ordinal1,d5_ordinal2,d21_ordinal2,t41_ordinal1,d21_ordinal2,t45_ordinal2,s2_ordinal1]),
    [file(ordinal2,t65_ordinal2),interesting(0.79)]).

fof(t5_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => r1_ordinal1(k3_tarski(A),A) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d4_tarski,d2_ordinal1]),
    [file(ordinal2,t5_ordinal2),interesting(0.78)]).

fof(t10_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k2_ordinal2(A) = A ) ),
    inference(mizar_proof,[status(thm)],[t23_ordinal1,d2_ordinal2]),
    [file(ordinal2,t10_ordinal2),interesting(0.77)]).

fof(t32_ordinal2,theorem,(
    ! [A] : r1_ordinal1(k6_ordinal2(A),k7_ordinal2(A)) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_ordinal2,t2_setfam_1,d7_ordinal2,t2_setfam_1,d3_tarski,d2_ordinal1,t2_setfam_1,d1_setfam_1]),
    [file(ordinal2,t32_ordinal2),interesting(0.76)]).

fof(t27_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( r2_hidden(A,B)
         => r2_hidden(A,k7_ordinal2(B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal2,d7_ordinal2]),
    [file(ordinal2,t27_ordinal2),interesting(0.76)]).

fof(t11_ordinal2,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => r1_tarski(k2_ordinal2(A),k2_ordinal2(B)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_ordinal2,d2_ordinal2]),
    [file(ordinal2,t11_ordinal2),interesting(0.75)]).

fof(t28_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( ! [C] : 
              ( v3_ordinal1(C)
             => ( r2_hidden(C,B)
               => r2_hidden(C,A) ) )
         => r1_ordinal1(k7_ordinal2(B),A) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_ordinal2,d2_ordinal2,d7_ordinal2]),
    [file(ordinal2,t28_ordinal2),interesting(0.74)]).

fof(t1_ordinal2,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)],[t33_ordinal1,t34_ordinal1]),
    [file(ordinal2,t1_ordinal2),interesting(0.74)]).

fof(t16_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ? [B] : 
          ( v3_ordinal1(B)
          & r2_hidden(A,B)
          & v4_ordinal1(B) ) ) ),
    inference(mizar_proof,[status(thm)],[t136_zfmisc_1,d2_ordinal2,d2_ordinal2,d3_tarski,t23_ordinal1,d2_ordinal2,d2_ordinal1,d2_ordinal2,t31_ordinal1,d2_ordinal2,d2_ordinal2,d3_tarski,t13_ordinal1,d2_ordinal1,d2_ordinal2,t41_ordinal1]),
    [file(ordinal2,t16_ordinal2),interesting(0.73)]).

fof(t25_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( r2_hidden(A,B)
         => r2_hidden(k6_ordinal2(B),B) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_ordinal1,d2_ordinal2,d2_ordinal2,d2_ordinal2,d2_ordinal2,d1_setfam_1]),
    [file(ordinal2,t25_ordinal2),interesting(0.71)]).

fof(t4_ordinal2,theorem,(
    v4_ordinal1(k1_xboole_0) ),
    inference(mizar_proof,[status(thm)],[t2_zfmisc_1,d6_ordinal1]),
    [file(ordinal2,t4_ordinal2),interesting(0.71)]).

fof(t13_ordinal2,theorem,(
    ! [A] : r1_tarski(k3_ordinal2(A),A) ),
    inference(mizar_proof,[status(thm)],[d3_ordinal2,d3_tarski]),
    [file(ordinal2,t13_ordinal2),interesting(0.70)]).

fof(t15_ordinal2,theorem,(
    ! [A] : r1_tarski(k3_ordinal2(A),k2_ordinal2(A)) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_ordinal2,d2_ordinal2]),
    [file(ordinal2,t15_ordinal2),interesting(0.70)]).

fof(t17_ordinal2,theorem,(
    ! [A] : 
      ( ! [B] : 
          ( r2_hidden(B,A)
         => v3_ordinal1(B) )
     => v3_ordinal1(k1_setfam_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[s1_ordinal1,d1_setfam_1,d1_setfam_1]),
    [file(ordinal2,t17_ordinal2),interesting(0.70)]).

fof(t30_ordinal2,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => r1_ordinal1(k7_ordinal2(A),k7_ordinal2(B)) ) ),
    inference(mizar_proof,[status(thm)],[d7_ordinal2,t11_ordinal2,d7_ordinal2,t1_xboole_1]),
    [file(ordinal2,t30_ordinal2),interesting(0.70)]).

fof(t9_ordinal2,theorem,(
    ! [A] : r1_tarski(k2_ordinal2(A),A) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal2,d3_tarski]),
    [file(ordinal2,t9_ordinal2),interesting(0.70)]).

fof(t7_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B) )
         => ( k1_relat_1(B) = k1_ordinal1(A)
           => k1_ordinal2(B) = k1_funct_1(B,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t2_ordinal2]),
    [file(ordinal2,t7_ordinal2),interesting(0.69)]).

fof(t31_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => k7_ordinal2(k1_tarski(A)) = k1_ordinal1(A) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_ordinal2,d1_tarski,t10_ordinal1,d1_tarski,d2_ordinal2,t33_ordinal1,d7_ordinal2]),
    [file(ordinal2,t31_ordinal2),interesting(0.69)]).

fof(t29_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : ~ ( r2_hidden(A,k7_ordinal2(B))
          & ! [C] : 
              ( v3_ordinal1(C)
             => ~ ( r2_hidden(C,B)
                  & r1_ordinal1(A,C) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t26_ordinal1,t28_ordinal2,t7_ordinal1]),
    [file(ordinal2,t29_ordinal2),interesting(0.68)]).

fof(t3_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => r1_tarski(k1_ordinal1(A),k1_zfmisc_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,t13_ordinal1,t23_ordinal1,d2_ordinal1]),
    [file(ordinal2,t3_ordinal2),interesting(0.68)]).

fof(t14_ordinal2,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => r1_tarski(k3_ordinal2(A),k3_ordinal2(B)) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d3_ordinal2,d3_ordinal2]),
    [file(ordinal2,t14_ordinal2),interesting(0.66)]).

fof(t22_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( r2_hidden(A,B)
         => r1_ordinal1(k6_ordinal2(B),A) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal2,t4_setfam_1]),
    [file(ordinal2,t22_ordinal2),interesting(0.65)]).

fof(t24_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B,C] : 
          ( ( r2_hidden(A,B)
            & r1_tarski(B,C) )
         => r1_ordinal1(k6_ordinal2(C),k6_ordinal2(B)) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_ordinal2,t11_ordinal2,t7_setfam_1]),
    [file(ordinal2,t24_ordinal2),interesting(0.64)]).

fof(t46_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( v4_ordinal1(A)
           => ( A = k1_xboole_0
              | ! [C] : 
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C) )
                 => ( ( k1_relat_1(C) = A
                      & ! [D] : 
                          ( v3_ordinal1(D)
                         => ( r2_hidden(D,A)
                           => k1_funct_1(C,D) = k14_ordinal2(B,D) ) ) )
                   => k14_ordinal2(B,A) = k8_ordinal2(C) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d18_ordinal2,s16_ordinal2]),
    [file(ordinal2,t46_ordinal2),interesting(0.63)]).

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) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s3_funct_1,d7_ordinal1,t31_zfmisc_1,t26_ordinal2]),
    [file(ordinal2,s2_ordinal2),interesting(0.62)]).

fof(t62_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( v4_ordinal1(A)
           => ( A = k1_xboole_0
              | ! [C] : 
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C) )
                 => ( ( k1_relat_1(C) = A
                      & ! [D] : 
                          ( v3_ordinal1(D)
                         => ( r2_hidden(D,A)
                           => k1_funct_1(C,D) = k16_ordinal2(B,D) ) ) )
                   => k16_ordinal2(B,A) = k12_ordinal2(C) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d20_ordinal2,s16_ordinal2]),
    [file(ordinal2,t62_ordinal2),interesting(0.62)]).

fof(t23_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( ! [C] : 
              ( v3_ordinal1(C)
             => ( r2_hidden(C,B)
               => r1_ordinal1(A,C) ) )
         => ( k2_ordinal2(B) = k1_xboole_0
            | r1_ordinal1(A,k6_ordinal2(B)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_tarski,d2_ordinal2,d2_ordinal2,d1_setfam_1]),
    [file(ordinal2,t23_ordinal2),interesting(0.61)]).

fof(s3_ordinal2,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & v1_ordinal2(A)
      & k1_relat_1(A) = f1_s3_ordinal2
      & ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(B,f1_s3_ordinal2)
           => k1_funct_1(A,B) = f2_s3_ordinal2(B) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s2_ordinal2,d8_ordinal2,d3_tarski,d5_funct_1,t23_ordinal1,d2_ordinal2,d7_ordinal2]),
    [file(ordinal2,s3_ordinal2),interesting(0.61)]).

fof(t54_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ( v4_ordinal1(A)
           => ( A = k1_xboole_0
              | ! [C] : 
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C) )
                 => ( ( k1_relat_1(C) = A
                      & ! [D] : 
                          ( v3_ordinal1(D)
                         => ( r2_hidden(D,A)
                           => k1_funct_1(C,D) = k15_ordinal2(D,B) ) ) )
                   => k15_ordinal2(A,B) = k3_tarski(k8_ordinal2(C)) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d19_ordinal2,s16_ordinal2]),
    [file(ordinal2,t54_ordinal2),interesting(0.60)]).

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) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_ordinal2]),
    [file(ordinal2,t19_ordinal2),interesting(0.57)]).

fof(t34_ordinal2,theorem,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ( r2_hidden(A,k1_relat_1(B))
           => v3_ordinal1(k1_funct_1(B,A)) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_funct_1,d8_ordinal2,t23_ordinal1]),
    [file(ordinal2,t34_ordinal2),interesting(0.56)]).

fof(s5_ordinal2,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & k1_relat_1(A) = f1_s5_ordinal2
      & ( r2_hidden(k1_xboole_0,f1_s5_ordinal2)
       => k1_funct_1(A,k1_xboole_0) = f2_s5_ordinal2 )
      & ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(k1_ordinal1(B),f1_s5_ordinal2)
           => k1_funct_1(A,k1_ordinal1(B)) = f3_s5_ordinal2(B,k1_funct_1(A,B)) ) )
      & ! [B] : 
          ( v3_ordinal1(B)
         => ( ( r2_hidden(B,f1_s5_ordinal2)
              & v4_ordinal1(B) )
           => ( B = k1_xboole_0
              | k1_funct_1(A,B) = f4_s5_ordinal2(B,k2_ordinal1(A,B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s4_ordinal2,s1_funct_1,d4_relat_1,d3_tarski,d10_xboole_0,d3_tarski,t23_ordinal1,d4_relat_1,t42_ordinal1,t42_ordinal1,t12_ordinal1,d4_relat_1,t23_ordinal1,t8_funct_1,t8_funct_1,t8_funct_1,t42_ordinal1,s2_funct_1,d7_ordinal1,t8_funct_1,d2_ordinal1,t91_relat_1,t23_ordinal1,t19_ordinal1,t72_funct_1,d2_ordinal1,t91_relat_1,t72_funct_1,t19_ordinal1,t10_ordinal1,t19_ordinal1,t19_ordinal1,t72_funct_1,d2_ordinal1,t82_funct_1,t19_ordinal1,t72_funct_1,t8_funct_1,t10_ordinal1,t42_ordinal1,t12_ordinal1,t72_funct_1,t9_funct_1,s2_ordinal1,t42_ordinal1,t10_ordinal1,t12_ordinal1,t72_funct_1,t42_ordinal1,s2_ordinal1]),
    [file(ordinal2,s5_ordinal2),interesting(0.47)]).

fof(s11_ordinal2,theorem,(
    ? [A] : 
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v5_ordinal1(A)
      & v1_ordinal2(A)
      & k1_relat_1(A) = f1_s11_ordinal2
      & ( r2_hidden(k1_xboole_0,f1_s11_ordinal2)
       => k1_funct_1(A,k1_xboole_0) = f2_s11_ordinal2 )
      & ! [B] : 
          ( v3_ordinal1(B)
         => ( r2_hidden(k1_ordinal1(B),f1_s11_ordinal2)
           => k1_funct_1(A,k1_ordinal1(B)) = f3_s11_ordinal2(B,k1_funct_1(A,B)) ) )
      & ! [B] : 
          ( v3_ordinal1(B)
         => ( ( r2_hidden(B,f1_s11_ordinal2)
              & v4_ordinal1(B) )
           => ( B = k1_xboole_0
              | k1_funct_1(A,B) = f4_s11_ordinal2(B,k2_ordinal1(A,B)) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s5_ordinal2,d8_ordinal2,d3_tarski,d5_funct_1,t23_ordinal1,t42_ordinal1,d2_ordinal2,d7_ordinal2,t31_zfmisc_1,t26_ordinal2]),
    [file(ordinal2,s11_ordinal2),interesting(0.44)]).

fof(s15_ordinal2,theorem,
    ( ! [A] : 
        ( v3_ordinal1(A)
       => ! [B] : 
            ( v3_ordinal1(B)
           => ( B = f4_s15_ordinal2(A)
            <=> ? [C] : 
                  ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v5_ordinal1(C)
                  & v1_ordinal2(C)
                  & B = k1_ordinal2(C)
                  & k1_relat_1(C) = k1_ordinal1(A)
                  & k1_funct_1(C,k1_xboole_0) = f1_s15_ordinal2
                  & ! [D] : 
                      ( v3_ordinal1(D)
                     => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                       => k1_funct_1(C,k1_ordinal1(D)) = f2_s15_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_s15_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
   => ! [A] : 
        ( v3_ordinal1(A)
       => f4_s15_ordinal2(k1_ordinal1(A)) = f2_s15_ordinal2(A,f4_s15_ordinal2(A)) ) ),
    inference(mizar_proof,[status(thm)],[s11_ordinal2,s12_ordinal2,t10_ordinal1,t19_ordinal1]),
    [file(ordinal2,s15_ordinal2),interesting(0.39)]).

fof(s14_ordinal2,theorem,
    ( ! [A] : 
        ( v3_ordinal1(A)
       => ! [B] : 
            ( v3_ordinal1(B)
           => ( B = f1_s14_ordinal2(A)
            <=> ? [C] : 
                  ( v1_relat_1(C)
                  & v1_funct_1(C)
                  & v5_ordinal1(C)
                  & v1_ordinal2(C)
                  & B = k1_ordinal2(C)
                  & k1_relat_1(C) = k1_ordinal1(A)
                  & k1_funct_1(C,k1_xboole_0) = f2_s14_ordinal2
                  & ! [D] : 
                      ( v3_ordinal1(D)
                     => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                       => k1_funct_1(C,k1_ordinal1(D)) = f3_s14_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_s14_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
   => f1_s14_ordinal2(k1_xboole_0) = f2_s14_ordinal2 ),
    inference(mizar_proof,[status(thm)],[s11_ordinal2,t7_ordinal2,t10_ordinal1]),
    [file(ordinal2,s14_ordinal2),interesting(0.39)]).

fof(s4_ordinal2,theorem,
    ( ( k1_relat_1(f5_s4_ordinal2) = f1_s4_ordinal2
      & ( r2_hidden(k1_xboole_0,f1_s4_ordinal2)
       => k1_funct_1(f5_s4_ordinal2,k1_xboole_0) = f2_s4_ordinal2 )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( r2_hidden(k1_ordinal1(A),f1_s4_ordinal2)
           => k1_funct_1(f5_s4_ordinal2,k1_ordinal1(A)) = f3_s4_ordinal2(A,k1_funct_1(f5_s4_ordinal2,A)) ) )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( ( r2_hidden(A,f1_s4_ordinal2)
              & v4_ordinal1(A) )
           => ( A = k1_xboole_0
              | k1_funct_1(f5_s4_ordinal2,A) = f4_s4_ordinal2(A,k2_ordinal1(f5_s4_ordinal2,A)) ) ) )
      & k1_relat_1(f6_s4_ordinal2) = f1_s4_ordinal2
      & ( r2_hidden(k1_xboole_0,f1_s4_ordinal2)
       => k1_funct_1(f6_s4_ordinal2,k1_xboole_0) = f2_s4_ordinal2 )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( r2_hidden(k1_ordinal1(A),f1_s4_ordinal2)
           => k1_funct_1(f6_s4_ordinal2,k1_ordinal1(A)) = f3_s4_ordinal2(A,k1_funct_1(f6_s4_ordinal2,A)) ) )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( ( r2_hidden(A,f1_s4_ordinal2)
              & v4_ordinal1(A) )
           => ( A = k1_xboole_0
              | k1_funct_1(f6_s4_ordinal2,A) = f4_s4_ordinal2(A,k2_ordinal1(f6_s4_ordinal2,A)) ) ) ) )
   => f5_s4_ordinal2 = f6_s4_ordinal2 ),
    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,t10_ordinal1,t72_funct_1,t42_ordinal1,t9_funct_1]),
    [file(ordinal2,s4_ordinal2),interesting(0.36)]).

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 ),
    inference(mizar_proof,[status(thm)],[s5_ordinal2,t7_ordinal2,t10_ordinal1]),
    [file(ordinal2,s8_ordinal2),interesting(0.28)]).

fof(s16_ordinal2,theorem,
    ( ( ! [A] : 
          ( v3_ordinal1(A)
         => ! [B] : 
              ( v3_ordinal1(B)
             => ( B = f3_s16_ordinal2(A)
              <=> ? [C] : 
                    ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C)
                    & B = k1_ordinal2(C)
                    & k1_relat_1(C) = k1_ordinal1(A)
                    & k1_funct_1(C,k1_xboole_0) = f4_s16_ordinal2
                    & ! [D] : 
                        ( v3_ordinal1(D)
                       => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                         => k1_funct_1(C,k1_ordinal1(D)) = f5_s16_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_s16_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
      & f2_s16_ordinal2 != k1_xboole_0
      & v4_ordinal1(f2_s16_ordinal2)
      & k1_relat_1(f1_s16_ordinal2) = f2_s16_ordinal2
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( r2_hidden(A,f2_s16_ordinal2)
           => k1_funct_1(f1_s16_ordinal2,A) = f3_s16_ordinal2(A) ) ) )
   => f3_s16_ordinal2(f2_s16_ordinal2) = f6_s16_ordinal2(f2_s16_ordinal2,f1_s16_ordinal2) ),
    inference(mizar_proof,[status(thm)],[s11_ordinal2,s12_ordinal2,t10_ordinal1,d2_ordinal1,t91_relat_1,t23_ordinal1,t19_ordinal1,t72_funct_1,t9_funct_1]),
    [file(ordinal2,s16_ordinal2),interesting(0.28)]).

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)) ) ),
    inference(mizar_proof,[status(thm)],[s5_ordinal2,s6_ordinal2,t10_ordinal1,t19_ordinal1]),
    [file(ordinal2,s9_ordinal2),interesting(0.27)]).

fof(s6_ordinal2,theorem,
    ( ( ! [A] : 
          ( v3_ordinal1(A)
         => ! [B] : 
              ( B = f2_s6_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_s6_ordinal2
                  & ! [D] : 
                      ( v3_ordinal1(D)
                     => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                       => k1_funct_1(C,k1_ordinal1(D)) = f5_s6_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_s6_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) )
      & k1_relat_1(f1_s6_ordinal2) = f3_s6_ordinal2
      & ( r2_hidden(k1_xboole_0,f3_s6_ordinal2)
       => k1_funct_1(f1_s6_ordinal2,k1_xboole_0) = f4_s6_ordinal2 )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( r2_hidden(k1_ordinal1(A),f3_s6_ordinal2)
           => k1_funct_1(f1_s6_ordinal2,k1_ordinal1(A)) = f5_s6_ordinal2(A,k1_funct_1(f1_s6_ordinal2,A)) ) )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( ( r2_hidden(A,f3_s6_ordinal2)
              & v4_ordinal1(A) )
           => ( A = k1_xboole_0
              | k1_funct_1(f1_s6_ordinal2,A) = f6_s6_ordinal2(A,k2_ordinal1(f1_s6_ordinal2,A)) ) ) ) )
   => ! [A] : 
        ( v3_ordinal1(A)
       => ( r2_hidden(A,k1_relat_1(f1_s6_ordinal2))
         => k1_funct_1(f1_s6_ordinal2,A) = f2_s6_ordinal2(A) ) ) ),
    inference(mizar_proof,[status(thm)],[t33_ordinal1,t91_relat_1,t7_ordinal2,t72_funct_1,t2_xboole_1,d8_xboole_0,t21_ordinal1,t72_funct_1,t10_ordinal1,t19_ordinal1,t72_funct_1,d2_ordinal1,t72_funct_1,t82_funct_1]),
    [file(ordinal2,s6_ordinal2),interesting(0.24)]).

fof(s12_ordinal2,theorem,
    ( ( ! [A] : 
          ( v3_ordinal1(A)
         => ! [B] : 
              ( v3_ordinal1(B)
             => ( B = f2_s12_ordinal2(A)
              <=> ? [C] : 
                    ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C)
                    & B = k1_ordinal2(C)
                    & k1_relat_1(C) = k1_ordinal1(A)
                    & k1_funct_1(C,k1_xboole_0) = f4_s12_ordinal2
                    & ! [D] : 
                        ( v3_ordinal1(D)
                       => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(A))
                         => k1_funct_1(C,k1_ordinal1(D)) = f5_s12_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_s12_ordinal2(D,k2_ordinal1(C,D)) ) ) ) ) ) ) )
      & k1_relat_1(f1_s12_ordinal2) = f3_s12_ordinal2
      & ( r2_hidden(k1_xboole_0,f3_s12_ordinal2)
       => k1_funct_1(f1_s12_ordinal2,k1_xboole_0) = f4_s12_ordinal2 )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( r2_hidden(k1_ordinal1(A),f3_s12_ordinal2)
           => k1_funct_1(f1_s12_ordinal2,k1_ordinal1(A)) = f5_s12_ordinal2(A,k1_funct_1(f1_s12_ordinal2,A)) ) )
      & ! [A] : 
          ( v3_ordinal1(A)
         => ( ( r2_hidden(A,f3_s12_ordinal2)
              & v4_ordinal1(A) )
           => ( A = k1_xboole_0
              | k1_funct_1(f1_s12_ordinal2,A) = f6_s12_ordinal2(A,k2_ordinal1(f1_s12_ordinal2,A)) ) ) ) )
   => ! [A] : 
        ( v3_ordinal1(A)
       => ( r2_hidden(A,k1_relat_1(f1_s12_ordinal2))
         => k1_funct_1(f1_s12_ordinal2,A) = f2_s12_ordinal2(A) ) ) ),
    inference(mizar_proof,[status(thm)],[t33_ordinal1,t91_relat_1,t7_ordinal2,t72_funct_1,t2_xboole_1,d8_xboole_0,t21_ordinal1,t72_funct_1,t10_ordinal1,t19_ordinal1,t72_funct_1,d2_ordinal1,t72_funct_1,t82_funct_1]),
    [file(ordinal2,s12_ordinal2),interesting(0.20)]).

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) ),
    inference(mizar_proof,[status(thm)],[s5_ordinal2,s6_ordinal2,t10_ordinal1,d2_ordinal1,t91_relat_1,t23_ordinal1,t19_ordinal1,t72_funct_1,t9_funct_1]),
    [file(ordinal2,s10_ordinal2),interesting(0.15)]).

fof(s7_ordinal2,theorem,
    ( ? [A,B] : 
        ( v1_relat_1(B)
        & v1_funct_1(B)
        & v5_ordinal1(B)
        & A = k1_ordinal2(B)
        & k1_relat_1(B) = k1_ordinal1(f1_s7_ordinal2)
        & k1_funct_1(B,k1_xboole_0) = f2_s7_ordinal2
        & ! [C] : 
            ( v3_ordinal1(C)
           => ( r2_hidden(k1_ordinal1(C),k1_ordinal1(f1_s7_ordinal2))
             => k1_funct_1(B,k1_ordinal1(C)) = f3_s7_ordinal2(C,k1_funct_1(B,C)) ) )
        & ! [C] : 
            ( v3_ordinal1(C)
           => ( ( r2_hidden(C,k1_ordinal1(f1_s7_ordinal2))
                & v4_ordinal1(C) )
             => ( C = k1_xboole_0
                | k1_funct_1(B,C) = f4_s7_ordinal2(C,k2_ordinal1(B,C)) ) ) ) )
    & ! [A,B] : ~ ( ? [C] : 
            ( v1_relat_1(C)
            & v1_funct_1(C)
            & v5_ordinal1(C)
            & A = k1_ordinal2(C)
            & k1_relat_1(C) = k1_ordinal1(f1_s7_ordinal2)
            & k1_funct_1(C,k1_xboole_0) = f2_s7_ordinal2
            & ! [D] : 
                ( v3_ordinal1(D)
               => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s7_ordinal2))
                 => k1_funct_1(C,k1_ordinal1(D)) = f3_s7_ordinal2(D,k1_funct_1(C,D)) ) )
            & ! [D] : 
                ( v3_ordinal1(D)
               => ( ( r2_hidden(D,k1_ordinal1(f1_s7_ordinal2))
                    & v4_ordinal1(D) )
                 => ( D = k1_xboole_0
                    | k1_funct_1(C,D) = f4_s7_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
        & ? [C] : 
            ( v1_relat_1(C)
            & v1_funct_1(C)
            & v5_ordinal1(C)
            & B = k1_ordinal2(C)
            & k1_relat_1(C) = k1_ordinal1(f1_s7_ordinal2)
            & k1_funct_1(C,k1_xboole_0) = f2_s7_ordinal2
            & ! [D] : 
                ( v3_ordinal1(D)
               => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s7_ordinal2))
                 => k1_funct_1(C,k1_ordinal1(D)) = f3_s7_ordinal2(D,k1_funct_1(C,D)) ) )
            & ! [D] : 
                ( v3_ordinal1(D)
               => ( ( r2_hidden(D,k1_ordinal1(f1_s7_ordinal2))
                    & v4_ordinal1(D) )
                 => ( D = k1_xboole_0
                    | k1_funct_1(C,D) = f4_s7_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
        & A != B ) ),
    inference(mizar_proof,[status(thm)],[s5_ordinal2,t2_xboole_1,d8_xboole_0,t21_ordinal1,s4_ordinal2]),
    [file(ordinal2,s7_ordinal2),interesting(0.05)]).

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(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) ) ) ) ) ) ) ),
    file(ordinal1,t32_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(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(t23_ordinal1,theorem,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ( r2_hidden(A,B)
       => v3_ordinal1(A) ) ) ),
    file(ordinal1,t23_ordinal1),
    [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(t10_ordinal1,theorem,(
    ! [A] : r2_hidden(A,k1_ordinal1(A)) ),
    file(ordinal1,t10_ordinal1),
    [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(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) ) ) ) ),
    file(ordinal1,t42_ordinal1),
    [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(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(t12_ordinal1,theorem,(
    ! [A,B] : 
      ( k1_ordinal1(A) = k1_ordinal1(B)
     => A = B ) ),
    file(ordinal1,t12_ordinal1),
    [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(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(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(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(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(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(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(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(t2_xboole_1,theorem,(
    ! [A] : r1_tarski(k1_xboole_0,A) ),
    file(xboole_1,t2_xboole_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(d8_ordinal2,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A) )
     => ( v1_ordinal2(A)
      <=> ? [B] : 
            ( v3_ordinal1(B)
            & r1_tarski(k2_relat_1(A),B) ) ) ) ),
    file(ordinal2,d8_ordinal2),
    [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(d2_ordinal2,definition,(
    ! [A,B] : 
      ( B = k2_ordinal2(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ( r2_hidden(C,A)
            & v3_ordinal1(C) ) ) ) ),
    file(ordinal2,d2_ordinal2),
    [interesting(0.00)]).

fof(d7_ordinal2,definition,(
    ! [A,B] : 
      ( v3_ordinal1(B)
     => ( B = k7_ordinal2(A)
      <=> ( r1_tarski(k2_ordinal2(A),B)
          & ! [C] : 
              ( v3_ordinal1(C)
             => ( r1_tarski(k2_ordinal2(A),C)
               => r1_ordinal1(B,C) ) ) ) ) ) ),
    file(ordinal2,d7_ordinal2),
    [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(s13_ordinal2,theorem,
    ( ? [A] : 
        ( v3_ordinal1(A)
        & ? [B] : 
            ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B)
            & A = k1_ordinal2(B)
            & k1_relat_1(B) = k1_ordinal1(f1_s13_ordinal2)
            & k1_funct_1(B,k1_xboole_0) = f2_s13_ordinal2
            & ! [C] : 
                ( v3_ordinal1(C)
               => ( r2_hidden(k1_ordinal1(C),k1_ordinal1(f1_s13_ordinal2))
                 => k1_funct_1(B,k1_ordinal1(C)) = f3_s13_ordinal2(C,k1_funct_1(B,C)) ) )
            & ! [C] : 
                ( v3_ordinal1(C)
               => ( ( r2_hidden(C,k1_ordinal1(f1_s13_ordinal2))
                    & v4_ordinal1(C) )
                 => ( C = k1_xboole_0
                    | k1_funct_1(B,C) = f4_s13_ordinal2(C,k2_ordinal1(B,C)) ) ) ) ) )
    & ! [A] : 
        ( v3_ordinal1(A)
       => ! [B] : 
            ( v3_ordinal1(B)
           => ~ ( ? [C] : 
                    ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C)
                    & A = k1_ordinal2(C)
                    & k1_relat_1(C) = k1_ordinal1(f1_s13_ordinal2)
                    & k1_funct_1(C,k1_xboole_0) = f2_s13_ordinal2
                    & ! [D] : 
                        ( v3_ordinal1(D)
                       => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s13_ordinal2))
                         => k1_funct_1(C,k1_ordinal1(D)) = f3_s13_ordinal2(D,k1_funct_1(C,D)) ) )
                    & ! [D] : 
                        ( v3_ordinal1(D)
                       => ( ( r2_hidden(D,k1_ordinal1(f1_s13_ordinal2))
                            & v4_ordinal1(D) )
                         => ( D = k1_xboole_0
                            | k1_funct_1(C,D) = f4_s13_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
                & ? [C] : 
                    ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v5_ordinal1(C)
                    & v1_ordinal2(C)
                    & B = k1_ordinal2(C)
                    & k1_relat_1(C) = k1_ordinal1(f1_s13_ordinal2)
                    & k1_funct_1(C,k1_xboole_0) = f2_s13_ordinal2
                    & ! [D] : 
                        ( v3_ordinal1(D)
                       => ( r2_hidden(k1_ordinal1(D),k1_ordinal1(f1_s13_ordinal2))
                         => k1_funct_1(C,k1_ordinal1(D)) = f3_s13_ordinal2(D,k1_funct_1(C,D)) ) )
                    & ! [D] : 
                        ( v3_ordinal1(D)
                       => ( ( r2_hidden(D,k1_ordinal1(f1_s13_ordinal2))
                            & v4_ordinal1(D) )
                         => ( D = k1_xboole_0
                            | k1_funct_1(C,D) = f4_s13_ordinal2(D,k2_ordinal1(C,D)) ) ) ) )
                & A != B ) ) ) ),
    inference(mizar_proof,[status(thm)],[s11_ordinal2,t2_xboole_1,d8_xboole_0,t21_ordinal1,t31_zfmisc_1,t26_ordinal2,t31_zfmisc_1,t26_ordinal2,s4_ordinal2]),
    [file(ordinal2,s13_ordinal2),interesting(0.00)]).

fof(t12_ordinal2,theorem,(
    $true ),
    file(ordinal2,t12_ordinal2),
    [interesting(0.00)]).

fof(d3_ordinal2,definition,(
    ! [A,B] : 
      ( B = k3_ordinal2(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> ( r2_hidden(C,A)
            & ? [D] : 
                ( v3_ordinal1(D)
                & C = D
                & v4_ordinal1(D) ) ) ) ) ),
    file(ordinal2,d3_ordinal2),
    [interesting(0.00)]).

fof(t136_zfmisc_1,theorem,(
    ! [A] : 
    ? [B] : 
      ( r2_hidden(A,B)
      & ! [C,D] : 
          ( ( r2_hidden(C,B)
            & r1_tarski(D,C) )
         => r2_hidden(D,B) )
      & ! [C] : 
          ( r2_hidden(C,B)
         => r2_hidden(k1_zfmisc_1(C),B) )
      & ! [C] : ~ ( r1_tarski(C,B)
          & ~ r2_tarski(C,B)
          & ~ r2_hidden(C,B) ) ) ),
    file(zfmisc_1,t136_zfmisc_1),
    [interesting(0.00)]).

fof(t31_ordinal1,theorem,(
    ! [A] : 
      ( ! [B] : 
          ( r2_hidden(B,A)
         => ( v3_ordinal1(B)
            & r1_tarski(B,A) ) )
     => v3_ordinal1(A) ) ),
    file(ordinal1,t31_ordinal1),
    [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(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(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(d1_setfam_1,definition,(
    ! [A,B] : 
      ( ( A != k1_xboole_0
       => ( B = k1_setfam_1(A)
        <=> ! [C] : 
              ( r2_hidden(C,B)
            <=> ! [D] : 
                  ( r2_hidden(D,A)
                 => r2_hidden(C,D) ) ) ) )
      & ( A = k1_xboole_0
       => ( B = k1_setfam_1(A)
        <=> B = k1_xboole_0 ) ) ) ),
    file(setfam_1,d1_setfam_1),
    [interesting(0.00)]).

fof(t18_ordinal2,theorem,(
    $true ),
    file(ordinal2,t18_ordinal2),
    [interesting(0.00)]).

fof(d5_ordinal2,definition,(
    ! [A] : 
      ( A = k5_ordinal2
    <=> ( r2_hidden(k1_xboole_0,A)
        & v4_ordinal1(A)
        & v3_ordinal1(A)
        & ! [B] : 
            ( v3_ordinal1(B)
           => ( ( r2_hidden(k1_xboole_0,B)
                & v4_ordinal1(B) )
             => r1_tarski(A,B) ) ) ) ) ),
    file(ordinal2,d5_ordinal2),
    [interesting(0.00)]).

fof(t20_ordinal2,theorem,(
    $true ),
    file(ordinal2,t20_ordinal2),
    [interesting(0.00)]).

fof(t21_ordinal2,theorem,(
    $true ),
    file(ordinal2,t21_ordinal2),
    [interesting(0.00)]).

fof(t4_setfam_1,theorem,(
    ! [A,B] : 
      ( r2_hidden(A,B)
     => r1_tarski(k1_setfam_1(B),A) ) ),
    file(setfam_1,t4_setfam_1),
    [interesting(0.00)]).

fof(t7_setfam_1,theorem,(
    ! [A,B] : 
      ( r1_tarski(A,B)
     => ( A = k1_xboole_0
        | r1_tarski(k1_setfam_1(B),k1_setfam_1(A)) ) ) ),
    file(setfam_1,t7_setfam_1),
    [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(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(d1_tarski,definition,(
    ! [A,B] : 
      ( B = k1_tarski(A)
    <=> ! [C] : 
          ( r2_hidden(C,B)
        <=> C = A ) ) ),
    file(tarski,d1_tarski),
    [interesting(0.00)]).

fof(t2_setfam_1,theorem,(
    k1_setfam_1(k1_xboole_0) = k1_xboole_0 ),
    file(setfam_1,t2_setfam_1),
    [interesting(0.00)]).

fof(t33_ordinal2,theorem,(
    $true ),
    file(ordinal2,t33_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(t36_ordinal2,theorem,(
    $true ),
    file(ordinal2,t36_ordinal2),
    [interesting(0.00)]).

fof(t37_ordinal2,theorem,(
    $true ),
    file(ordinal2,t37_ordinal2),
    [interesting(0.00)]).

fof(t38_ordinal2,theorem,(
    $true ),
    file(ordinal2,t38_ordinal2),
    [interesting(0.00)]).

fof(t39_ordinal2,theorem,(
    $true ),
    file(ordinal2,t39_ordinal2),
    [interesting(0.00)]).

fof(t40_ordinal2,theorem,(
    $true ),
    file(ordinal2,t40_ordinal2),
    [interesting(0.00)]).

fof(t41_ordinal2,theorem,(
    $true ),
    file(ordinal2,t41_ordinal2),
    [interesting(0.00)]).

fof(t42_ordinal2,theorem,(
    $true ),
    file(ordinal2,t42_ordinal2),
    [interesting(0.00)]).

fof(t43_ordinal2,theorem,(
    $true ),
    file(ordinal2,t43_ordinal2),
    [interesting(0.00)]).

fof(d18_ordinal2,definition,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( C = k14_ordinal2(A,B)
              <=> ? [D] : 
                    ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v5_ordinal1(D)
                    & v1_ordinal2(D)
                    & C = k1_ordinal2(D)
                    & k1_relat_1(D) = k1_ordinal1(B)
                    & k1_funct_1(D,k1_xboole_0) = A
                    & ! [E] : 
                        ( v3_ordinal1(E)
                       => ( r2_hidden(k1_ordinal1(E),k1_ordinal1(B))
                         => k1_funct_1(D,k1_ordinal1(E)) = k1_ordinal1(k1_funct_1(D,E)) ) )
                    & ! [E] : 
                        ( v3_ordinal1(E)
                       => ( ( r2_hidden(E,k1_ordinal1(B))
                            & v4_ordinal1(E) )
                         => ( E = k1_xboole_0
                            | k1_funct_1(D,E) = k8_ordinal2(k2_ordinal1(D,E)) ) ) ) ) ) ) ) ) ),
    file(ordinal2,d18_ordinal2),
    [interesting(0.00)]).

fof(t2_zfmisc_1,theorem,(
    k3_tarski(k1_xboole_0) = k1_xboole_0 ),
    file(zfmisc_1,t2_zfmisc_1),
    [interesting(0.00)]).

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

fof(d19_ordinal2,definition,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( C = k15_ordinal2(A,B)
              <=> ? [D] : 
                    ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v5_ordinal1(D)
                    & v1_ordinal2(D)
                    & C = k1_ordinal2(D)
                    & k1_relat_1(D) = k1_ordinal1(A)
                    & k1_funct_1(D,k1_xboole_0) = k1_xboole_0
                    & ! [E] : 
                        ( v3_ordinal1(E)
                       => ( r2_hidden(k1_ordinal1(E),k1_ordinal1(A))
                         => k1_funct_1(D,k1_ordinal1(E)) = k14_ordinal2(k1_funct_1(D,E),B) ) )
                    & ! [E] : 
                        ( v3_ordinal1(E)
                       => ( ( r2_hidden(E,k1_ordinal1(A))
                            & v4_ordinal1(E) )
                         => ( E = k1_xboole_0
                            | k1_funct_1(D,E) = k3_tarski(k8_ordinal2(k2_ordinal1(D,E))) ) ) ) ) ) ) ) ) ),
    file(ordinal2,d19_ordinal2),
    [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(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) ) ),
    inference(mizar_proof,[status(thm)],[t10_ordinal1,t42_ordinal1,s2_ordinal1]),
    [file(ordinal2,s1_ordinal2),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(t3_xboole_1,theorem,(
    ! [A] : 
      ( r1_tarski(A,k1_xboole_0)
     => A = k1_xboole_0 ) ),
    file(xboole_1,t3_xboole_1),
    [interesting(0.00)]).

fof(d20_ordinal2,definition,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( v3_ordinal1(B)
         => ! [C] : 
              ( v3_ordinal1(C)
             => ( C = k16_ordinal2(A,B)
              <=> ? [D] : 
                    ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v5_ordinal1(D)
                    & v1_ordinal2(D)
                    & C = k1_ordinal2(D)
                    & k1_relat_1(D) = k1_ordinal1(B)
                    & k1_funct_1(D,k1_xboole_0) = k4_ordinal2
                    & ! [E] : 
                        ( v3_ordinal1(E)
                       => ( r2_hidden(k1_ordinal1(E),k1_ordinal1(B))
                         => k1_funct_1(D,k1_ordinal1(E)) = k15_ordinal2(A,k1_funct_1(D,E)) ) )
                    & ! [E] : 
                        ( v3_ordinal1(E)
                       => ( ( r2_hidden(E,k1_ordinal1(B))
                            & v4_ordinal1(E) )
                         => ( E = k1_xboole_0
                            | k1_funct_1(D,E) = k12_ordinal2(k2_ordinal1(D,E)) ) ) ) ) ) ) ) ) ),
    file(ordinal2,d20_ordinal2),
    [interesting(0.00)]).

fof(d13_ordinal2,definition,(
    ! [A] : 
      ( v3_ordinal1(A)
     => ! [B] : 
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v5_ordinal1(B)
            & v1_ordinal2(B) )
         => ( ( A = k1_xboole_0
             => ( r1_ordinal2(A,B)
              <=> ? [C] : 
                    ( v3_ordinal1(C)
                    & r2_hidden(C,k1_relat_1(B))
                    & ! [D] : 
                        ( v3_ordinal1(D)
                       => ( ( r1_ordinal1(C,D)
                            & r2_hidden(D,k1_relat_1(B)) )
                         => k1_funct_1(B,D) = k1_xboole_0 ) ) ) ) )
            & ( A != k1_xboole_0
             => ( r1_ordinal2(A,B)
              <=> ! [C] : 
                    ( v3_ordinal1(C)
                   => ! [D] : 
                        ( v3_ordinal1(D)
                       => ~ ( r2_hidden(C,A)
                            & r2_hidden(A,D)
                            & ! [E] : 
                                ( v3_ordinal1(E)
                               => ~ ( r2_hidden(E,k1_relat_1(B))
                                    & ! [F] : 
                                        ( v3_ordinal1(F)
                                       => ( ( r1_ordinal1(E,F)
                                            & r2_hidden(F,k1_relat_1(B)) )
                                         => ( r2_hidden(C,k1_funct_1(B,F))
                                            & r2_hidden(k1_funct_1(B,F),D) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
    file(ordinal2,d13_ordinal2),
    [interesting(0.00)]).

fof(d14_ordinal2,definition,(
    ! [A] : 
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v5_ordinal1(A)
        & v1_ordinal2(A) )
     => ( ? [B] : 
            ( v3_ordinal1(B)
            & r1_ordinal2(B,A) )
       => ! [B] : 
            ( v3_ordinal1(B)
           => ( B = k12_ordinal2(A)
            <=> r1_ordinal2(B,A) ) ) ) ) ),
    file(ordinal2,d14_ordinal2),
    [interesting(0.00)]).

fof(t64_ordinal2,theorem,(
    $true ),
    file(ordinal2,t64_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(t6_ordinal2,theorem,(
    $true ),
    file(ordinal2,t6_ordinal2),
    [interesting(0.00)]).

fof(t8_ordinal2,theorem,(
    $true ),
    file(ordinal2,t8_ordinal2),
    [interesting(0.00)]).