fof(t1_mcart_1,theorem,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & r1_xboole_0(B,A) ) ) ),
    inference(mizar_proof,[status(thm)],[t7_tarski,t3_xboole_0]),
    [file(mcart_1,t1_mcart_1),interesting(1.00)]).

fof(t2_mcart_1,theorem,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & ! [C] : 
              ( r2_hidden(C,B)
             => r1_xboole_0(C,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0,t15_xboole_1,t1_mcart_1,d4_tarski,d2_xboole_0,t3_xboole_0,d2_xboole_0,t70_xboole_1]),
    [file(mcart_1,t2_mcart_1),interesting(0.88)]).

fof(t49_mcart_1,theorem,(
    ! [A,B,C] : 
      ( ~ ( ~ r1_tarski(A,k3_zfmisc_1(A,B,C))
          & ~ r1_tarski(A,k3_zfmisc_1(B,C,A))
          & ~ r1_tarski(A,k3_zfmisc_1(C,A,B)) )
     => A = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_1,t35_mcart_1,t29_mcart_1,t48_mcart_1,d3_zfmisc_1,t135_zfmisc_1,t29_mcart_1,t48_mcart_1]),
    [file(mcart_1,t49_mcart_1),interesting(0.84)]).

fof(t38_mcart_1,theorem,(
    ! [A,B] : 
      ( k3_zfmisc_1(A,A,A) = k3_zfmisc_1(B,B,B)
     => A = B ) ),
    inference(mizar_proof,[status(thm)],[t36_mcart_1]),
    [file(mcart_1,t38_mcart_1),interesting(0.82)]).

fof(t58_mcart_1,theorem,(
    ! [A,B] : 
      ( k4_zfmisc_1(A,A,A,A) = k4_zfmisc_1(B,B,B,B)
     => A = B ) ),
    inference(mizar_proof,[status(thm)],[t56_mcart_1]),
    [file(mcart_1,t58_mcart_1),interesting(0.81)]).

fof(t4_mcart_1,theorem,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D,E] : 
              ( ( r2_hidden(C,D)
                & r2_hidden(D,E)
                & r2_hidden(E,B) )
             => r1_xboole_0(C,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0,s1_xboole_0,s1_xboole_0,t4_xboole_1,t4_xboole_1,t15_xboole_1,t1_mcart_1,d4_tarski,d2_xboole_0,d2_xboole_0,d2_xboole_0,t3_xboole_0,d2_xboole_0,t4_xboole_1,d4_tarski,d2_xboole_0,t3_xboole_0,t70_xboole_1,d2_xboole_0,d4_tarski,d2_xboole_0,t3_xboole_0,d2_xboole_0,t70_xboole_1]),
    [file(mcart_1,t4_mcart_1),interesting(0.81)]).

fof(t20_mcart_1,theorem,(
    ! [A] : 
      ( ? [B,C] : A = k4_tarski(B,C)
     => ( A != k1_mcart_1(A)
        & A != k2_mcart_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_tarski,d1_tarski,d2_tarski,d1_mcart_1,d2_tarski,d2_mcart_1]),
    [file(mcart_1,t20_mcart_1),interesting(0.79)]).

fof(t34_mcart_1,theorem,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D,E,F] : ~ ( ( r2_hidden(C,A)
                | r2_hidden(D,A) )
              & B = k4_mcart_1(C,D,E,F) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t6_mcart_1,d2_tarski,t3_xboole_0,d5_tarski,d5_tarski,d5_tarski,d2_tarski]),
    [file(mcart_1,t34_mcart_1),interesting(0.79)]).

fof(t29_mcart_1,theorem,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D,E] : ~ ( ( r2_hidden(C,A)
                | r2_hidden(D,A) )
              & B = k3_mcart_1(C,D,E) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t4_mcart_1,d2_tarski,t3_xboole_0,d5_tarski,d5_tarski,d2_tarski]),
    [file(mcart_1,t29_mcart_1),interesting(0.78)]).

fof(t28_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( k3_mcart_1(A,B,C) = k3_mcart_1(D,E,F)
     => ( A = D
        & B = E
        & C = F ) ) ),
    inference(mizar_proof,[status(thm)],[t33_zfmisc_1,t33_zfmisc_1]),
    [file(mcart_1,t28_mcart_1),interesting(0.77)]).

fof(t8_mcart_1,theorem,(
    ! [A] : 
      ( ? [B,C] : A = k4_tarski(B,C)
     => k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) = A ) ),
    inference(mizar_proof,[status(thm)],[d1_mcart_1,d2_mcart_1]),
    [file(mcart_1,t8_mcart_1),interesting(0.76)]).

fof(t10_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(B,C))
     => ( r2_hidden(k1_mcart_1(A),B)
        & r2_hidden(k2_mcart_1(A),C) ) ) ),
    inference(mizar_proof,[status(thm)],[d2_zfmisc_1,d1_mcart_1,d2_mcart_1]),
    [file(mcart_1,t10_mcart_1),interesting(0.76)]).

fof(t54_mcart_1,theorem,(
    ! [A,B,C,D] : k3_zfmisc_1(k2_zfmisc_1(A,B),C,D) = k4_zfmisc_1(A,B,C,D) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t53_mcart_1]),
    [file(mcart_1,t54_mcart_1),interesting(0.76)]).

fof(t77_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D)
        & r1_tarski(E,F) )
     => r1_tarski(k3_zfmisc_1(A,C,E),k3_zfmisc_1(B,D,F)) ) ),
    inference(mizar_proof,[status(thm)],[t119_zfmisc_1,d3_zfmisc_1,t119_zfmisc_1]),
    [file(mcart_1,t77_mcart_1),interesting(0.76)]).

fof(t53_mcart_1,theorem,(
    ! [A,B,C,D] : k4_zfmisc_1(A,B,C,D) = k2_zfmisc_1(k2_zfmisc_1(k2_zfmisc_1(A,B),C),D) ),
    inference(mizar_proof,[status(thm)],[d4_zfmisc_1,d3_zfmisc_1]),
    [file(mcart_1,t53_mcart_1),interesting(0.75)]).

fof(t3_mcart_1,theorem,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D] : 
              ( ( r2_hidden(C,D)
                & r2_hidden(D,B) )
             => r1_xboole_0(C,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0,s1_xboole_0,t15_xboole_1,t15_xboole_1,t1_mcart_1,t4_xboole_1,d4_tarski,d2_xboole_0,d2_xboole_0,t3_xboole_0,d2_xboole_0,d4_tarski,d2_xboole_0,t3_xboole_0,d2_xboole_0,t70_xboole_1,t70_xboole_1]),
    [file(mcart_1,t3_mcart_1),interesting(0.74)]).

fof(t6_mcart_1,theorem,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D,E,F,G] : 
              ( ( r2_hidden(C,D)
                & r2_hidden(D,E)
                & r2_hidden(E,F)
                & r2_hidden(F,G)
                & r2_hidden(G,B) )
             => r1_xboole_0(C,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0,s1_xboole_0,s1_xboole_0,s1_xboole_0,s1_xboole_0,t4_xboole_1,t4_xboole_1,t4_xboole_1,t4_xboole_1,t15_xboole_1,t1_mcart_1,d4_tarski,d2_xboole_0,d2_xboole_0,d2_xboole_0,t3_xboole_0,t70_xboole_1,t70_xboole_1,d2_xboole_0,t4_xboole_1,t4_xboole_1,t4_xboole_1,d4_tarski,d2_xboole_0,t3_xboole_0,t70_xboole_1,t70_xboole_1,t70_xboole_1,d2_xboole_0,t4_xboole_1,t4_xboole_1,d4_tarski,d2_xboole_0,d2_xboole_0,d2_xboole_0,t3_xboole_0,d2_xboole_0,t4_xboole_1,d4_tarski,d2_xboole_0,d2_xboole_0,t3_xboole_0,d2_xboole_0,d4_tarski,d2_xboole_0,t3_xboole_0,d2_xboole_0,t70_xboole_1]),
    [file(mcart_1,t6_mcart_1),interesting(0.73)]).

fof(t63_mcart_1,theorem,(
    ! [A,B,C,D] : 
      ( ~ ( ~ r1_tarski(A,k4_zfmisc_1(A,B,C,D))
          & ~ r1_tarski(A,k4_zfmisc_1(B,C,D,A))
          & ~ r1_tarski(A,k4_zfmisc_1(C,D,A,B))
          & ~ r1_tarski(A,k4_zfmisc_1(D,A,B,C)) )
     => A = k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[t3_xboole_1,t55_mcart_1,t34_mcart_1,t60_mcart_1,t54_mcart_1,t49_mcart_1,t54_mcart_1,t49_mcart_1,t34_mcart_1,t60_mcart_1]),
    [file(mcart_1,t63_mcart_1),interesting(0.72)]).

fof(t73_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( r2_hidden(k3_mcart_1(A,B,C),k3_zfmisc_1(D,E,F))
    <=> ( r2_hidden(A,D)
        & r2_hidden(B,E)
        & r2_hidden(C,F) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t106_zfmisc_1,t106_zfmisc_1]),
    [file(mcart_1,t73_mcart_1),interesting(0.72)]).

fof(t25_mcart_1,theorem,(
    ! [A,B,C,D] : k2_zfmisc_1(k2_tarski(A,B),k2_tarski(C,D)) = k2_enumset1(k4_tarski(A,C),k4_tarski(A,D),k4_tarski(B,C),k4_tarski(B,D)) ),
    inference(mizar_proof,[status(thm)],[t132_zfmisc_1,t36_zfmisc_1,t36_zfmisc_1,t45_enumset1]),
    [file(mcart_1,t25_mcart_1),interesting(0.69)]).

fof(t72_mcart_1,theorem,(
    ! [A,B,C,D] : ~ ( r2_hidden(A,k3_zfmisc_1(B,C,D))
      & ! [E,F,G] : ~ ( r2_hidden(E,B)
          & r2_hidden(F,C)
          & r2_hidden(G,D)
          & A = k3_mcart_1(E,F,G) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,d2_zfmisc_1,d2_zfmisc_1]),
    [file(mcart_1,t72_mcart_1),interesting(0.67)]).

fof(t23_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(B,C))
     => A = k4_tarski(k1_mcart_1(A),k2_mcart_1(A)) ) ),
    inference(mizar_proof,[status(thm)],[t102_zfmisc_1,t8_mcart_1]),
    [file(mcart_1,t23_mcart_1),interesting(0.66)]).

fof(t39_mcart_1,theorem,(
    ! [A,B,C] : k3_zfmisc_1(k1_tarski(A),k1_tarski(B),k1_tarski(C)) = k1_tarski(k3_mcart_1(A,B,C)) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t35_zfmisc_1,t35_zfmisc_1]),
    [file(mcart_1,t39_mcart_1),interesting(0.66)]).

fof(t67_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(B,C))
     => ( A != k1_mcart_1(A)
        & A != k2_mcart_1(A) ) ) ),
    inference(mizar_proof,[status(thm)],[t66_mcart_1]),
    [file(mcart_1,t67_mcart_1),interesting(0.66)]).

fof(t55_mcart_1,theorem,(
    ! [A,B,C,D] : 
      ( ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0
        & D != k1_xboole_0 )
    <=> k4_zfmisc_1(A,B,C,D) != k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[d4_zfmisc_1,t35_mcart_1,t113_zfmisc_1]),
    [file(mcart_1,t55_mcart_1),interesting(0.65)]).

fof(t35_mcart_1,theorem,(
    ! [A,B,C] : 
      ( ( A != k1_xboole_0
        & B != k1_xboole_0
        & C != k1_xboole_0 )
    <=> k3_zfmisc_1(A,B,C) != k1_xboole_0 ) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t113_zfmisc_1,t113_zfmisc_1]),
    [file(mcart_1,t35_mcart_1),interesting(0.63)]).

fof(t36_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( k3_zfmisc_1(A,B,C) = k3_zfmisc_1(D,E,F)
     => ( A = k1_xboole_0
        | B = k1_xboole_0
        | C = k1_xboole_0
        | ( A = D
          & B = E
          & C = F ) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t113_zfmisc_1,d3_zfmisc_1,t134_zfmisc_1,t134_zfmisc_1]),
    [file(mcart_1,t36_mcart_1),interesting(0.62)]).

fof(t52_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( ~ r1_xboole_0(k3_zfmisc_1(A,B,C),k3_zfmisc_1(D,E,F))
     => ( ~ r1_xboole_0(A,D)
        & ~ r1_xboole_0(B,E)
        & ~ r1_xboole_0(C,F) ) ) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t127_zfmisc_1,t127_zfmisc_1]),
    [file(mcart_1,t52_mcart_1),interesting(0.62)]).

fof(t5_mcart_1,theorem,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D,E,F] : 
              ( ( r2_hidden(C,D)
                & r2_hidden(D,E)
                & r2_hidden(E,F)
                & r2_hidden(F,B) )
             => r1_xboole_0(C,A) ) ) ) ),
    inference(mizar_proof,[status(thm)],[s1_xboole_0,s1_xboole_0,s1_xboole_0,s1_xboole_0,t4_xboole_1,t4_xboole_1,t4_xboole_1,t15_xboole_1,t1_mcart_1,d4_tarski,d2_xboole_0,d2_xboole_0,d2_xboole_0,d2_xboole_0,t3_xboole_0,d2_xboole_0,t4_xboole_1,t4_xboole_1,d4_tarski,d2_xboole_0,t3_xboole_0,t70_xboole_1,t70_xboole_1,d2_xboole_0,t4_xboole_1,d4_tarski,d2_xboole_0,t3_xboole_0,t70_xboole_1,d2_xboole_0,d4_tarski,d2_xboole_0,t3_xboole_0,d2_xboole_0,t70_xboole_1]),
    [file(mcart_1,t5_mcart_1),interesting(0.62)]).

fof(t66_mcart_1,theorem,(
    ! [A,B] : 
      ( k2_zfmisc_1(A,B) != k1_xboole_0
     => ! [C] : 
          ( m1_subset_1(C,k2_zfmisc_1(A,B))
         => ( C != k1_mcart_1(C)
            & C != k2_mcart_1(C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t113_zfmisc_1,t26_mcart_1]),
    [file(mcart_1,t66_mcart_1),interesting(0.62)]).

fof(t9_mcart_1,theorem,(
    ! [A] : ~ ( A != k1_xboole_0
      & ! [B] : ~ ( r2_hidden(B,A)
          & ! [C,D] : ~ ( ( r2_hidden(C,A)
                | r2_hidden(D,A) )
              & B = k4_tarski(C,D) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t2_mcart_1,d2_tarski,t3_xboole_0,d5_tarski,d2_tarski]),
    [file(mcart_1,t9_mcart_1),interesting(0.62)]).

fof(t33_mcart_1,theorem,(
    ! [A,B,C,D,E,F,G,H] : 
      ( k4_mcart_1(A,B,C,D) = k4_mcart_1(E,F,G,H)
     => ( A = E
        & B = F
        & C = G
        & D = H ) ) ),
    inference(mizar_proof,[status(thm)],[t33_zfmisc_1,t28_mcart_1]),
    [file(mcart_1,t33_mcart_1),interesting(0.60)]).

fof(t37_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( k3_zfmisc_1(A,B,C) = k3_zfmisc_1(D,E,F)
     => ( k3_zfmisc_1(A,B,C) = k1_xboole_0
        | ( A = D
          & B = E
          & C = F ) ) ) ),
    inference(mizar_proof,[status(thm)],[t35_mcart_1,t36_mcart_1]),
    [file(mcart_1,t37_mcart_1),interesting(0.60)]).

fof(t64_mcart_1,theorem,(
    ! [A,B,C,D,E,F,G,H] : 
      ( ~ r1_xboole_0(k4_zfmisc_1(A,B,C,D),k4_zfmisc_1(E,F,G,H))
     => ( ~ r1_xboole_0(A,E)
        & ~ r1_xboole_0(B,F)
        & ~ r1_xboole_0(C,G)
        & ~ r1_xboole_0(D,H) ) ) ),
    inference(mizar_proof,[status(thm)],[t53_mcart_1,t127_zfmisc_1,t127_zfmisc_1,t127_zfmisc_1]),
    [file(mcart_1,t64_mcart_1),interesting(0.60)]).

fof(t88_mcart_1,theorem,(
    ! [A,B,C,D,E,F,G,H] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D)
        & r1_tarski(E,F)
        & r1_tarski(G,H) )
     => r1_tarski(k4_zfmisc_1(A,C,E,G),k4_zfmisc_1(B,D,F,H)) ) ),
    inference(mizar_proof,[status(thm)],[t77_mcart_1,d4_zfmisc_1,t119_zfmisc_1]),
    [file(mcart_1,t88_mcart_1),interesting(0.60)]).

fof(t26_mcart_1,theorem,(
    ! [A,B] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & ~ ! [C] : 
            ( m1_subset_1(C,k2_zfmisc_1(A,B))
           => ( C != k1_mcart_1(C)
              & C != k2_mcart_1(C) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t24_mcart_1,t20_mcart_1]),
    [file(mcart_1,t26_mcart_1),interesting(0.59)]).

fof(t7_mcart_1,theorem,(
    ! [A,B] : 
      ( k1_mcart_1(k4_tarski(A,B)) = A
      & k2_mcart_1(k4_tarski(A,B)) = B ) ),
    inference(mizar_proof,[status(thm)],[d1_mcart_1,d2_mcart_1]),
    [file(mcart_1,t7_mcart_1),interesting(0.59)]).

fof(t57_mcart_1,theorem,(
    ! [A,B,C,D,E,F,G,H] : 
      ( k4_zfmisc_1(A,B,C,D) = k4_zfmisc_1(E,F,G,H)
     => ( k4_zfmisc_1(A,B,C,D) = k1_xboole_0
        | ( A = E
          & B = F
          & C = G
          & D = H ) ) ) ),
    inference(mizar_proof,[status(thm)],[t55_mcart_1,t56_mcart_1]),
    [file(mcart_1,t57_mcart_1),interesting(0.57)]).

fof(l27_mcart_1,theorem,(
    ! [A,B] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & ? [C] : 
          ( m1_subset_1(C,k2_zfmisc_1(A,B))
          & ! [D] : 
              ( m1_subset_1(D,A)
             => ! [E] : 
                  ( m1_subset_1(E,B)
                 => C != k4_tarski(D,E) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t113_zfmisc_1,t10_mcart_1,t10_mcart_1,t24_mcart_1]),
    [file(mcart_1,l27_mcart_1),interesting(0.57)]).

fof(t84_mcart_1,theorem,(
    ! [A,B,C,D,E,F,G,H] : 
      ( r2_hidden(k4_mcart_1(A,B,C,D),k4_zfmisc_1(E,F,G,H))
    <=> ( r2_hidden(A,E)
        & r2_hidden(B,F)
        & r2_hidden(C,G)
        & r2_hidden(D,H) ) ) ),
    inference(mizar_proof,[status(thm)],[t54_mcart_1,t106_zfmisc_1,t73_mcart_1]),
    [file(mcart_1,t84_mcart_1),interesting(0.56)]).

fof(t43_mcart_1,theorem,(
    ! [A,B,C,D,E] : k3_zfmisc_1(k2_tarski(A,B),k2_tarski(C,D),k1_tarski(E)) = k2_enumset1(k3_mcart_1(A,C,E),k3_mcart_1(B,C,E),k3_mcart_1(A,D,E),k3_mcart_1(B,D,E)) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t25_mcart_1,t45_enumset1,t120_zfmisc_1,t36_zfmisc_1,t36_zfmisc_1,t45_enumset1,t104_enumset1]),
    [file(mcart_1,t43_mcart_1),interesting(0.55)]).

fof(t42_mcart_1,theorem,(
    ! [A,B,C,D] : k3_zfmisc_1(k1_tarski(A),k1_tarski(B),k2_tarski(C,D)) = k2_tarski(k3_mcart_1(A,B,C),k3_mcart_1(A,B,D)) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t35_zfmisc_1,t36_zfmisc_1]),
    [file(mcart_1,t42_mcart_1),interesting(0.53)]).

fof(t51_mcart_1,theorem,(
    ! [A,B,C] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & ~ ! [D] : 
            ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
           => ( D != k5_mcart_1(A,B,C,D)
              & D != k6_mcart_1(A,B,C,D)
              & D != k7_mcart_1(A,B,C,D) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t48_mcart_1,d5_tarski,d5_tarski,d2_tarski,t4_ordinal1,t50_mcart_1,t20_mcart_1]),
    [file(mcart_1,t51_mcart_1),interesting(0.53)]).

fof(t85_mcart_1,theorem,(
    ! [A,B,C,D,E] : 
      ( ! [F] : 
          ( r2_hidden(F,A)
        <=> ? [G,H,I,J] : 
              ( r2_hidden(G,B)
              & r2_hidden(H,C)
              & r2_hidden(I,D)
              & r2_hidden(J,E)
              & F = k4_mcart_1(G,H,I,J) ) )
     => A = k4_zfmisc_1(B,C,D,E) ) ),
    inference(mizar_proof,[status(thm)],[t73_mcart_1,d2_zfmisc_1,d2_zfmisc_1,t72_mcart_1,t2_tarski,d4_zfmisc_1]),
    [file(mcart_1,t85_mcart_1),interesting(0.53)]).

fof(t74_mcart_1,theorem,(
    ! [A,B,C,D] : 
      ( ! [E] : 
          ( r2_hidden(E,A)
        <=> ? [F,G,H] : 
              ( r2_hidden(F,B)
              & r2_hidden(G,C)
              & r2_hidden(H,D)
              & E = k3_mcart_1(F,G,H) ) )
     => A = k3_zfmisc_1(B,C,D) ) ),
    inference(mizar_proof,[status(thm)],[d2_zfmisc_1,d2_zfmisc_1,d2_zfmisc_1,d2_zfmisc_1,t2_tarski,d3_zfmisc_1]),
    [file(mcart_1,t74_mcart_1),interesting(0.51)]).

fof(l44_mcart_1,theorem,(
    ! [A,B,C] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & ? [D] : 
          ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
          & ! [E] : 
              ( m1_subset_1(E,A)
             => ! [F] : 
                  ( m1_subset_1(F,B)
                 => ! [G] : 
                      ( m1_subset_1(G,C)
                     => D != k3_mcart_1(E,F,G) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t113_zfmisc_1,d3_zfmisc_1,l27_mcart_1,l27_mcart_1]),
    [file(mcart_1,l44_mcart_1),interesting(0.50)]).

fof(t83_mcart_1,theorem,(
    ! [A,B,C,D,E] : ~ ( r2_hidden(A,k4_zfmisc_1(B,C,D,E))
      & ! [F,G,H,I] : ~ ( r2_hidden(F,B)
          & r2_hidden(G,C)
          & r2_hidden(H,D)
          & r2_hidden(I,E)
          & A = k4_mcart_1(F,G,H,I) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_zfmisc_1,d2_zfmisc_1,t72_mcart_1]),
    [file(mcart_1,t83_mcart_1),interesting(0.50)]).

fof(t12_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(k1_tarski(B),C))
     => ( k1_mcart_1(A) = B
        & r2_hidden(k2_mcart_1(A),C) ) ) ),
    inference(mizar_proof,[status(thm)],[t10_mcart_1,d1_tarski]),
    [file(mcart_1,t12_mcart_1),interesting(0.49)]).

fof(t13_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(B,k1_tarski(C)))
     => ( r2_hidden(k1_mcart_1(A),B)
        & k2_mcart_1(A) = C ) ) ),
    inference(mizar_proof,[status(thm)],[t10_mcart_1,d1_tarski]),
    [file(mcart_1,t13_mcart_1),interesting(0.49)]).

fof(t24_mcart_1,theorem,(
    ! [A,B] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & ~ ! [C] : 
            ( m1_subset_1(C,k2_zfmisc_1(A,B))
           => C = k4_tarski(k1_mcart_1(C),k2_mcart_1(C)) ) ) ),
    inference(mizar_proof,[status(thm)],[t113_zfmisc_1,t23_mcart_1]),
    [file(mcart_1,t24_mcart_1),interesting(0.49)]).

fof(t65_mcart_1,theorem,(
    ! [A,B,C,D] : k4_zfmisc_1(k1_tarski(A),k1_tarski(B),k1_tarski(C),k1_tarski(D)) = k1_tarski(k4_mcart_1(A,B,C,D)) ),
    inference(mizar_proof,[status(thm)],[t54_mcart_1,t35_zfmisc_1,t39_mcart_1]),
    [file(mcart_1,t65_mcart_1),interesting(0.49)]).

fof(t87_mcart_1,theorem,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k1_zfmisc_1(A))
     => ! [F] : 
          ( m1_subset_1(F,k1_zfmisc_1(B))
         => ! [G] : 
              ( m1_subset_1(G,k1_zfmisc_1(C))
             => ! [H] : 
                  ( m1_subset_1(H,k1_zfmisc_1(D))
                 => ! [I] : 
                      ( m1_subset_1(I,k4_zfmisc_1(A,B,C,D))
                     => ( r2_hidden(I,k4_zfmisc_1(E,F,G,H))
                       => ( r2_hidden(k8_mcart_1(A,B,C,D,I),E)
                          & r2_hidden(k9_mcart_1(A,B,C,D,I),F)
                          & r2_hidden(k10_mcart_1(A,B,C,D,I),G)
                          & r2_hidden(k11_mcart_1(A,B,C,D,I),H) ) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t55_mcart_1,t86_mcart_1]),
    [file(mcart_1,t87_mcart_1),interesting(0.49)]).

fof(t40_mcart_1,theorem,(
    ! [A,B,C,D] : k3_zfmisc_1(k2_tarski(A,B),k1_tarski(C),k1_tarski(D)) = k2_tarski(k3_mcart_1(A,C,D),k3_mcart_1(B,C,D)) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t36_zfmisc_1,t36_zfmisc_1]),
    [file(mcart_1,t40_mcart_1),interesting(0.47)]).

fof(t41_mcart_1,theorem,(
    ! [A,B,C,D] : k3_zfmisc_1(k1_tarski(A),k2_tarski(B,C),k1_tarski(D)) = k2_tarski(k3_mcart_1(A,B,D),k3_mcart_1(A,C,D)) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t36_zfmisc_1,t36_zfmisc_1]),
    [file(mcart_1,t41_mcart_1),interesting(0.47)]).

fof(t44_mcart_1,theorem,(
    ! [A,B,C,D,E] : k3_zfmisc_1(k2_tarski(A,B),k1_tarski(C),k2_tarski(D,E)) = k2_enumset1(k3_mcart_1(A,C,D),k3_mcart_1(B,C,D),k3_mcart_1(A,C,E),k3_mcart_1(B,C,E)) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t36_zfmisc_1,t25_mcart_1,t104_enumset1]),
    [file(mcart_1,t44_mcart_1),interesting(0.46)]).

fof(t45_mcart_1,theorem,(
    ! [A,B,C,D,E] : k3_zfmisc_1(k1_tarski(A),k2_tarski(B,C),k2_tarski(D,E)) = k2_enumset1(k3_mcart_1(A,B,D),k3_mcart_1(A,C,D),k3_mcart_1(A,B,E),k3_mcart_1(A,C,E)) ),
    inference(mizar_proof,[status(thm)],[d3_zfmisc_1,t36_zfmisc_1,t25_mcart_1,t104_enumset1]),
    [file(mcart_1,t45_mcart_1),interesting(0.46)]).

fof(t11_mcart_1,theorem,(
    ! [A,B,C] : 
      ( ( r2_hidden(k1_mcart_1(A),B)
        & r2_hidden(k2_mcart_1(A),C) )
     => ( ! [D,E] : A != k4_tarski(D,E)
        | r2_hidden(A,k2_zfmisc_1(B,C)) ) ) ),
    inference(mizar_proof,[status(thm)],[t8_mcart_1,d2_zfmisc_1]),
    [file(mcart_1,t11_mcart_1),interesting(0.44)]).

fof(t14_mcart_1,theorem,(
    ! [A,B,C] : 
      ( r2_hidden(A,k2_zfmisc_1(k1_tarski(B),k1_tarski(C)))
     => ( k1_mcart_1(A) = B
        & k2_mcart_1(A) = C ) ) ),
    inference(mizar_proof,[status(thm)],[t10_mcart_1,d1_tarski]),
    [file(mcart_1,t14_mcart_1),interesting(0.44)]).

fof(t46_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : k3_zfmisc_1(k2_tarski(A,B),k2_tarski(C,D),k2_tarski(E,F)) = k6_enumset1(k3_mcart_1(A,C,E),k3_mcart_1(A,D,E),k3_mcart_1(A,C,F),k3_mcart_1(A,D,F),k3_mcart_1(B,C,E),k3_mcart_1(B,D,E),k3_mcart_1(B,C,F),k3_mcart_1(B,D,F)) ),
    inference(mizar_proof,[status(thm)],[t25_mcart_1,t104_enumset1,t25_mcart_1,t104_enumset1,d3_zfmisc_1,t25_mcart_1,t45_enumset1,t120_zfmisc_1,t65_enumset1]),
    [file(mcart_1,t46_mcart_1),interesting(0.42)]).

fof(t15_mcart_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(A,k2_zfmisc_1(k2_tarski(B,C),D))
     => ( ( k1_mcart_1(A) = B
          | k1_mcart_1(A) = C )
        & r2_hidden(k2_mcart_1(A),D) ) ) ),
    inference(mizar_proof,[status(thm)],[t10_mcart_1,d2_tarski]),
    [file(mcart_1,t15_mcart_1),interesting(0.40)]).

fof(t16_mcart_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(A,k2_zfmisc_1(B,k2_tarski(C,D)))
     => ( r2_hidden(k1_mcart_1(A),B)
        & ( k2_mcart_1(A) = C
          | k2_mcart_1(A) = D ) ) ) ),
    inference(mizar_proof,[status(thm)],[t10_mcart_1,d2_tarski]),
    [file(mcart_1,t16_mcart_1),interesting(0.40)]).

fof(t76_mcart_1,theorem,(
    ! [A,B,C,D] : 
      ( m1_subset_1(D,k1_zfmisc_1(A))
     => ! [E] : 
          ( m1_subset_1(E,k1_zfmisc_1(B))
         => ! [F] : 
              ( m1_subset_1(F,k1_zfmisc_1(C))
             => ! [G] : 
                  ( m1_subset_1(G,k3_zfmisc_1(A,B,C))
                 => ( r2_hidden(G,k3_zfmisc_1(D,E,F))
                   => ( r2_hidden(k5_mcart_1(A,B,C,G),D)
                      & r2_hidden(k6_mcart_1(A,B,C,G),E)
                      & r2_hidden(k7_mcart_1(A,B,C,G),F) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t35_mcart_1,t75_mcart_1]),
    [file(mcart_1,t76_mcart_1),interesting(0.40)]).

fof(t86_mcart_1,theorem,(
    ! [A,B,C,D,E,F,G,H] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & E != k1_xboole_0
      & F != k1_xboole_0
      & G != k1_xboole_0
      & H != k1_xboole_0
      & ? [I] : 
          ( m1_subset_1(I,k4_zfmisc_1(A,B,C,D))
          & ? [J] : 
              ( m1_subset_1(J,k4_zfmisc_1(E,F,G,H))
              & I = J
              & ~ ( k8_mcart_1(A,B,C,D,I) = k8_mcart_1(E,F,G,H,J)
                  & k9_mcart_1(A,B,C,D,I) = k9_mcart_1(E,F,G,H,J)
                  & k10_mcart_1(A,B,C,D,I) = k10_mcart_1(E,F,G,H,J)
                  & k11_mcart_1(A,B,C,D,I) = k11_mcart_1(E,F,G,H,J) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t61_mcart_1,t61_mcart_1,t61_mcart_1,t61_mcart_1,t61_mcart_1,t61_mcart_1,t61_mcart_1,t61_mcart_1]),
    [file(mcart_1,t86_mcart_1),interesting(0.40)]).

fof(t48_mcart_1,theorem,(
    ! [A,B,C] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & ~ ! [D] : 
            ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
           => D = k3_mcart_1(k5_mcart_1(A,B,C,D),k6_mcart_1(A,B,C,D),k7_mcart_1(A,B,C,D)) ) ) ),
    inference(mizar_proof,[status(thm)],[l44_mcart_1,d5_mcart_1,d6_mcart_1,d7_mcart_1]),
    [file(mcart_1,t48_mcart_1),interesting(0.39)]).

fof(t62_mcart_1,theorem,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ~ ! [E] : 
            ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
           => ( E != k8_mcart_1(A,B,C,D,E)
              & E != k9_mcart_1(A,B,C,D,E)
              & E != k10_mcart_1(A,B,C,D,E)
              & E != k11_mcart_1(A,B,C,D,E) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t113_zfmisc_1,t60_mcart_1,d5_tarski,d5_tarski,d5_tarski,d2_tarski,t6_ordinal1,t54_mcart_1,t61_mcart_1,t50_mcart_1,t61_mcart_1,t50_mcart_1,t51_mcart_1]),
    [file(mcart_1,t62_mcart_1),interesting(0.38)]).

fof(t75_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & E != k1_xboole_0
      & F != k1_xboole_0
      & ? [G] : 
          ( m1_subset_1(G,k3_zfmisc_1(A,B,C))
          & ? [H] : 
              ( m1_subset_1(H,k3_zfmisc_1(D,E,F))
              & G = H
              & ~ ( k5_mcart_1(A,B,C,G) = k5_mcart_1(D,E,F,H)
                  & k6_mcart_1(A,B,C,G) = k6_mcart_1(D,E,F,H)
                  & k7_mcart_1(A,B,C,G) = k7_mcart_1(D,E,F,H) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t50_mcart_1,t50_mcart_1,t50_mcart_1,t50_mcart_1,t50_mcart_1,t50_mcart_1]),
    [file(mcart_1,t75_mcart_1),interesting(0.36)]).

fof(t17_mcart_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(A,k2_zfmisc_1(k2_tarski(B,C),k1_tarski(D)))
     => ( ( k1_mcart_1(A) = B
          | k1_mcart_1(A) = C )
        & k2_mcart_1(A) = D ) ) ),
    inference(mizar_proof,[status(thm)],[t10_mcart_1,d1_tarski,d2_tarski]),
    [file(mcart_1,t17_mcart_1),interesting(0.30)]).

fof(t18_mcart_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(A,k2_zfmisc_1(k1_tarski(B),k2_tarski(C,D)))
     => ( k1_mcart_1(A) = B
        & ( k2_mcart_1(A) = C
          | k2_mcart_1(A) = D ) ) ) ),
    inference(mizar_proof,[status(thm)],[t10_mcart_1,d1_tarski,d2_tarski]),
    [file(mcart_1,t18_mcart_1),interesting(0.30)]).

fof(t19_mcart_1,theorem,(
    ! [A,B,C,D,E] : 
      ( r2_hidden(A,k2_zfmisc_1(k2_tarski(B,C),k2_tarski(D,E)))
     => ( ( k1_mcart_1(A) = B
          | k1_mcart_1(A) = C )
        & ( k2_mcart_1(A) = D
          | k2_mcart_1(A) = E ) ) ) ),
    inference(mizar_proof,[status(thm)],[t10_mcart_1,d2_tarski]),
    [file(mcart_1,t19_mcart_1),interesting(0.30)]).

fof(t60_mcart_1,theorem,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ~ ! [E] : 
            ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
           => E = k4_mcart_1(k8_mcart_1(A,B,C,D,E),k9_mcart_1(A,B,C,D,E),k10_mcart_1(A,B,C,D,E),k11_mcart_1(A,B,C,D,E)) ) ) ),
    inference(mizar_proof,[status(thm)],[l68_mcart_1,d8_mcart_1,d9_mcart_1,d10_mcart_1,d11_mcart_1]),
    [file(mcart_1,t60_mcart_1),interesting(0.26)]).

fof(t50_mcart_1,theorem,(
    ! [A,B,C] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & ~ ! [D] : 
            ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
           => ( k5_mcart_1(A,B,C,D) = k1_mcart_1(k1_mcart_1(D))
              & k6_mcart_1(A,B,C,D) = k2_mcart_1(k1_mcart_1(D))
              & k7_mcart_1(A,B,C,D) = k2_mcart_1(D) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_mcart_1,d1_mcart_1,t48_mcart_1,d2_mcart_1,d1_mcart_1,t48_mcart_1,d2_mcart_1,t48_mcart_1]),
    [file(mcart_1,t50_mcart_1),interesting(0.22)]).

fof(t47_mcart_1,theorem,(
    ! [A,B,C] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & ? [D] : 
          ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
          & ? [E,F,G] : 
              ( D = k3_mcart_1(E,F,G)
              & ~ ( k5_mcart_1(A,B,C,D) = E
                  & k6_mcart_1(A,B,C,D) = F
                  & k7_mcart_1(A,B,C,D) = G ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_mcart_1,d6_mcart_1,d7_mcart_1]),
    [file(mcart_1,t47_mcart_1),interesting(0.13)]).

fof(t68_mcart_1,theorem,(
    ! [A,B,C,D] : 
      ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
     => ~ ( A != k1_xboole_0
          & B != k1_xboole_0
          & C != k1_xboole_0
          & ? [E,F,G] : 
              ( D = k3_mcart_1(E,F,G)
              & ~ ( k5_mcart_1(A,B,C,D) = E
                  & k6_mcart_1(A,B,C,D) = F
                  & k7_mcart_1(A,B,C,D) = G ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d5_mcart_1,d6_mcart_1,d7_mcart_1]),
    [file(mcart_1,t68_mcart_1),interesting(0.13)]).

fof(t61_mcart_1,theorem,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ~ ! [E] : 
            ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
           => ( k8_mcart_1(A,B,C,D,E) = k1_mcart_1(k1_mcart_1(k1_mcart_1(E)))
              & k9_mcart_1(A,B,C,D,E) = k2_mcart_1(k1_mcart_1(k1_mcart_1(E)))
              & k10_mcart_1(A,B,C,D,E) = k2_mcart_1(k1_mcart_1(E))
              & k11_mcart_1(A,B,C,D,E) = k2_mcart_1(E) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d1_mcart_1,d1_mcart_1,d1_mcart_1,t60_mcart_1,d2_mcart_1,d1_mcart_1,d1_mcart_1,t60_mcart_1,d2_mcart_1,d1_mcart_1,t60_mcart_1,d2_mcart_1,t60_mcart_1]),
    [file(mcart_1,t61_mcart_1),interesting(0.02)]).

fof(d1_mcart_1,definition,(
    ! [A] : 
      ( ? [B,C] : A = k4_tarski(B,C)
     => ! [B] : 
          ( B = k1_mcart_1(A)
        <=> ! [C,D] : 
              ( A = k4_tarski(C,D)
             => B = C ) ) ) ),
    file(mcart_1,d1_mcart_1),
    [interesting(0.00)]).

fof(d2_mcart_1,definition,(
    ! [A] : 
      ( ? [B,C] : A = k4_tarski(B,C)
     => ! [B] : 
          ( B = k2_mcart_1(A)
        <=> ! [C,D] : 
              ( A = k4_tarski(C,D)
             => B = D ) ) ) ),
    file(mcart_1,d2_mcart_1),
    [interesting(0.00)]).

fof(d2_zfmisc_1,definition,(
    ! [A,B,C] : 
      ( C = k2_zfmisc_1(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ? [E,F] : 
              ( r2_hidden(E,A)
              & r2_hidden(F,B)
              & D = k4_tarski(E,F) ) ) ) ),
    file(zfmisc_1,d2_zfmisc_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_tarski,definition,(
    ! [A,B,C] : 
      ( C = k2_tarski(A,B)
    <=> ! [D] : 
          ( r2_hidden(D,C)
        <=> ( D = A
            | D = B ) ) ) ),
    file(tarski,d2_tarski),
    [interesting(0.00)]).

fof(t21_mcart_1,theorem,(
    $true ),
    file(mcart_1,t21_mcart_1),
    [interesting(0.00)]).

fof(t22_mcart_1,theorem,(
    $true ),
    file(mcart_1,t22_mcart_1),
    [interesting(0.00)]).

fof(t27_mcart_1,theorem,(
    $true ),
    file(mcart_1,t27_mcart_1),
    [interesting(0.00)]).

fof(t30_mcart_1,theorem,(
    $true ),
    file(mcart_1,t30_mcart_1),
    [interesting(0.00)]).

fof(t31_mcart_1,theorem,(
    ! [A,B,C,D] : k4_mcart_1(A,B,C,D) = k4_tarski(k4_tarski(k4_tarski(A,B),C),D) ),
    file(mcart_1,t31_mcart_1),
    [interesting(0.00)]).

fof(t32_mcart_1,theorem,(
    ! [A,B,C,D] : k4_mcart_1(A,B,C,D) = k3_mcart_1(k4_tarski(A,B),C,D) ),
    file(mcart_1,t32_mcart_1),
    [interesting(0.00)]).

fof(t33_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( k4_tarski(A,B) = k4_tarski(C,D)
     => ( A = C
        & B = D ) ) ),
    file(zfmisc_1,t33_zfmisc_1),
    [interesting(0.00)]).

fof(d3_zfmisc_1,definition,(
    ! [A,B,C] : k3_zfmisc_1(A,B,C) = k2_zfmisc_1(k2_zfmisc_1(A,B),C) ),
    file(zfmisc_1,d3_zfmisc_1),
    [interesting(0.00)]).

fof(t113_zfmisc_1,theorem,(
    ! [A,B] : 
      ( k2_zfmisc_1(A,B) = k1_xboole_0
    <=> ( A = k1_xboole_0
        | B = k1_xboole_0 ) ) ),
    file(zfmisc_1,t113_zfmisc_1),
    [interesting(0.00)]).

fof(t134_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( k2_zfmisc_1(A,B) = k2_zfmisc_1(C,D)
     => ( A = k1_xboole_0
        | B = k1_xboole_0
        | ( A = C
          & B = D ) ) ) ),
    file(zfmisc_1,t134_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(t15_xboole_1,theorem,(
    ! [A,B] : 
      ( k2_xboole_0(A,B) = k1_xboole_0
     => A = k1_xboole_0 ) ),
    file(xboole_1,t15_xboole_1),
    [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(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(t4_xboole_1,theorem,(
    ! [A,B,C] : k2_xboole_0(k2_xboole_0(A,B),C) = k2_xboole_0(A,k2_xboole_0(B,C)) ),
    file(xboole_1,t4_xboole_1),
    [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(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(t70_xboole_1,theorem,(
    ! [A,B,C] : 
      ( ~ ( ~ r1_xboole_0(A,k2_xboole_0(B,C))
          & r1_xboole_0(A,B)
          & r1_xboole_0(A,C) )
      & ~ ( ~ ( r1_xboole_0(A,B)
              & r1_xboole_0(A,C) )
          & r1_xboole_0(A,k2_xboole_0(B,C)) ) ) ),
    file(xboole_1,t70_xboole_1),
    [interesting(0.00)]).

fof(t36_zfmisc_1,theorem,(
    ! [A,B,C] : 
      ( k2_zfmisc_1(k1_tarski(A),k2_tarski(B,C)) = k2_tarski(k4_tarski(A,B),k4_tarski(A,C))
      & k2_zfmisc_1(k2_tarski(A,B),k1_tarski(C)) = k2_tarski(k4_tarski(A,C),k4_tarski(B,C)) ) ),
    file(zfmisc_1,t36_zfmisc_1),
    [interesting(0.00)]).

fof(t35_zfmisc_1,theorem,(
    ! [A,B] : k2_zfmisc_1(k1_tarski(A),k1_tarski(B)) = k1_tarski(k4_tarski(A,B)) ),
    file(zfmisc_1,t35_zfmisc_1),
    [interesting(0.00)]).

fof(t132_zfmisc_1,theorem,(
    ! [A,B,C] : 
      ( k2_zfmisc_1(k2_tarski(A,B),C) = k2_xboole_0(k2_zfmisc_1(k1_tarski(A),C),k2_zfmisc_1(k1_tarski(B),C))
      & k2_zfmisc_1(C,k2_tarski(A,B)) = k2_xboole_0(k2_zfmisc_1(C,k1_tarski(A)),k2_zfmisc_1(C,k1_tarski(B))) ) ),
    file(zfmisc_1,t132_zfmisc_1),
    [interesting(0.00)]).

fof(t45_enumset1,theorem,(
    ! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_xboole_0(k2_tarski(A,B),k2_tarski(C,D)) ),
    file(enumset1,t45_enumset1),
    [interesting(0.00)]).

fof(t120_zfmisc_1,theorem,(
    ! [A,B,C] : 
      ( k2_zfmisc_1(k2_xboole_0(A,B),C) = k2_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,C))
      & k2_zfmisc_1(C,k2_xboole_0(A,B)) = k2_xboole_0(k2_zfmisc_1(C,A),k2_zfmisc_1(C,B)) ) ),
    file(zfmisc_1,t120_zfmisc_1),
    [interesting(0.00)]).

fof(t104_enumset1,theorem,(
    ! [A,B,C,D] : k2_enumset1(A,B,C,D) = k2_enumset1(A,C,B,D) ),
    file(enumset1,t104_enumset1),
    [interesting(0.00)]).

fof(t65_enumset1,theorem,(
    ! [A,B,C,D,E,F,G,H] : k6_enumset1(A,B,C,D,E,F,G,H) = k2_xboole_0(k2_enumset1(A,B,C,D),k2_enumset1(E,F,G,H)) ),
    file(enumset1,t65_enumset1),
    [interesting(0.00)]).

fof(d5_mcart_1,definition,(
    ! [A,B,C] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & ~ ! [D] : 
            ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
           => ! [E] : 
                ( m1_subset_1(E,A)
               => ( E = k5_mcart_1(A,B,C,D)
                <=> ! [F,G,H] : 
                      ( D = k3_mcart_1(F,G,H)
                     => E = F ) ) ) ) ) ),
    file(mcart_1,d5_mcart_1),
    [interesting(0.00)]).

fof(d6_mcart_1,definition,(
    ! [A,B,C] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & ~ ! [D] : 
            ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
           => ! [E] : 
                ( m1_subset_1(E,B)
               => ( E = k6_mcart_1(A,B,C,D)
                <=> ! [F,G,H] : 
                      ( D = k3_mcart_1(F,G,H)
                     => E = G ) ) ) ) ) ),
    file(mcart_1,d6_mcart_1),
    [interesting(0.00)]).

fof(d7_mcart_1,definition,(
    ! [A,B,C] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & ~ ! [D] : 
            ( m1_subset_1(D,k3_zfmisc_1(A,B,C))
           => ! [E] : 
                ( m1_subset_1(E,C)
               => ( E = k7_mcart_1(A,B,C,D)
                <=> ! [F,G,H] : 
                      ( D = k3_mcart_1(F,G,H)
                     => E = H ) ) ) ) ) ),
    file(mcart_1,d7_mcart_1),
    [interesting(0.00)]).

fof(t127_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( ( r1_xboole_0(A,B)
        | r1_xboole_0(C,D) )
     => r1_xboole_0(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ),
    file(zfmisc_1,t127_zfmisc_1),
    [interesting(0.00)]).

fof(d4_zfmisc_1,definition,(
    ! [A,B,C,D] : k4_zfmisc_1(A,B,C,D) = k2_zfmisc_1(k3_zfmisc_1(A,B,C),D) ),
    file(zfmisc_1,d4_zfmisc_1),
    [interesting(0.00)]).

fof(t56_mcart_1,theorem,(
    ! [A,B,C,D,E,F,G,H] : 
      ( k4_zfmisc_1(A,B,C,D) = k4_zfmisc_1(E,F,G,H)
     => ( A = k1_xboole_0
        | B = k1_xboole_0
        | C = k1_xboole_0
        | D = k1_xboole_0
        | ( A = E
          & B = F
          & C = G
          & D = H ) ) ) ),
    inference(mizar_proof,[status(thm)],[d4_zfmisc_1,t35_mcart_1,d4_zfmisc_1,t134_zfmisc_1,t36_mcart_1]),
    [file(mcart_1,t56_mcart_1),interesting(0.00)]).

fof(d8_mcart_1,definition,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ~ ! [E] : 
            ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
           => ! [F] : 
                ( m1_subset_1(F,A)
               => ( F = k8_mcart_1(A,B,C,D,E)
                <=> ! [G,H,I,J] : 
                      ( E = k4_mcart_1(G,H,I,J)
                     => F = G ) ) ) ) ) ),
    file(mcart_1,d8_mcart_1),
    [interesting(0.00)]).

fof(d9_mcart_1,definition,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ~ ! [E] : 
            ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
           => ! [F] : 
                ( m1_subset_1(F,B)
               => ( F = k9_mcart_1(A,B,C,D,E)
                <=> ! [G,H,I,J] : 
                      ( E = k4_mcart_1(G,H,I,J)
                     => F = H ) ) ) ) ) ),
    file(mcart_1,d9_mcart_1),
    [interesting(0.00)]).

fof(d10_mcart_1,definition,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ~ ! [E] : 
            ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
           => ! [F] : 
                ( m1_subset_1(F,C)
               => ( F = k10_mcart_1(A,B,C,D,E)
                <=> ! [G,H,I,J] : 
                      ( E = k4_mcart_1(G,H,I,J)
                     => F = I ) ) ) ) ) ),
    file(mcart_1,d10_mcart_1),
    [interesting(0.00)]).

fof(d11_mcart_1,definition,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ~ ! [E] : 
            ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
           => ! [F] : 
                ( m1_subset_1(F,D)
               => ( F = k11_mcart_1(A,B,C,D,E)
                <=> ! [G,H,I,J] : 
                      ( E = k4_mcart_1(G,H,I,J)
                     => F = J ) ) ) ) ) ),
    file(mcart_1,d11_mcart_1),
    [interesting(0.00)]).

fof(t59_mcart_1,theorem,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ? [E] : 
          ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
          & ? [F,G,H,I] : 
              ( E = k4_mcart_1(F,G,H,I)
              & ~ ( k8_mcart_1(A,B,C,D,E) = F
                  & k9_mcart_1(A,B,C,D,E) = G
                  & k10_mcart_1(A,B,C,D,E) = H
                  & k11_mcart_1(A,B,C,D,E) = I ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_mcart_1,d9_mcart_1,d10_mcart_1,d11_mcart_1]),
    [file(mcart_1,t59_mcart_1),interesting(0.00)]).

fof(t102_zfmisc_1,theorem,(
    ! [A,B,C] : ~ ( r2_hidden(A,k2_zfmisc_1(B,C))
      & ! [D,E] : k4_tarski(D,E) != A ) ),
    file(zfmisc_1,t102_zfmisc_1),
    [interesting(0.00)]).

fof(l68_mcart_1,theorem,(
    ! [A,B,C,D] : ~ ( A != k1_xboole_0
      & B != k1_xboole_0
      & C != k1_xboole_0
      & D != k1_xboole_0
      & ? [E] : 
          ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
          & ! [F] : 
              ( m1_subset_1(F,A)
             => ! [G] : 
                  ( m1_subset_1(G,B)
                 => ! [H] : 
                      ( m1_subset_1(H,C)
                     => ! [I] : 
                          ( m1_subset_1(I,D)
                         => E != k4_mcart_1(F,G,H,I) ) ) ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t35_mcart_1,d4_zfmisc_1,l27_mcart_1,l44_mcart_1]),
    [file(mcart_1,l68_mcart_1),interesting(0.00)]).

fof(d5_tarski,definition,(
    ! [A,B] : k4_tarski(A,B) = k2_tarski(k2_tarski(A,B),k1_tarski(A)) ),
    file(tarski,d5_tarski),
    [interesting(0.00)]).

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) ) ),
    file(ordinal1,t6_ordinal1),
    [interesting(0.00)]).

fof(t4_ordinal1,theorem,(
    ! [A,B,C,D] : ~ ( r2_hidden(A,B)
      & r2_hidden(B,C)
      & r2_hidden(C,D)
      & r2_hidden(D,A) ) ),
    file(ordinal1,t4_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(t135_zfmisc_1,theorem,(
    ! [A,B] : 
      ( ( r1_tarski(A,k2_zfmisc_1(A,B))
        | r1_tarski(A,k2_zfmisc_1(B,A)) )
     => A = k1_xboole_0 ) ),
    file(zfmisc_1,t135_zfmisc_1),
    [interesting(0.00)]).

fof(t69_mcart_1,theorem,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k3_zfmisc_1(A,B,C))
     => ( ! [F] : 
            ( m1_subset_1(F,A)
           => ! [G] : 
                ( m1_subset_1(G,B)
               => ! [H] : 
                    ( m1_subset_1(H,C)
                   => ( E = k3_mcart_1(F,G,H)
                     => D = F ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k5_mcart_1(A,B,C,E) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t48_mcart_1]),
    [file(mcart_1,t69_mcart_1),interesting(0.00)]).

fof(t70_mcart_1,theorem,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k3_zfmisc_1(A,B,C))
     => ( ! [F] : 
            ( m1_subset_1(F,A)
           => ! [G] : 
                ( m1_subset_1(G,B)
               => ! [H] : 
                    ( m1_subset_1(H,C)
                   => ( E = k3_mcart_1(F,G,H)
                     => D = G ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k6_mcart_1(A,B,C,E) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t48_mcart_1]),
    [file(mcart_1,t70_mcart_1),interesting(0.00)]).

fof(t71_mcart_1,theorem,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k3_zfmisc_1(A,B,C))
     => ( ! [F] : 
            ( m1_subset_1(F,A)
           => ! [G] : 
                ( m1_subset_1(G,B)
               => ! [H] : 
                    ( m1_subset_1(H,C)
                   => ( E = k3_mcart_1(F,G,H)
                     => D = H ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k7_mcart_1(A,B,C,E) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t48_mcart_1]),
    [file(mcart_1,t71_mcart_1),interesting(0.00)]).

fof(t2_tarski,theorem,(
    ! [A,B] : 
      ( ! [C] : 
          ( r2_hidden(C,A)
        <=> r2_hidden(C,B) )
     => A = B ) ),
    file(tarski,t2_tarski),
    [interesting(0.00)]).

fof(t78_mcart_1,theorem,(
    ! [A,B,C,D,E] : 
      ( m1_subset_1(E,k4_zfmisc_1(A,B,C,D))
     => ~ ( A != k1_xboole_0
          & B != k1_xboole_0
          & C != k1_xboole_0
          & D != k1_xboole_0
          & ? [F,G,H,I] : 
              ( E = k4_mcart_1(F,G,H,I)
              & ~ ( k8_mcart_1(A,B,C,D,E) = F
                  & k9_mcart_1(A,B,C,D,E) = G
                  & k10_mcart_1(A,B,C,D,E) = H
                  & k11_mcart_1(A,B,C,D,E) = I ) ) ) ) ),
    inference(mizar_proof,[status(thm)],[d8_mcart_1,d9_mcart_1,d10_mcart_1,d11_mcart_1]),
    [file(mcart_1,t78_mcart_1),interesting(0.00)]).

fof(t79_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( m1_subset_1(F,k4_zfmisc_1(A,B,C,D))
     => ( ! [G] : 
            ( m1_subset_1(G,A)
           => ! [H] : 
                ( m1_subset_1(H,B)
               => ! [I] : 
                    ( m1_subset_1(I,C)
                   => ! [J] : 
                        ( m1_subset_1(J,D)
                       => ( F = k4_mcart_1(G,H,I,J)
                         => E = G ) ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k1_xboole_0
          | E = k8_mcart_1(A,B,C,D,F) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t60_mcart_1]),
    [file(mcart_1,t79_mcart_1),interesting(0.00)]).

fof(t80_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( m1_subset_1(F,k4_zfmisc_1(A,B,C,D))
     => ( ! [G] : 
            ( m1_subset_1(G,A)
           => ! [H] : 
                ( m1_subset_1(H,B)
               => ! [I] : 
                    ( m1_subset_1(I,C)
                   => ! [J] : 
                        ( m1_subset_1(J,D)
                       => ( F = k4_mcart_1(G,H,I,J)
                         => E = H ) ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k1_xboole_0
          | E = k9_mcart_1(A,B,C,D,F) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t60_mcart_1]),
    [file(mcart_1,t80_mcart_1),interesting(0.00)]).

fof(t81_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( m1_subset_1(F,k4_zfmisc_1(A,B,C,D))
     => ( ! [G] : 
            ( m1_subset_1(G,A)
           => ! [H] : 
                ( m1_subset_1(H,B)
               => ! [I] : 
                    ( m1_subset_1(I,C)
                   => ! [J] : 
                        ( m1_subset_1(J,D)
                       => ( F = k4_mcart_1(G,H,I,J)
                         => E = I ) ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k1_xboole_0
          | E = k10_mcart_1(A,B,C,D,F) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t60_mcart_1]),
    [file(mcart_1,t81_mcart_1),interesting(0.00)]).

fof(t82_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( m1_subset_1(F,k4_zfmisc_1(A,B,C,D))
     => ( ! [G] : 
            ( m1_subset_1(G,A)
           => ! [H] : 
                ( m1_subset_1(H,B)
               => ! [I] : 
                    ( m1_subset_1(I,C)
                   => ! [J] : 
                        ( m1_subset_1(J,D)
                       => ( F = k4_mcart_1(G,H,I,J)
                         => E = J ) ) ) ) )
       => ( A = k1_xboole_0
          | B = k1_xboole_0
          | C = k1_xboole_0
          | D = k1_xboole_0
          | E = k11_mcart_1(A,B,C,D,F) ) ) ) ),
    inference(mizar_proof,[status(thm)],[t60_mcart_1]),
    [file(mcart_1,t82_mcart_1),interesting(0.00)]).

fof(t106_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( r2_hidden(k4_tarski(A,B),k2_zfmisc_1(C,D))
    <=> ( r2_hidden(A,C)
        & r2_hidden(B,D) ) ) ),
    file(zfmisc_1,t106_zfmisc_1),
    [interesting(0.00)]).

fof(t119_zfmisc_1,theorem,(
    ! [A,B,C,D] : 
      ( ( r1_tarski(A,B)
        & r1_tarski(C,D) )
     => r1_tarski(k2_zfmisc_1(A,C),k2_zfmisc_1(B,D)) ) ),
    file(zfmisc_1,t119_zfmisc_1),
    [interesting(0.00)]).

fof(t89_mcart_1,theorem,(
    ! [A,B,C,D,E,F] : 
      ( k17_mcart_1(k4_tarski(k4_tarski(A,B),C)) = A
      & k18_mcart_1(k4_tarski(k4_tarski(A,B),C)) = B
      & k19_mcart_1(k4_tarski(F,k4_tarski(D,E))) = D
      & k20_mcart_1(k4_tarski(F,k4_tarski(D,E))) = E ) ),
    inference(mizar_proof,[status(thm)],[t7_mcart_1]),
    [file(mcart_1,t89_mcart_1),interesting(0.00)]).