reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

theorem Th17:
  ex A st Tarski-Class(X,A) = Tarski-Class(X,succ A)
proof
  assume
A1: for A holds Tarski-Class(X,A) <> Tarski-Class(X,succ A);
  defpred P[object] means ex A st $1 in Tarski-Class(X,A);
  consider Z such that
A2: for x being object holds x in Z iff x in Tarski-Class X & P[x]
from XBOOLE_0:sch 1;
  defpred P[object,object] means
  ex A st $2 = A & $1 in Tarski-Class(X,succ A) & not $1 in Tarski-Class(X,A);
A3: for x,y,z being object st P[x,y] & P[x,z] holds y = z
  proof
    let x,y,z be object;
    given A such that
A4: y = A and
A5: x in Tarski-Class(X,succ A) and
A6: not x in Tarski-Class(X,A);
    given B such that
A7: z = B and
A8: x in Tarski-Class(X,succ B) and
A9: not x in Tarski-Class(X,B);
    assume
A10: y <> z;
 A c= B or B c= A;
then A11: A c< B or B c< A by A4,A7,A10;
 now
      assume A c< B;
then   succ A c= B by ORDINAL1:11,21;
then   Tarski-Class(X,succ A) c= Tarski-Class(X,B) by Th16;
      hence contradiction by A5,A9;
    end;
then  succ B c= A by ORDINAL1:11,21,A11;
then  Tarski-Class(X,succ B) c= Tarski-Class(X,A) by Th16;
    hence contradiction by A6,A8;
  end;
  consider Y such that
A12: for x being object holds x in Y iff
  ex y being object st y in Z & P[y,x] from TARSKI:sch 1(A3);
 now
    let A;
A13: Tarski-Class(X,A) c= Tarski-Class(X,succ A) by Th15;
    consider x being object such that
A14: not (x in Tarski-Class(X,A) iff x in Tarski-Class(X,succ A)) by A1,
TARSKI:2;
 x in Z by A2,A14;
    hence A in Y by A12,A13,A14;
  end;
  hence contradiction by ORDINAL1:26;
end;
