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
  ex A st Tarski-Class(X,A) = Tarski-Class X &
  for B st B in A holds Tarski-Class(X,B) <> Tarski-Class X
proof
  defpred P[Ordinal] means Tarski-Class(X,$1) = Tarski-Class X;
A1: ex A st P[A] by Th19;
  consider A such that
A2: P[A] & for B st P[B] holds A c= B from ORDINAL1:sch 1(A1);
  take A;
  thus Tarski-Class(X,A) = Tarski-Class X by A2;
  let B;
  assume B in A;
  hence thesis by A2,ORDINAL1:5;
end;
