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
  Y <> X & Y in Tarski-Class X implies
  ex A st not Y in Tarski-Class(X,A) & Y in Tarski-Class(X,succ A)
proof
  assume that
A1: Y <> X and
A2: Y in Tarski-Class X;
  defpred P[Ordinal] means Y in Tarski-Class(X,$1);
 ex A st Tarski-Class(X,A) = Tarski-Class X by Th19;
then A3: ex A st P[A] by A2;
  consider A such that
A4: P[A] & for B st P[B] holds A c= B from ORDINAL1:sch 1(A3);
A5: not Y in { X } by A1,TARSKI:def 1;
A6: Tarski-Class(X,{}) = { X } by Lm1;
 now
    assume A is limit_ordinal;
then  ex B st B in A & Y in Tarski-Class(X,B) by A4,A5,A6,Th13;
    hence contradiction by A4,ORDINAL1:5;
  end;
  then consider B such that
A7: A = succ B by ORDINAL1:29;
  take B;
 not A c= B by A7,ORDINAL1:5,6;
  hence thesis by A4,A7;
end;
