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;
reserve n for Element of omega;

theorem Th69:
  the_rank_of X c= A iff for Y st Y in X holds the_rank_of Y in A
proof
  set R = the_rank_of X;
A1: X c= Rank R by Def9;
  thus
  the_rank_of X c= A implies for Y st Y in X holds the_rank_of Y in A
  by A1,Th66;
  assume
A2: for Y st Y in X holds the_rank_of Y in A;
 X c= Rank A
  proof
    let x be object;
            reconsider xx=x as set by TARSKI:1;
    assume x in X;
then  the_rank_of xx in A by A2;
    hence thesis by Th66;
  end;
  hence thesis by Def9;
end;
