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
  the_rank_of X = {} iff X = {}
proof
  thus the_rank_of X = {} implies X = {}
  proof
    assume the_rank_of X = {};
then  X c= Rank {} by Def9;
    hence thesis by Lm2;
  end;
  assume X = {};
then  for Y st Y in X holds the_rank_of Y in {};
  hence the_rank_of X c= {} by Th69;
  thus {} c= the_rank_of X;
end;
