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 Tarski-Class X <> {} &
  the_rank_of Tarski-Class X is limit_ordinal
proof
A1: Tarski-Class X c= Rank the_rank_of Tarski-Class X by Def9;
  thus the_rank_of Tarski-Class X <> {}
  proof
    assume the_rank_of Tarski-Class X = {};
then  Tarski-Class X c= {} by Def9,Lm2;
    hence contradiction;
  end;
  assume not thesis;
  then consider A such that
A2: the_rank_of Tarski-Class X = succ A by ORDINAL1:29;
  consider Y such that
A3: Y in Tarski-Class X and
A4: the_rank_of Y = A by A2,Th72;
A5: bool Y in Tarski-Class X by A3,Th4;
A6: the_rank_of bool Y = succ A by A4,Th63;
 bool Y c= Rank A by A1,A2,A5,Th32;
then  succ A c= A by A6,Def9;
then  A in A by ORDINAL1:21;
  hence contradiction;
end;
