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 Th70:
  A c= the_rank_of X iff for B st B in A ex Y st Y in X & B c= the_rank_of Y
proof
  thus A c= the_rank_of X implies
  for B st B in A ex Y st Y in X & B c= the_rank_of Y
  proof
    assume
A1: A c= the_rank_of X;
    let B;
    assume B in A;
then  not the_rank_of X c= B by A1,ORDINAL1:5;
then  not X c= Rank B by Def9;
then A2: X \ Rank B <> {} by XBOOLE_1:37;
    set x = the Element of X \ Rank B;
    take x;
A3: not x in Rank B by A2,XBOOLE_0:def 5;
    thus x in X by A2,XBOOLE_0:def 5;
    thus thesis by ORDINAL1:16,A3,Th66;
  end;
  assume
A4: for B st B in A ex Y st Y in X & B c= the_rank_of Y;
  let x be object;
  assume
A5: x in A;
  then reconsider x as Ordinal;
  consider Y such that
A6: Y in X and
A7: x c= the_rank_of Y by A4,A5;
 the_rank_of Y in the_rank_of X by A6,Th68;
  hence thesis by A7,ORDINAL1:12;
end;
