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
  X c= Rank A iff bool X c= Rank succ A
proof
  thus X c= Rank A implies bool X c= Rank succ A
  proof
    assume
A1: X c= Rank A;
    let x be object;
            reconsider xx=x as set by TARSKI:1;
    assume x in bool X;
then A2: xx c= Rank A by A1;
 Rank succ A = bool Rank A by Lm2;
    hence thesis by A2;
  end;
  assume
A3: bool X c= Rank succ A;
  let x be object;
  assume x in X;
then  { x } c= X by ZFMISC_1:31;
then A4: { x } in bool X;
 Rank succ A = bool Rank A & x in { x } by Lm2,TARSKI:def 1;
  hence thesis by A3,A4;
end;
