reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;
reserve f,g for Function,
  L for Sequence,
  F for Cardinal-Function;

theorem Th30:
  W is Tarski & W is epsilon-transitive implies W c= Rank card W
proof
  assume that
A1: W is Tarski and
A2: W is epsilon-transitive;
  let x be object;
            reconsider xx=x as set by TARSKI:1;
  assume x in W;
  then the_rank_of xx in W by A1,A2,Th29;
  then
A3: the_rank_of xx in On W by ORDINAL1:def 9;
  On W = card W by A1,Th9;
  then
A4: Rank the_rank_of xx in Rank card W by A3,CLASSES1:36;
  xx c= Rank the_rank_of xx by CLASSES1:def 9;
  hence thesis by A4,CLASSES1:41;
end;
