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 Th42:
  W is Tarski & X is epsilon-transitive & X in W implies X in Rank card W
proof
  assume
A1: W is Tarski;
  assume
A2: X is epsilon-transitive;
  assume X in W;
  then card X in card W by A1,Th1;
  then
A3: card the_rank_of X in card W by A2,Th41,ORDINAL1:12;
  card card W = card W;
  then the_rank_of X in card W by A3,CARD_3:43;
  then
A4: Rank the_rank_of X in Rank card W by CLASSES1:36;
  X c= Rank the_rank_of X by CLASSES1:def 9;
  hence thesis by A4,CLASSES1:41;
end;
