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

theorem Th5:
  W is Tarski & A in W implies succ A in W & A c= W
proof
  assume that
A1: for X,Y st X in W & Y c= X holds Y in W and
A2: for X st X in W holds bool X in W and
  for X st X c= W holds X,W are_equipotent or X in W and
A3: A in W;
  bool A in W by A2,A3;
  hence succ A in W by A1,ORDINAL2:3;
  let x be object;
  assume
A4: x in A;
  then reconsider B = x as Ordinal;
  B c= A by A4,ORDINAL1:def 2;
  hence thesis by A1,A3;
end;
