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

theorem Th1:
  W is subset-closed & X in W implies not X,W are_equipotent & card X in card W
proof
  assume
A1: W is subset-closed;
  assume
A2: X in W;
  bool X c= W
  by A1,A2;
  then
A3: card bool X c= card W by CARD_1:11;
A4: card X in card bool X by CARD_1:14;
  then card X in card W by A3;
  then card X <> card W;
  hence thesis by A4,A3,CARD_1:5;
end;
