
theorem
  for X being set holds singletons X misses 2Set X
proof
  let X be set;
  assume singletons X meets 2Set X;
  then consider x being object such that
    A1: x in (singletons X) /\ 2Set X by XBOOLE_0:4;
  A2: x in singletons X & x in 2Set X by A1, XBOOLE_0:def 4;
  then consider Y being Subset of X such that
    A3: x = Y & Y is 1-element;
  consider Z being Subset of X such that
    A4: x = Z & card Z = 2 by A2;
  thus contradiction by A3, A4, CARD_1:def 7;
end;
