theorem
  X <> {} implies {x1} is Subset of X
proof
  assume X <> {};
  then
A1: x1 in X by Def1;
  {x1} c= X by A1,TARSKI:def 1;
  then {x1} in bool X by ZFMISC_1:def 1;
  hence thesis by Def1;
end;
