reserve E,X,Y,x for set;
reserve A,B,C for Subset of E;
reserve x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 for Element of X;

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;
