reserve i for object, I for set,
  f for Function,
  x, x1, x2, y, A, B, X, Y, Z for ManySortedSet of I;

theorem     :: ZFMISC_1:68
  {x} (\) X = EmptyMS I iff x in X
proof
  thus {x} (\) X = EmptyMS I implies x in X
  proof
    assume
A1: {x} (\) X = EmptyMS I;
    let i;
    assume
A2: i in I;
    then {x.i} \ X.i = {x}.i \ X.i by Def1
      .= ({x} (\) X).i by A2,PBOOLE:def 6
      .= {} by A1,PBOOLE:5;
    hence thesis by ZFMISC_1:60;
  end;
  assume
A3: x in X;
  now
    let i be object;
    assume
A4: i in I;
    then
A5: x.i in X.i by A3;
    thus ({x} (\) X).i = {x}.i \ X.i by A4,PBOOLE:def 6
      .= {x.i} \ X.i by A4,Def1
      .= {} by A5,ZFMISC_1:60
      .= EmptyMS I.i by PBOOLE:5;
  end;
  hence thesis;
end;
