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_1:73
  {x,y} (\) X = EmptyMS I iff x in X & y in X
proof
  thus {x,y} (\) X = EmptyMS I implies x in X & y in X
  proof
    assume
A1: {x,y} (\) X = EmptyMS I;
    thus x in X
    proof
      let i;
      assume
A2:   i in I;
      then {x.i,y.i} \ X.i = {x,y}.i \ X.i by Def2
        .= ({x,y} (\) X).i by A2,PBOOLE:def 6
        .= {} by A1,PBOOLE:5;
      hence thesis by ZFMISC_1:64;
    end;
    let i;
    assume
A3: i in I;
    then {x.i,y.i} \ X.i = {x,y}.i \ X.i by Def2
      .= ({x,y} (\) X).i by A3,PBOOLE:def 6
      .= {} by A1,PBOOLE:5;
    hence thesis by ZFMISC_1:64;
  end;
  assume that
A4: x in X and
A5: y in X;
  now
    let i be object;
    assume
A6: i in I;
    then
A7: x.i in X.i by A4;
A8: y.i in X.i by A5,A6;
    thus ({x,y} (\) X).i = {x,y}.i \ X.i by A6,PBOOLE:def 6
      .= {x.i,y.i} \ X.i by A6,Def2
      .= {} by A7,A8,ZFMISC_1:64
      .= EmptyMS I.i by PBOOLE:5;
  end;
  hence thesis;
end;
