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_1:72
  I is non empty & {x,y} (\) X = {x,y} implies not x in X & not y in X
proof
  assume I is non empty;
  then consider i being object such that
A1: i in I by XBOOLE_0:def 1;
    assume
A2: {x,y} (\) X = {x,y};
    thus not x in X
    proof
      assume
A3:   x in X;
      {x.i,y.i} \ X.i = {x,y}.i \ X.i by A1,Def2
        .= ({x,y} (\) X).i by A1,PBOOLE:def 6
        .= {x.i,y.i} by A1,A2,Def2;
      then not x.i in X.i by ZFMISC_1:63;
      hence contradiction by A1,A3;
    end;
    assume
A4: y in X;
    {x.i,y.i} \ X.i = {x,y}.i \ X.i by A1,Def2
      .= ({x,y} (\) X).i by A1,PBOOLE:def 6
      .= {x.i,y.i} by A1,A2,Def2;
    then not y.i in X.i by ZFMISC_1:63;
    hence contradiction by A1,A4;
end;
