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_1:67
  I is non empty & {x} (\) X = {x} implies not x in X
proof
  assume that
A1: I is non empty and
A2: {x} (\) X = {x} and
A3: x in X;
  consider i being object such that
A4: i in I by A1,XBOOLE_0:def 1;
  {x.i} \ X.i = {x}.i \ X.i by A4,Def1
    .= ({x} (\) X).i by A4,PBOOLE:def 6
    .= {x.i} by A2,A4,Def1;
  then not x.i in X.i by ZFMISC_1:59;
  hence contradiction by A3,A4;
end;
