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_1:130
  X is non-empty implies [|{x},X|] is non-empty & [|X,{x}|] is non-empty
proof
  assume
A1: X is non-empty;
  thus [|{x},X|] is non-empty
  proof
    let i be object;
    assume
A2: i in I;
    then X.i is non empty by A1;
    then [:{x.i},X.i:] is non empty by ZFMISC_1:107;
    then [:{x}.i,X.i:] is non empty by A2,Def1;
    hence thesis by A2,PBOOLE:def 16;
  end;
  let i be object;
  assume
A3: i in I;
  then X.i is non empty by A1;
  then [:X.i,{x.i}:] is non empty by ZFMISC_1:107;
  then [:X.i,{x}.i:] is non empty by A3,Def1;
  hence thesis by A3,PBOOLE:def 16;
end;
