reserve E for non empty set;
reserve a for Element of E;
reserve A, B for Subset of E;
reserve Y for set;
reserve p for FinSequence;

theorem Th1:
  for e being non empty Subset of E holds e is Singleton of E iff for Y
  holds (Y c= e iff Y = {} or Y = e)
proof
  let e be non empty Subset of E;
  thus e is Singleton of E implies for Y holds (Y c= e iff Y = {} or Y = e)
  proof
    assume
A1: e is Singleton of E;
    let Y;
    ex x being object st e = {x} by A1,ZFMISC_1:131;
    hence thesis by ZFMISC_1:33;
  end;
  assume
A2: for Y holds Y c= e iff Y = {} or Y = e;
  consider x being object such that
A3: x in e by XBOOLE_0:def 1;
  {x} c= e by A3,ZFMISC_1:31;
  hence thesis by A2;
end;
