When I took a math foundations class we only did naive set theory and took as an axiom that the empty set is a member of every set. The book had formal set theory and thus listed the ZFC axioms. One of them was that the empty set exists and that it was a member of every set. I've looked at a couple of listing of the axioms on the net and they only give the existence of the empty set. To prepare for graduate analysis (I start grad school in a week), I've been reading through Rudin's Principles of Mathematical Analysis. I noticed today that an exercise in the book is to prove that the empty set is a member of every set. I'm not asking for the solution, but I was wondering how one would go about proving this. Whenever you prove subsets, you chase elements. But the empty set has no elements. Are there other methods of proving subsets?