empty set?

I doubt you have used it correctly becuase you appear to have several lines of proof. It is not a definition that the empty set is a subset of everything, it is a (vacuously) true deduction (in any reasonable logic). If it weren't a subset of X, say, then there is an e in E (empty set) that is not in X. But that is false - there is no e in E since E is empty. I don't see that there is anything to prove at all. The complement of U is the set of u in U that are not in U - this is vacuously empty, and it is vacuously (and I am using the word deliberately, pun intended) a subset of U as well.