Here is what I did:
1.
2.
3.
4.
EDIT: S = the universe.
I just want to verify that this works, particularly the third step.
Indeed, the empty set's members have EVERY property, even mutually exclusive ones (for example, it is a subset of both of two disjoint sets, and lies in the intersection of even and odd integers, for example). Indeed, the essence of a set lies in *distinction* of otherwise formless form: as soon as we say: "x is in S", we have the notion of "inside" (membership) and "outside" (exclusion), a duality which in classical logic is borne by the idea of "not" (complementation induces disjoint union). This idea is turned into a (somewhat) elegant formal propositional arithmetic in a charming book "The Laws Of Form" (whose author escapes me at the moment).
In other words, "somethingness" is something LESS than "nothingness", for when we say it is this or that, we are diminishing the limitless possibility of it being "anything" (even an impossible thing like triple-headed pink flying elephants). I believe this seeming paradox lies at the heart of all linguistic systems (including, for example, English in everyday discourse), where what we say is always somewhat less than what actually is, we filter to give meaning (or context) (this is the purpose of the "universal set of discourse", S: to "set the stage for future reference"...it is a bit distressing that an "all-encompassing context" (at least for "enough" mathematics) does not, properly speaking, exist, but it's not a perfect world, is it?).
On topic, I agree with romsek, there is no need to introduce a second dummy index, the original alpha will serve throughout:
where the second implication is because (the first and third implications are actually "two-way" (iff) although that is not necessary to note).
To a certain extent, this is a minor quibble, but the point is: in mathematics, one often strives for CLARITY, so keeping "variables" to a minimum helps avoid confusion.