Def: An alphabet $\Sigma$ is (finite or infinite) set of symbols $s_i$.
The symbols $s_i$ are not sets themselves, since a symbol cannot contain any elements. Thus, an alphabet $\Sigma$ cannot be a set in terms of ZFC, since all entities in the universe of discourse are sets.
Am I right making this assertion?
An idea: No, the assertion is not true, since only the special case, in which we choose the symbols not to contain any other symbols is not a set (e.g.: Latin alphabet, Binary, ...).
Thank you in advance.
In ZFC, any variable represents a set, and $\in$ may be put between any two variables (the resulting statement may be true or false, but it makes sense). When building higher theories upon ZFC, we are free to "forget" that some sets are indeed sets (also, which sets they were before you forgot depends entirely on how you model your theory), and using $\in$ is thus more restricted, so that statements like $s_1\in s_2$ in your case or $3\in 4$ in number theory are not only false, but they don't even make sense. As ZFC statements they make sense (and may even be true).
Technically, it's not even the same $\in$. One comes from ZFC and has its behaviour strictly defined there, while the other comes from what is probably some implicitly defined naive set theory that helps you argue about the higher theory.
So when a mathematician is doing, for instance, number theory and arguing about prime numbers and divisibility, he doesn't consciously do that on the ZFC level. He may know that it could be embedded in ZFC if he so wished, and thus it stands on as firm a logical ground as anything we do in mathematics. But once that's cleared up and out of the way, he forgets about whether $4$ is really the same thing as $\{0,1,2,3\}$, and just uses the axioms and developed theory of the higher field. As such, it is not really built on ZFC in the sense that a given number is really a concrete set. Rather it could be if we wanted it to, and that's enough for most of us.
The above paragraph should apply to all higher theories of mathematics, including formal languages.