I know that in set theory, $\forall A:\emptyset \subseteq A$
My question is, does this apply to formal languages? In my mind, formal languages are just a set of strings that are over some set of letters, which can be looked at as a set of strings with size 1. If I look at it strictly from the formal language side, all non-empty alphabets cannot have an "empty" letter and there would not? be an empty set in every language. Note I am not looking at the set containing only the empty string, which I understand is not in every language.
However, would a predicate $P(w)=w\in AB \iff \exists a:\exists b:a \in A \land b \in B \land ab=w$ that tests the membership of string $w$ in the concatenation of two finite (non empty) formal languages ($A,B$) always be true if there is no string $w$ at all ergo $w$ is the empty set? I think but am not sure if it is just vacuously true that if there is no string at all, $P$ would be true by the fact that $\forall A:\emptyset \subseteq A$ is vacuously true, so for any set product $\forall A:\forall B:\emptyset \subseteq AB$ would also be vacuously true
Perhaps my confusion is due to misunderstanding the empty set with regards to language theory.
Languages don't contain sets, they contain strings, It's true that for any language $L$, we have that the empty set is a subset of $L$, i.e. $\emptyset \subseteq L$, since this is true of any set. So if this is what you're asking, then yes, that is trivially true.
However, if you're asking whether $\emptyset \in L$ for every language, then this question doesn't really make sense, since languages don't contain sets. The analogous notion is the empty string $\varepsilon$, so it makes sense to ask whether $\varepsilon \in L$ for any language $L$, but the answer is of course no.