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.