As a working mathematician, not as a logician what I am not, set theory is the set (!) of rules in order to use the symbol $\in$, meaning "belongs to". That tells me when and how I can use it.
EDIT: Someone downvoted my (short) contribution. I would like to tell him that what I said is not as trivial as he may think. This is what my colleague Luck Darnière specialist of theories of models (in France) says himself:
Le langage des ensembles est constitué d'un unique symbole de relation binaire, "appartient à" [$\in$]. C'est dans ce langage que sont exprimés tous les axiomes de la théorie des ensembles.
I suggest my downvoter to think about that...