About the notation of Axiom of Regularity

Question

Axiom of Regularity/Foundation

∀ ( ≠ ∅ → ∃ ( ∈ ∧ ⋂ = ∅))

This says that every non-empty set must contain a set that shares no elements with it.

So, how do we understand axiom of regularity is talking about a set with "y" just from the notation?

(I think using intersection ⋂ gives a clue that y is a set because only sets intersects not elements but not sure if its right inference.)

Answer 1

There's less here than meets the eye. In $\mathsf{ZFC}$, everything is a set. So $y$ is a set because it can't not be.

(This is the source of the most common criticism of $\mathsf{ZFC}$, incidentally: that formalizing mathematics in $\mathsf{ZFC}$ inevitably leads to "junk theorems." See here for example.)