How to write "There is v in set W and P, such that v is in set P"? How to write "There is v in set W and P, such that v is in P"?

I thought of the following possibilities:

∃ v ∈ W : ( ∃ P : v ∈ P)


or

∃ v ∈ W, P : v ∈ P


Is either of these correct? If not, how should I write it?
This should suffice:
$$\exists v \in W : v \in P$$
Another, equivalent expression is:
$$\exists v \in W \cap P : v \in P$$
You can write 'there is no such P' in several ways. Directly
$$\not\exists P : v\in P$$
or 'it's not true there exist such P':
$$\neg\left(\exists P : v\in P\right)$$
or even by transferring the negation into the predicate, as 'for each P, v can't belong to P':
$$\forall P : v\notin P$$