Can I have help with the following question related to the concept vector 'subspaces'?

Question

Question : Let X and Y be vector subspaces of a vector space V over a field F, (V,+,•,F).
Prove that the intersection of X and Y is also a vector subspace of V.

(1)So when we take any two vectors u,v€(X^Y) (€denote belongs to a set ans ^ denote intersection),

Then we can show that the vector u+v€ (X^Y)

2) If we consider any scalar A€F, any vector u€(X^Y),
we can show that the vector Au€(X^Y)

My question is, do we need to show that the identity element of V(say 0), also belong to X^Y, to show that X^Y is a vector subspace of V?
Or is proving the conditions (1) & (2) above would be sufficient to prove that X^Y is a vector subspace of V?

Thank you!