What is the correct answer to the following blanks to prove that 𝒫(A) ∩ 𝒫(B) ⊆ 𝒫(A ∩ B)?

Question

Suppose A and B are any sets.

1. Let X be any element in 𝒫(A) ∩ 𝒫(B).
2. Then X ∈ 𝒫(A) and X ∈ 𝒫(B) by definition of intersection
3. Hence, X ⊆ A and X ⊆ B by definition of power set
4. So, _____ by definition of subset
5. Thus, _____ by _____
6. _____ by _____
7. Therefore, X ∈ _____. By _____
8. Since X could be any element in _____ it follows that every element in _____ is in _____.
9. Therefore, 𝒫(A) ∩ 𝒫(B) ⊆ 𝒫(A ∩ B).

Answer 1

Once again, not sure about every blank, but here's a proof:

Let . Then, and by definition of intersection. Thus, and by definition of power set. Since , it follows that every element of is in .; a symmetric argument holds for the and so every element of is in . Because every element of is in and in , every element of is in ; so, , which shows that .