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

Question

Suppose A and B are any sets.

1. Let X be any element in 𝒫(A ∩ B).
2. Then X⊆A ∩ B by definition of power set
3. So, _____ by definition of subset
4. Thus, _____ by definition of intersection
5. _____ 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 ∩ B) ⊆ 𝒫(A) ∩ 𝒫(B).

Answer 1

I'm not 100% sure about every blank, but here's a proof:

Let . Then, by definition of power set. Thus, every element of is in by definition of subset. It follows that every element of is in and every element of is in . Since every element of is in , ; a symmetric argument holds for and so . Hence, and by definition of power set. Therefore, .