# Thread: Does this proof work?

1. ## Does this proof work?

Q:Prove that P(A∩B)=P(A)∩P(B).

A:Let x∈P(A∩B) be arbitrary ⇒ x⊆A∩B
⇔ ∀y(y∈x⇒y∈(A∩B))
⇔ ∀y(y∈x⇒y∈A and y∈B)
⇔ x⊆A and x⊆B
⇔ x∈P(A) and x∈P(B)
⇔ x∈P(A)∩P(B) ■

Does this work? I'm not sure if I have used the logical connectives correctly, such as breaking up the subset definition the way I did. I'm just trying prove things using iff statements for the practice.

Thanks.

2. It works, but has more steps than necessary.
But it does work.