Is it okay?

For sets A,B:

Prove that P(power set)(A) U P(B) is contained in P(AUB)

I wrote:

Let S be a subset and let S belong to P(AUB)

Then S belongs to P(AUB) iff S is contained in (AUB)

and S is contained in (AUB) iff S is contained in A and S is contained in B

iff S belongs to the power set of A and the power set of B, which is equivalent to S belongs to P(A)U P(B)

Is it okay?