Attachment 21586

Attached is a proof that $\displaystyle A \subseteq \wp{(A)} \rightarrow \wp{(A)} \subseteq \wp{(\wp{(A)})}$. I would like to verify that the proof is correct. Specifically I would like to know for sure that moving the universal quantifier $\displaystyle \forall z$ from within the parenthesis to outside of them is OK. Sorry to bore you guys with all this simple stuff!