Verify my proof: $A \subseteq B$ iff $\mathscr P(A) \subseteq \mathscr P(B)$ I'm self learning from the book "How to Prove it" by Velleman (3rd edition). I don't have access to a math professor, so I need a little help from the community. Please verify my proof.
Problem 8 pag. 140

Prove that  $A \subseteq B$ iff $\mathscr P(A) \subseteq \mathscr P(B)$

Proof. $(\to)$ Suppose $ A \subseteq B$. Let's take an arbitrary subset $x$ from $A$, then $x \subseteq B$. But also $x \in \mathscr P(A)$ and $x \in \mathscr P(B)$. But this is exactly what $ \mathscr P(A) \subseteq \mathscr P(B)$ means.
$(\gets)$ Suppose $ \mathscr P(A) \subseteq \mathscr P(B)$. Because $A \in \mathscr P(A)$, it means that $A \in \mathscr P(B)$ and also $A \subseteq B$ $\blacksquare$
 A: You can be more economic on words which would simplify your reasoning.
When we want to prove inclusions $A \subset B$, the usual procedure is to take an arbitrary $x \in A$ and show that $x \in B$. This would imply that every element of $A$ will be an element of $B$, thus the inclusion.
In your case:
We want to prove that $\mathcal{P}(A) \subset \mathcal{P}(B)$. Lets take $x \in \mathcal{P}(A)$ arbitrary and show that $x \in \mathcal{P}(B)$. Indeed, let $x$ be such element in $\mathcal{P}(A)$. Now, $x \in \mathcal{P}(B)$, means that $x \subset B$. Well, our hypothesis says that $A \subset B$, so any subset of $A$ will be a subset of $B$. And so, since $x \subset A$, this implies that $x \subset B$. Therefore, $x \in \mathcal{P}(B)$.
For the other direction we want to take $x \in A$ and try to prove $x \in B$ given $\mathcal{P}(A) \subset \mathcal{P}(B)$. I will leave this for you!
Good luck!
A: As several others have commented, your proof is unnecessarily wordy.  There is no need to start each part of the proof with "We must prove that ..."  A literate reader knows what you need to prove.  There is no need to say "The premise $A \subseteq B$ means that ..."  A literate reader knows what it means.
A more serious problems is that you seem to have the parts of the proof backwards.  In the part labeled "$(\to)$", you seem to be assuming $\mathscr{P}(A)\subseteq\mathscr{P}(B)$ and trying to prove $A \subseteq B$, which is backwards.  Also, you write "Now let say there is a set $y$, such that $x \in y$.  If $y \subseteq A$, then ..."  But what if there is no such set $y$?  What if it's not the case that $y \subseteq A$?  You can't just assume things that you want to be true.
For the $(\to)$ part, your proof should have this form:  "Suppose $A \subseteq B$.  Suppose $x \in \mathscr{P}(A)$. ... [now prove $x \in \mathscr{P}(B)$]."  See if you can finish it now.
