Suppose $A \subseteq P(A)$. Prove that $P(A)\subseteq P(P(A))$

It is perfectly true that if $A\subseteq\wp(A)$ and $z\in Y\in X\in\wp(A)$, then $z\in A$, but this does not show that $\wp(A)\subseteq\wp(\wp(A))$. To show that $\wp(A)\subseteq\wp(\wp(A))$, you need to let $X$ be an arbitrary element of $\wp(A)$ and show that $X\in\wp(\wp(A))$. And all of the necessary work is actually there in your argument, along with a fair bit of irrelevant material: if $X\in\wp(A)$, then $X\subseteq A\subseteq\wp(A)$, so $X\subseteq\wp(A)$, and therefore by definition $X\in\wp(\wp(A))$. And that’s it: $X$ was an arbitrary member of $\wp(A)$, so we’ve shown that $\wp(A)\subseteq\wp(\wp(A))$.