I am a little confused by this question: Suppose $\displaystyle A \subseteq \mathcal{P}(A)$, Prove that $\displaystyle \mathcal{P}(A) \subseteq \mathcal{P}(\mathcal{P}A)$.

By definition, I know that $\displaystyle A\in \mathcal{P}(A)$ but not $\displaystyle A \subseteq \mathcal{P}(A)$. I believe only when $\displaystyle A=\varnothing$ that it can be a subset of $\displaystyle \mathcal{P}(A)$. I am right?