While working through Velleman, I proved that if $A \subseteq P(A)$, then $P(A) \subseteq P(P(A))$.
One example where this may be the case is when $A = \emptyset$. Another may be when $\emptyset \in A$.
I cannot think of any other example though. Supposing that $x$ and $y$ are two arbitrary elements of $A$, then $P(A)$ will always enclose those elements in a new set, thus $x,y \notin P(A) $
Thus my question is:
Is there an example of a set, $A$, where $A \subseteq P(A)$ wand $A$ is non-empty and does not contain the empty set.