Suppose that $ A \subseteq \mathscr{P}(A)$. Prove $\mathscr{P}(A) \subseteq\mathscr{P}(\mathscr{P}(A)) $

Question

Well i've been having problems trying to prove that $\mathscr{P}(A) \subseteq\mathscr{P}(\mathscr{P}(A)) $ if $ A \subseteq \mathscr{P}(A)$ What i need is to get a proof by using quantifiers

Answer 1

$$X \subseteq Y \implies \mathscr{P}(X) \subseteq \mathscr{P}(Y)$$

take $x \in \mathscr{P}(X)$ so $x \subseteq X$ and since $X \subseteq Y$ then $x \subseteq Y$ so finaly $x \in \mathscr{P}(Y)$

Answer 2

Before I actually do the problem, we should at least think about what we are really trying to prove here. The problem can be translated into words to mean :

If $A$ is a transitive set, then $\mathcal{P}(A)$ is a transitive set. (assuming that our universe contains only sets)

Anyway, here is the proof:

Let $x\in \mathcal{P}(A)$, then by definition of the power set $x \subseteq A$. But by assumption $A\subseteq \mathcal{P}(A)$ thus we see that $x\subseteq A \subseteq \mathcal{P}(A)$ and so $x\in \mathcal{P}(\mathcal{P}(A))$

I'm not sure how detailed "proof by quantifiers" is perhaps something like

$$\forall x \in \mathcal{P}(A) (x\subseteq A) \land (A \subseteq \mathcal{P}(A)) \implies \forall x\in \mathcal{P}(A) (x\subseteq \mathcal{P}(A))$$