Using only logical symbols and $\in$, and $=$ translate into first-order logic: "$A$ is the power set of $B$"

Question

Using only logical symbols and $\in$, and $=$ translate into first-order logic: "$A$ is the power set of $B$"

I've attempted to solve this, but I am not sure whether my solution contains no errors. I'd be glad if you gave me some kind of feedback.

$$(\forall x)(x\in A \Rightarrow ((\forall y)(y \in x \Rightarrow y \in B))$$

My reasoning:
If $A$ is the power set of $B$ then every element $x$ of $A$ will be a subset of $B$, and so any object in $x$ will also be in $B$.

Answer 1

Basically correct, but your first $\Rightarrow$ needs to be a $\Leftrightarrow$. Your formula needs to say that all subsets of $B$ are elements of $A$, not just that all elements of $A$ are subsets of $B$.