Proving iff statement concerning transitivity I am asked to show that $A$ is transitive $iff$ $A\subset\ P(A)$. My definition is that a set A is transitive if whenever $x\in\ a\in\ A$, then $x\in A$.
My attempt at a proof: 
I started with let $x\in\ a\in\ A$. By the Power Set Axiom, it follows that for the set $A$, there exists a set $P(A)$ such that for all $a$, $a\in\ P(A)$ $iff$ $a\subset\ A$. I am not too sure if this is the correct way to approach this side of the proof. 
For the other direction, I have let $A\subset\ P(A)$, then if $a\in\ A$ it follows that $a\in\ P(A)$. Then, I am not too sure how to continue to get to the fact of being transitive. I think I should also be using the Power set axiom here as well but I am not sure. 
For reference, the Power Set Axiom reads: For any set A, there exists a set B such that for all x, $x\in\ b$ $iff$ $x\subset\ A$. Our B is what we call the Power set. 
 A: I don't know if I'd say you're "using the power set axiom" which establishes the existence of a power set for every set. I guess you use it to prove the question isn't referring to a nonexistent object, but otherwise you're just using the definition of a power set.
I'm not sure I follow your reasoning for the first direction. You are trying to show $A\subset P(A).$ The standard way to establish that is to take an arbitrary $a\in A$ and show that $a$ must also be an element of $P(A).$ So let $a\in A.$ By the definition of the power set, $a\in P(A)$ if and only if $a\subset A,$ so we need to show $a\subset A.$ Again, the way we will show this is to take an arbitrary $x\in a$ and show that $x\in A.$ So let $x\in a.$ We have $x\in a$ and $a\in A$ so by the definition of transitivity $x\in A.$ Thus we have shown $a\subset A$ and thus $a\in P(A).$ Since we started with $a\in A$ and showed this implies $a\in P(A),$ we have established that $A\subset P(A).$
For the other way, to show transitivity, you need to show that if $x\in a$ and $a\in A$ then it follows that $x\in A.$ So assume $x\in a$ and $a\in A$ and set out to prove $x\in A.$ We use the assumption that $A\subset P(A)$ which as you indicated implies $a\in P(A)$ which in turn by definition of power set implies $a\subset A.$ Since we have $x\in a,$ by the definition of "$a\subset A$", we have $x\in A.$ And we're done.
