Let ∼ be an equivalence relation on a set $A$, and let $a, b ∈ A$. Prove that $a ∈ [b]$ iff $b ∈ [a]$.

Question

I have the following proof outline, but I am not sure how to get started proving this. Can anyone point me in the right directon?

Proof.
Suppose that $\sim$ is an equivalence relation on a set $A$, and that $a,b \in A$.
[$\Rightarrow$] Suppose $a ∈ [b]$.
Then $a\sim b$.
Since $\sim$ is symmetric, $b\sim a$. Therefore $b \in [a]$.
[$\Leftarrow$] On the other hand, suppose $b ∈ [a]$.
Then $b\sim a$.
Since $\sim$ is symmetric, then $a\sim b$.
Therefore $a \in [b]$.

Answer 1

Is it not as simple as

$$x\in[y]\iff x\sim y\iff y\sim x\iff y\in[x]$$

?

It'd help if we knew precisely what definition you are using for $x\in[y]$. There are a lot of equivalent ways to phrase it but you need to use the definition you were provided with for the proof