Prove $[A]_\sim = \{ Y \space| \space Y^c \in [A]_\sim \}$

Question

This is a follow-up of this question

Prove that $[A^c]_\sim = \{ Y \space| \space Y^c \in [A]_\sim \}$

I tried to prove this by using the definition of equivalence class. So take an arbitrary set $B \in [A^c]_\sim$, then $A^c \sim B$. By the equivalence relation we see that $A^c\Delta B$ is finite, but $B^c\Delta A$ is also finite as it is symmetric, so $B^c \sim A$ and $B^c \in [A]_\sim$.

Is this proof sufficient?

Answer 1

Hint \begin{equation} A^c\setminus B^c = \{x\in X, x\not\in A \text{ and } x\in B\} = B\setminus A \end{equation}