Proof that $-(-A) = A$ I have tried to prove that, given a set $A$, $-(-A)=A$. Are there flaws in my logic? I am very new to proof writing and set theory so any tips on structuring the proof would be greatly appreciated. My proof is:

If $S$ is the space and $A\subset S$, then $-(-A)=S-(-A)$.
Suppose that $x \in S-(-A) \iff x \notin-A \iff x\notin S-A \iff x\in A. $ Therefore, $-(-A)=A$

My main queries with my proof are:
1)Is it necessary to introduce the space $S$ and state $-(-A)=S-(-A)$?
2)Is my use of $\iff$ correct?

An easy way out:
$$x\in A\implies x\notin -A\implies x\in-(-A)\implies A\subset -(-A)$$
and on the other hand
$$x\in-(-A)\implies x\notin -A\implies x\in A\implies -(-A)\subset A$$
