proof involving double complement

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Prove $(A^c)^c$ = $A$

$(A^c)^c = \{x|x\in (A^c)^c\}$

$(A^c)^c = \{x|x\notin A^c\}$ by definition of complement

$(A^c)^c = \{x|\ (x \in A)\}$ by definition of complement

Therefore

$(A^c)^c=A$

Is this proof ok, I'm not sure if its valid? thanks