Proving that the Empty Set $\emptyset$ is Unique Verification I know that this question has been asked before, but I am presenting my proof for verification. I do not want anyone to give me a proof or suggest a better one; I only want to know if the following proof is legitimate.
Let $A$ and $B$ be empty sets. Let $U$ be the universe in which $A$ and $B$ abide. Clearly, $U = \overline{A}$ and $U = \overline{B}$. However, this implies that  $\overline{U} = \bar{\bar{A}} = A$  and $\overline{U} = \bar{\bar{B}} = B$. Since $\overline{U} = \overline{U}$, we conclude that $A=B$, that is, the empty set is unique.

As Rob Arthan alluded to in his answer, you have not really proved uniqueness of the empty set. You have shown that for any set $U$, there is at most one empty subset. But you cannot take $U$ to be "all sets" because this is not a set in ZF theory. You would be best to try to prove this without referring to any such $U$, and in particular avoiding use of complements (which require a base space to define).
