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.

Your proof is not legitimate in ZF (in which there is no "universe" in which $A$ and $B$ abide). (I would say more but you explicitly asked me not to.)
