I'm trying to prove that the empty set is unique.

**Proof:**

Let $U = \{ a \}$ be the universal set.

Assume $a \not\in \emptyset '$ and $a \not\in \emptyset$.

Without loss of generality, since $a \not\in \emptyset'$, $\emptyset '$ does not contain any elements. Since $\emptyset '$ does not contain any elements, it must by default be a subset of $\emptyset$, since the conditional statement 

$$a \in \emptyset ' \Rightarrow \emptyset ' \subseteq \emptyset$$

is vacuously true.

Therefore, since $\emptyset' \subseteq \emptyset$ and $\emptyset \subseteq \emptyset '$, we have that $\emptyset ' = \emptyset$. $\tag\*{$\blacksquare$}$

I would appreciate it if people could please provide feedback as to the correctness of my proof.

EDIT: Please be specific about what is incorrect and why. That way, I can learn what I did wrong and improve much more effectively.