Empty Set is Unique/Proof 1

Proof
Let $\O$ and $\O'$ both be empty sets.

From Empty Set is Subset of All Sets, $\O \subseteq \O'$, because $\O$ is empty.

Likewise, we have $\O' \subseteq \O$, since $\O'$ is empty.

Together, by the definition of set equality, this implies that $\O = \O'$.

Thus there is only one empty set.