Empty Set is Unique

Theorem
The empty set is unique.

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

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

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

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

Thus there is only one empty set.