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.