Empty Set is Unique

Theorem
The empty set is unique.

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

Since an empty set is a 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.