Empty Set is Unique

Theorem
The empty set is unique.

Proof
Suppose there were more than one empty set.

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

Now $$\varnothing \subseteq \varnothing'$$ (as $$\varnothing$$ is a subset of every set.

Also $$\varnothing' \subseteq \varnothing$$ (as $$\varnothing'$$ is also an empty set, and therefore a subset of every set).

Therefore, as $$\varnothing \subseteq \varnothing' \land \varnothing' \subseteq \varnothing$$, we see that $$\varnothing = \varnothing'$$ from the definition of set equality.

So $$\varnothing$$ and $$\varnothing'$$ are the same, and by the definition of equality, that means they are the same object.

Thus there is only one empty set.