Singleton Class of Empty Set is Supercomplete

Theorem
Let $\O$ denote the empty set.

Then the singleton $\set \O$ is supercomplete.

Proof
Let $x \in \set \O$ be any element of $\set \O$.

Then it has to be the case that $x = \O$.

Then every element of $\O$ is an element of $\set \O$ vacuously.

That is, $\set \O$ is swelled.

There is one element of $\set \O$, and that is $\O$.

This is a subclass of $\set \O$.

That is, $\set \O$ is transitive.

The result follows by definition of supercomplete class.