# Empty Class is Supercomplete

## Theorem

The empty class is supercomplete.

## Proof

Vacuously, every element of $\O$ is also a subclass of $\O$.

Hence $\O$ is transitive by definition.

Vacuously, every subclass of every element of $\O$ is also an element of $\O$.

Hence $\O$ is swelled by definition.

The result follows by definition of supercomplete.

$\blacksquare$