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.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 3$ Axiom of the empty set: Note $2$.