Empty Class is Well-Ordered

Theorem
Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation.

Let $\O$ denote the empty class.

Then $\O$ is well-ordered under $\RR$.

Proof
We have that $\O$ is well-ordered under $\RR$ every non-empty subclass of $\O$ has a smallest element under $\RR$.

But $\O$ has no non-empty subclass.

Hence this condition is satisfied vacuously.

The result follows.