Empty Class is Well-Ordered

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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 $\RR$ if and only if 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.

$\blacksquare$