Empty Set is Well-Ordered/Proof 1

Theorem

Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a relation on $S$.

Let $\O$ denote the empty set.

Let $\RR_\O$ denote the restriction of $\RR$ to $\O$.

Then $\struct {\O, \RR_\O}$ is a well-ordered set.

Proof

We have that $\O$ is well-ordered under $\RR$ if and only if every non-empty subset of $\O$ has a smallest element under $\RR$.

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

Hence this condition is satisfied vacuously.

The result follows.

$\blacksquare$