Empty Set is Well-Ordered/Proof 2

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

Let $V$ be a basic universe.

By definition of basic universe, $\O$ is an element of $V$.

By the Axiom of Transitivity, $\O$ is a class.

The result follows from Empty Class is Well-Ordered.

$\blacksquare$