Initial Segment of Canonical Order is Set

Theorem
Let $R_0$ denote the canonical ordering of $(\operatorname{On} \times \operatorname{On})$.

Then, for all $(x,y) \in ( \operatorname{On} \times \operatorname{On} )$, the $R_0$-initial segment is a set.

Proof
Let $z = \max(x,y)^+$. Then, if $(v,w) R_0 (x,y)$, then:

Thus, the initial segment $(\operatorname{On} \times \operatorname{On})_{(x,y)} \subseteq ( z \times z )$.

By Axiom of Subsets Equivalents, the initial segment of $(x,y)$ is a set.