Canonical Order Well-Orders Ordered Pairs of Ordinals

Theorem
The canonical order, $R_0$ strictly well-orders the ordered pairs of ordinal numbers.

Strict Ordering
Assume $(x,y) R_0 (x,y)$. Then, $\max(x,y) < \max(x,y) \lor (x,y) \operatorname{Le} (x,y)$. Both lead to contradictions, so $\neg (x,y) R_0 (x,y)$ and $R_0$ is irreflexive.

Assume $\left(\alpha,\beta\right) R_0 \left(\gamma,\delta\right) \land \left(\gamma,\delta\right) R_0\left(\epsilon,\zeta\right)$.

There are two cases: $\max(\alpha, \beta) < \max(\gamma,\delta)$ or $\max(\alpha,\beta)=\max(\gamma,\delta)$.

In either case, $(\alpha,\beta) R_0 (\epsilon,\zeta)$ and $R_0$ is transitive.

Strict Total Ordering
Assume $\neg (\alpha,\beta) R_0 (\gamma,\delta) \land \neg (\gamma,\delta) R_0 (\alpha,\beta)$.

Then, $\max(\alpha,\beta) \le \max(\gamma,\delta) \land \max(\gamma,\delta)\le\max(\alpha,\beta)$.

So $\max(\alpha,\beta)=\max(\gamma,\delta)$.

Therefore, $\neg (\alpha,\beta) \operatorname{Le} (\gamma,\delta) \land \neg (\gamma,\delta) \operatorname{Le} (\alpha,\beta)$.

By the fact that Lexicographic Order Forms Well-Ordering on Ordered Pairs of Ordinals, $(\alpha,\beta) = (\gamma,\delta)$.

Well-Ordering
Take any nonempty subset $A$ of $\left( \operatorname{On} \times \operatorname{On} \right)$. We shall allow $A$ to be any class. This isn't strictly necessary, but it will not alter the proof. The $\max$ mapping sends each element in $A$ to an element of $\operatorname{On}$, so the image of $\max$ has a minimal element, $N$.

Take $B$ to be the class of all ordered pairs $(x,y)$, such that $\max(x,y) = N$.

Let the $\operatorname{Le}$-minimal element of $B$ be denoted $C$. Then, $\max(C) = N$ and $C$ is $\operatorname{Le}$-minimal.

Therefore, $C$ is the $R_0$-minimal element of $A$.