Relation between Two Ordinals/Corollary

Theorem
Let $S$ and $T$ be ordinals.

If $S \ne T$, then either $S$ is an initial segment of $T$, or vice versa.

Proof
By Ordinal Membership Trichotomy, either $S \in T$ or $T \in S$.

By definition, every element of an ordinal is an initial segment; hence the result.