Relation between Two Ordinals/Corollary/Proof 3

Corollary to Relation between Two Ordinals

Let $S$ and $T$ be ordinals.

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

Proof

We have that $S \ne T$

Therefore, from Relation between Two Ordinals either $S \subset T$ or $T \subset S$.

By Ordering on Ordinal is Subset Relation or Transitive Set is Proper Subset of Ordinal iff Element of Ordinal, either $S \in T$ or $T \in S$.

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

$\blacksquare$