# Relation between Two Ordinals/Corollary/Proof 1

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## 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

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

By definition, every element of an ordinal is an initial segment.

Hence the result.

$\blacksquare$