Initial Segment of Ordinal is Ordinal

Theorem
Let $S$ be an ordinal.

Let $a \in S$.

Then the initial segment $S_a$ of $S$ determined by $a$ is also an ordinal.

Proof
If $S_a$ is empty, then the result trivially holds.

Otherwise, let $b \in S_a$.

From Ordering on an Ordinal is Subset Relation, and the definition of an initial segment, it follows that $b \subset a$.

Then:

The result follows from the definition of an ordinal.