Initial Segment of Ordinal is Ordinal

Theorem
Let $S$ be an ordinal.

Let $a \in S$.

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

Proof
Let $b \in S_a$.

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

Then:

The result follows from the definition of an ordinal.