Initial Segment of Ordinal is Ordinal

Theorem
Let $S$ be an ordinal, and suppose that $a \in S$.

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

In other words, every element of an ordinal is also an ordinal.

Proof
By Subset of Well-Ordered Set is Well-Ordered, $S_a$ is well-ordered.

Suppose that $b \in S_a$.

From Ordering on 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.