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.