# 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

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:

 $\displaystyle \paren {S_a}_b$ $=$ $\displaystyle \set {x \in S_a: x \subset b}$ Definition of Initial Segment $\displaystyle$ $=$ $\displaystyle \set {x \in S: x \subset a \land x \subset b}$ Definition of Initial Segment $\displaystyle$ $=$ $\displaystyle \set {x \in S: x \subset b}$ as $b \subset a$ $\displaystyle$ $=$ $\displaystyle S_b$ Definition of Initial Segment $\displaystyle$ $=$ $\displaystyle b$ as $S$ is an ordinal

The result follows from the definition of an ordinal.

$\blacksquare$