\begin{definition}[Definition:Ordinal/Definition 3]
An '''ordinal''' is a strictly well-ordered set $\struct {S, \prec}$ such that:
:$\forall a \in S: S_a = a$
where $S_a$ is the initial segment of $S$ determined by $a$.
From the definition of an initial segment, and Ordering on Ordinal is Subset Relation, we have that:
:$S_a = \set {x \in S: x \subsetneqq a}$
From Initial Segment of Ordinal is Ordinal we have that $S_a$ is itself an ordinal.
\end{definition}