Definition:Ordinal/Definition 1

Definition
Let $S$ be a set.

Let $\Epsilon \! \restriction_S$ be the restriction of the epsilon relation on $S$.

Then $S$ is an ordinal :
 * $S$ is a transitive set
 * $\Epsilon \! \restriction_S$ strictly well-orders $S$.

Also see

 * Equivalence of Definitions of Ordinal