Alternative Definition of Ordinal

Theorem
A set $S$ is an ordinal $S$ is transitive and is strictly well-ordered by the $\in$-relation.

Necessary Condition
Suppose that $S$ is an ordinal.

Then $S$ is transitive.

By definition, the strict well-ordering on $S$ is given by the $\in$-relation.

Hence, the necessary condition is satisfied.

Sufficient Condition
Suppose that $S$ is a transitive set that is strictly well-ordered by the $\in$-relation.

Let $a \in S$.

Then:

That is, $S$ is an ordinal.

Also see

 * Alternative Definition of Ordinal in Well-Founded Theory