Alternative Definition of Ordinal

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

A set $S$ is an ordinal iff $S$ is transitive and $\forall x,y \in S: ( x \in y \lor x = y \lor y \in x )$

Proof
For the first, biconditional statement

Necessary Condition
Suppose that $S$ is an ordinal.

Then $S$ is transitive and, 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.

The sufficient condition in the second statement follows immediately from the definition of a well-ordering. The necessary condition follows from the fact that Epsilon is Foundational.

Source

 * : $\S 7.3$, $\S 7.4$