Ordering on Ordinal is Subset Relation

Theorem
Let $\left({S, \preceq}\right)$ be an ordinal.

Then $\forall x, y \in S:$
 * $x \prec y \iff S_x \subset S_y \iff x \subset y$

where $S_x$ and $S_y$ are the initial segments of $S$ determined by $x$ and $y$ respectively.

Thus there is no need to specify what the ordering on an ordinal is - it's always the subset relation.

Proof
The first equivalence holds for any woset by Woset Isomorphic to Set of Its Sections.

The second equivalence holds by definition of an ordinal.

It follows from Ordering Equivalent to a Subset Relation and Order Isomorphism Between Wosets is Unique that this ordering is the only one.