Ordering on Ordinal is Subset Relation

Theorem
Let $\left({S, \preceq}\right)$ be an ordinal by Definition 3.

Then $\forall x, y \in S:$
 * $x \preceq y \iff S_x \subseteq S_y \iff x \subseteq y$

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 (defined by Definition 3 is&mdash;it is always the subset relation.

Proof
The first equivalence holds for any well-ordered set by Woset Isomorphic to Set of its Sections.

The second equivalence holds by definition of an ordinal.

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

Comment
Some sources refer to this result as Copi's identity for Irving Copi, from his statement of this in his 1954 book, but it may have been known about earlier than that and may not be direcly attributable to Copi.