Ordering on Ordinal is Subset Relation

Theorem
Let $\struct {S, \prec}$ be an ordinal.

Then $\forall x, y \in S:$


 * $x \in y \iff x \prec y \iff S_x \subsetneqq S_y \iff x \subsetneqq y$

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 is always the subset relation.

Proof
The first equivalence is an immediate consequence of Equivalence of Definitions of Ordinal.

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

The third equivalence holds by definition of an ordinal.

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

Also known as
Some sources refer to this result as Copi's identity for, from his statement of this in his $1954$ book.

However, it may have been known about earlier than that and may not be direcly attributable to.