Equivalence of Definitions of Ordinal

Theorem
The following definitions of an ordinal are equivalent:

Definition 1 is equivalent to Definition 2
This follows immediately from the definition of a strict well-ordering.

Definition 1 implies Definition 3
Let $S$ be an ordinal according to Definition 1.

Let $a \in S$.

Then:

Definition 3 implies Definition 1
Let $\left({S, \prec}\right)$ be an ordinal according to Definition 3.

Let $a \in S$.

Then $a = S_a \subseteq S$ and so $S$ is transitive.

Also, by the definition of set equality:

It should be noted that as $\prec$ is only defined on $S$, it follows that:
 * $x \prec a \implies x \in S$

It is seen that:
 * $\left({x \in S \land x \prec a}\right) \implies x \prec a$

and:
 * $x \prec a \implies \left({x \in S \land x \prec a}\right)$

Thus:
 * $x \prec a \iff \left({x \in S \land x \prec a}\right)$

It has been shown that if $a \in S$ then:
 * $\forall x: x \in a \iff x \prec a$

Therefore, $\prec \; = \left({S, S, R}\right)$ where

Hence $\prec \; = \Epsilon {\restriction_S}$.