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

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

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