Equivalence of Definitions of Ordinal

Theorem
Let $\alpha$ be a set.

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

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

Let $\beta \in \alpha$.

Then:

Definition 3 implies Definition 1
Let $\struct {\alpha, \prec}$ be an ordinal according to Definition 3.

Let $\beta \in \alpha$.

Then $\beta = \alpha_\beta \subseteq \alpha$ and so $\alpha$ is transitive.

Also, by the definition of set equality:

It has been shown that if $x, \beta \in \alpha$ then:
 * $x \in \beta \iff x \prec \beta$

Therefore, ${\prec} = \struct {\alpha, \alpha, \RR}$ where:

Hence ${\prec} = \Epsilon {\restriction_\alpha}$.