Ordinal is Transitive/Proof 1

Theorem

Every ordinal is a transitive set.

Proof

Let $\alpha$ be an ordinal by Definition 1:

$\alpha$ is an ordinal if and only if it fulfils the following conditions:

\((1)\)	$:$		$\alpha$ is a transitive set
\((2)\)	$:$		$\Epsilon {\restriction_\alpha}$ strictly well-orders $\alpha$

where $\Epsilon {\restriction_\alpha}$ is the restriction of the epsilon relation to $\alpha$.

Thus $\alpha$ is a priori transitive.

$\blacksquare$