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.

$\blacksquare$

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