# Ordinal is Transitive/Proof 1

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## 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.

$\blacksquare$