Ordinal is Transitive/Proof 2
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Theorem
Every ordinal is a transitive set.
Proof
Let $\alpha$ be an ordinal by Definition 2:
$\alpha$ is an ordinal if and only if it fulfils the following conditions:
\((1)\) | $:$ | $\alpha$ is a transitive set | |||||||
\((2)\) | $:$ | the epsilon relation is connected on $\alpha$: | \(\ds \forall x, y \in \alpha: x \ne y \implies x \in y \lor y \in x \) | ||||||
\((3)\) | $:$ | $\alpha$ is well-founded. |
Thus $\alpha$ is a priori transitive.
$\blacksquare$