Ordinals Are Downward Closed

Proof

Let \(\alpha\in\beta\). In order to show that \(\alpha\) is an ordinal, we have to show that (1) \(\alpha\) is transitive and (2) each element of \(\alpha\) is also transitive.

(1) According to the definition of ordinal numbers, as an element of an ordinal, \(\alpha\) is identified as a transitive set.

(2) Because \(\beta\) is transitive, it follows:

\[\alpha\in \beta\Longrightarrow \alpha\subseteq \beta=(\gamma\in \alpha\Longrightarrow \gamma\in \beta),\]

i.e. for each element \(\gamma\in\alpha\) it is \(\gamma\in \beta\). As an element of an ordinal \(\beta\), \(\gamma\) is also identified as a transitive set.

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