Element of Ordinal is Ordinal

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Theorem

Let $n$ be an ordinal.

Let $m \in n$.


Then $m$ is also an ordinal.


Proof

By the definition of ordinal, $n$ is transitive.

Thus $m \subseteq n$.

By Subset of Strictly Well-Ordered Set is Strictly Well-Ordered, it follows that $m$ is strictly well-ordered by the epsilon restriction $\Epsilon {\restriction_m}$.


It is now to be shown that $m$ is transitive.

If $m = \varnothing$ then the result follows by Empty Set is Transitive.


If $m \ne \varnothing$, then let $x \in m$.

If $x = \varnothing$, then $x \subseteq m$ by Empty Set is Subset of All Sets.


If $x \ne \varnothing$, then let $y \in x$.

It suffices to show that $y \in m$.


Since $m \subseteq n$, it follows that $x \in n$.

Also, $y \in x \land x \in n \implies y \in n$ because $n$ is transitive.

And so $x \in n$, $y \in n$, and $m \in n$.


A strict well-ordering is transitive by definition.

Therefore:

$y \in x \land x \in m \implies y \in m$


Hence the result.

$\blacksquare$