Ordinal Subset is Well-Ordered

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Theorem

Let $S$ be a class.

Let every element of $S$ be an ordinal.


Then $\struct {S, \in}$ is a strict well-ordering.


Proof

Let $\On$ denote the class of ordinals.

By definition of subset, $S \subseteq \On$.


But by Ordinal Class is Ordinal $\On$ is an ordinal.

Therefore $\On$ is well-ordered by $\in$.


This means that $S$ is also well-ordered by $\in$.

$\blacksquare$