Unique Isomorphism between Ordinal Subset and Unique Ordinal
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Theorem
Let $\operatorname{On}$ be the class of ordinals.
Let $S \subset \operatorname{On}$ where $S$ is a set.
Then there exists a unique mapping $\phi$ and a unique ordinal $x$ such that $\phi : x \to S$ is an order isomorphism.
Proof
Since $S \subset \operatorname{On}$, $\left({S, \in}\right)$ is a strict well-ordering.
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The result follows directly from Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.52$
