# Unique Isomorphism between Ordinal Subset and Unique Ordinal

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## Theorem

Let $\On$ be the class of all ordinals.

Let $S \subset \On$ 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 $S \subset \On$, $\struct {S, \in}$ 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*: $\S 7.52$