# Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping

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## Theorem

Let $S$ be a set.

Let $\left({S, \prec}\right)$ be a strict well-ordering.

Then there exists a unique ordinal $x$ and unique mapping $f$ such that $f: x \to S$ is an order isomorphism.

## Proof

The existence of $x$ and $f$ follows from Woset is Isomorphic to Unique Ordinal.

The uniqueness of $x$ follows from Woset is Isomorphic to Unique Ordinal.

The uniqueness of $f$ follows from Order Isomorphism between Wosets is Unique.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 7.51$