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

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This article, or a section of it, needs explaining.In particular: Why strict well-ordering? Order isomorphism is not defined in terms of strict orderings. Pieces here don't match up properly.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |

## Theorem

Let $S$ be a set.

Let $\struct {S, \prec}$ 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 the Counting Theorem.

The uniqueness of $x$ follows from the Counting Theorem.

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*: $\S 7.51$