Strict Well-Ordering Isomorphic to Unique Ordinal under Unique Mapping

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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: $\S 7.51$