# Union of Nest of Ordinal Sequences which is Proper Class

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## Theorem

Let $N$ be a nest of ordinal sequences such that $N$ is a proper class.

Let $\bigcup N$ denote the union of $N$.

Then $\bigcup N$ is a mapping whose domain is the class of all ordinals $\On$.

## Proof

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## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 5$ Transfinite recursion theorems: Lemma $5.2 \ (2)$