Union of Subset of Ordinals is Ordinal/Corollary

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Theorem

Let $y$ be a set.

Let $\operatorname{On}$ be the class of all ordinals.

Let $F: y \to \operatorname{On}$ be a mapping.


Then:

$\displaystyle \bigcup F \left({y}\right) \in \operatorname{On}$

where $F \left({y}\right)$ is the image of $y$ under $F$.


Proof

By the Axiom of Replacement, $F \left({y}\right)$ is a set.

Thus by the Axiom of Union, $\bigcup F \left({y}\right)$ is a set.

By Union of Subset of Ordinals is Ordinal, $\bigcup F \left({y}\right)$ is transitive.

By the epsilon relation $\bigcup F \left({y}\right)$ is well-ordered.

Thus $\bigcup F \left({y}\right)$ is a member of $\operatorname{On}$, the ordinal class.

$\blacksquare$