Union of Subset of Ordinals is Ordinal/Corollary

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Theorem

Let $y$ be a set.

Let $\On$ be the class of all ordinals.

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


Then:

$\displaystyle \bigcup \map F y \in \On$

where $\map F y$ is the image of $y$ under $F$.


Proof

By the Axiom of Replacement, $\map F y$ is a set.

Thus by the Axiom of Unions, $\bigcup \map F y$ is a set.

By Union of Subset of Ordinals is Ordinal, $\bigcup \map F y$ is transitive.

By the epsilon relation $\bigcup \map F y$ is well-ordered.

Thus $\bigcup \map F y$ is a member of $\On$, the ordinal class.

$\blacksquare$