Union of Ordinals is Ordinal

Theorem
Let $y$ be a set.

Let $\map F z$ be a mapping such that:


 * $\ds \forall z \in y: \map F z \in \On$

Then:


 * $\ds \bigcup_{z \mathop \in y} \map F z \in \On$

Proof
So:
 * $\ds \bigcup_{z \mathop \in y} \map F z \subseteq \On$

So $\ds \bigcup_{z \mathop \in y} \map F z$ is a transitive set.

Therefore $\ds \bigcup_{z \mathop \in y} \map F z$ is an ordinal.


 * $\ds \bigcup_{z \mathop \in y} \map F z = \bigcup \Img y$

Let $U$ denote the universal class.

Therefore $\ds \bigcup_{z \mathop \in y} \map F z$ is a set, so it is a member of the ordinal class.