Grothendieck Universe is Closed under Binary Union

Theorem
Let $\mathbb U$ be a Grothendieck universe.

Let $u, v \in \mathbb U$.

Then $u \cup v \in \mathbb U$.

Proof
If $\mathbb U = \O$, the claim is true.

Assume $\mathbb U \ne \O$.

By Nonempty Grothendieck Universe contains Von Neumann Natural Numbers, every von Neumann natural number is an element of $\mathbb U$.

In particular:
 * $2 = \set {\O, \set \O} \in \mathbb U$

Using $2$ as an indexing set, we remember that $0 = \O$ and $1 = \set \O$, and define:

This sets up the structures needed to exploit Grothendieck Universe: Axiom $(4)$ below.

Then: