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 $2 = \set {\O, \set \O} \in \mathbb U$.

Define $w_0 := u$ and $w_1 := v$, where $0 = \O$ and $1 = \set \O$.

Then: