Grothendieck Universe is Closed under Binary Union

Theorem

Let $\mathbb U$ be a Grothendieck universe.

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

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

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

Assume $\mathbb U \ne \O$.

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:

$\ds w_0$

$:=$

$\ds u$

$\ds w_1$

$:=$

$\ds v$

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

Then:

$\ds u, v$

$\in$

$\ds \mathbb U$

$\ds \leadsto \ \ $

$\ds \set {u, v}$

$\in$

$\ds \mathbb U$

$\ds \leadsto \ \ $

$\ds \bigcup_{i \mathop \in 2} w_i = u \cup v$

$\in$

$\ds \mathbb U$

$\blacksquare$