Empty Set is Element of Nonempty Grothendieck Universe

Theorem

Let $\mathbb U$ be a non-empty Grothendieck universe.

Then $\O \in \mathbb U$.

Proof

Let $A \in \mathbb U$ be an arbitrary element.

\(\ds \O\)

\(\subseteq\)

\(\ds A\)

Empty Set is Subset of All Sets

\(\ds \leadsto \ \ \)

\(\ds \O\)

\(\in\)

\(\ds \mathbb U\)

Grothendieck Universe is Closed under Subset

$\blacksquare$