Nonempty Grothendieck Universe contains Von Neumann Natural Numbers

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

Let $\N$ denote the set of von Neumann natural numbers.

Then $\N$ is a subset of $\mathbb U$.

Proof
We prove the claim by induction.

Basis for the Induction
By Empty Set is Element of Nonempty Grothendieck Universe $\O \in \mathbb U$.

Induction hypothesis
For some fixed $n \in \N$, we have $n \in \mathbb U$.

Induction Step
We have to show, that $n + 1 \in \mathbb U$.