propositional logic

Theorem. Let $\mathcal{P} \subset \mathbb{N} ^{*}$ be the set of  prime numbers. If $\# \mathcal{P} < \infty$, there exists $n_{0} \in \mathbb{N}$ such that for all $n \geq n_{0}$, $10^{n}$ is not a prime.

The proof is elementary.