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.