Baire category theorem (statement)

Theorem. A complete metric space (e.g. $\mathbb{R}$) is not a countable union of nowhere dense sets.

Consequence. $\mathbb{R}$ is not a countable union of nowhere dense sets. Since a singleton is nowhere dense, $\mathbb{R}$ is not a countable union of singletons, i.e. $\mathbb{R}$ is uncountable.

Conclusion. Baire category implies uncountability of complete metric spaces.