Probably the most famous example of a proof, where consideration of cardinalities is used to show existence of some object, it the Cantor's proof that there exist transcendental numbers.
What are some other interesting examples of proofs in the similar spirit.
Although my motivation for asking this question is to have something you can show to students, to show them what nice things they already can prove with the results they've learned about cardinals, feel free to add answers on any level.
EDIT: As pointed out by Arthur Fischer in his comment, there already is a rather similar question: Looking for a problem where one could use a cardinality argument to find a solution.
A difference is that the question asks about examples of the type $A\setminus B$, where $B$ is countable and $A$ is uncountable. (But as you can see some answers, including the answer that is accepted, are about larger cardinalities, too.)