If we have a set $S$ such that $\#(S)<\#(\Bbb N)$, where $\Bbb N$ is the set of the natural numbers, how would one prove that $S$ must be finite?
Here is a proof using the Axiom of Choice:
Proof: From $\#(S)<\#(\mathbb{N})$, we know that there exists an injection $f:S\to \mathbb{N}$. On the other hand, $S$ cannot be an infinite set: If $S$ is infinite, it contains a copy of $\mathbb{N}$ by the Axiom of choice. Therefore, we have an injection $g:\mathbb{N}\to S$. By the Schröder–Bernstein Theorem, $\#(S)=\#(\mathbb{N})$, which is a contradiction.
Can we prove the theorem without the Axiom of Choice?