When completing the proof (unfortunately, at the same time presenting it as Euclid's and performing it ad absurdum) of the infinitude of prime numbers, my algebra professor stated "...and thus the set of the primes is not finite. By the axiom of infinity, we conclude it is infinite. "
But obviously such a set is a subset of $\mathbb{N}$, and this exists exactly thanks to the axiom of infinity. So why isn't this axiom implicit in the proof? That is, why isn't the last sentence redundant?
No, you can easily prove that there are infinitely many prime numbers without appealing to the axiom of infinity. The usual proof by Euclid does just that.
The axiom of infinity is used to show that the collection of natural numbers is a set, and therefore the collection of prime numbers is a set as well.