Well, maybe it has to do with the ways of "proving" infinitude of primes, of that it is a "high level description" and more importantly, writing it so that other, future theorems can avail to it.

Think of all you need to prove that: the natural numbers, the definition of addition, subtraction, multiplication, division (and remainder), definition of cardinality/infinity (might not be necessary, if you just need proof that N doesn't divide any numbers between 2 and N-1) and a lot of formal logic