1. 1+Πp is congruent to 1 (mod p) for every prime p.

2. This conflicts with the definition of "prime number"; all composite numbers must be congruent to 0 (mod p) for some prime p. Only the empty product can be congruent to 1 (mod p) for every p.

This is already sufficient to prove that the number of primes is not finite -- we derived a contradiction from that premise without demonstrating any additional primes.

(Note that -- within the proof, where we've assumed a finite list of primes -- it's easy to show that 1+Πp is itself prime (since it has no prime divisors less than itself). You can then declare a contradiction with the premise, as it isn't in the original list. However, you can only show that it's prime by using the earlier contradiction, so while this step makes the proof more intuitive, it isn't actually necessary.)

> Yes that is correct but that's not what I was talking about in earlier comments, which were exclusively about example-based proof/disproof.[0]

In reply, thaumasiotes said:

> If you're not talking about proof/disproof, you might want to look for a different word.[1]

Now you say:

> I was unaware there is only one type of proof or that it is not allowed to talk only about specific types of proof.[2]

Based on your comment [0] above, your comments were apparently about a specific type of proof, and yet the words you used did not make that qualification clear. In particular, statement [0] says you were restricting the type of proof you were considering.

Reply [1] above is suggesting that if you are only talking about a restricted type of proof then you might want to use a word other that "proof", because you are not talking about proofs in general.

Your comment [2] seems to have missed the point. You yourself in [0] said you were restricting your type of proof. You're allowed to talk about anything you like, but if you use the word "proof" without qualifying it, but you intend only to be talking about a specific type of proof, don't then be surprised if people misunderstand you.