[red_trumpet]

You are right, but I wouldn't emphasize this in the proof, for the following reasons:

1. Since you assume a contradictory statement, you can actually derive everything you want. So saying "it's true in that context" is pretty meaningless.

2. I think it adds an unnecessary step in the proof. "This new number is not divisible by any prime, therefore it is prime, contradiction as it is not in the list" compared to "It is not divisible by any prime, contradiction since any number is divisible by a prime". I think that is confusing.

3. For didactical reasons. It can leave the reader/student with the wrong impression that multiplying the first n primes and adding one always creates a new prime.