Earlier quoted context omitted.
Technically, the parent post literally said "No prime divides the new number ... so you've just produced a new prime". It does not make the direct claim that the new prime is equal to the new number. It is possible that the OP did not make the mistake that you are attributing to them, although best case is that the exposition was simply unclear.
Thanks, very charitable reading but doesn't work because next sentence I said that the new prime is larger than largest. As other poster points out if 2,7 are all your primes you produce 15, whose primes are not larger than 7 so the "new prime" wasn't referring to one of them. But that's not how I argued. Instead I showed that 15 is the new prime because it's relatively prime to all the primes. (By the assumption we…
"Any number that is relatively prime to all other primes is itself a prime"
You would need to first state and prove this in order for your original post to constitute a rigorous proof.