Can't understand the logical structure of Euclid's infinitely many primes proof in Rosen's book. I'm reading the book Discrete Mathematics and It's Applications written by Rosen. At page 260, there is a proof using the technique similar to Euclid's. The following is my trying to understand the proof.



Assume that there are only finitely many primes, $p_1, p_2, ..., p_n$, I'll call it prime list later on. Let 

$$Q = p_1p_2\cdots p_n+1.$$

By the fundamental theorem of arithmetic, $Q$ is prime or else it can be written as the product of two or more primes.

Let $p_j$ be an arbitrary prime in the prime list. Then $p_j$ can't divide $Q$, since:


If $p_j \mid Q$, then $p_j \mid (Q - p_1p_2\cdots p_n=1)$:

The is because $p_j \mid (p_1p_2\cdots p_n)$, so $p_j \mid (1\cdot Q+(-1)\cdot(p_1p_2\cdots p_n))$.
Since $p_j \mid (Q - p_1p_2\cdots p_n=1)$ is false, $p_j \not\mid Q$:

The only integer which divides all integer is $1$, but $p_j$ is a prime.


Since $p_j \not\mid Q$ also means that $Q$ is not $p_j$, $Q$ is not in the prime list. And since there is no prime in the prime list divides $Q$, $Q$ can't be written as the product of two or more primes. Finally, $Q$ has to be a prime, which is not in the prime list. We've arrived a contradiction.



I'm not sure whether it's correct, but that's how I understand the proof. And the following is the proof in my book:




I don't understand why it arrives the conclusion "Hence, there is a prime not in the list".

Please give me some hints...
If $p$ is a prime number such that $p\mid Q$ then, since $Q-p_1p_2\ldots p_n=1$, $p\notin\{p_1,p_2,\ldots,p_n\}$ because otherwise $p\mid1$.
I was going through checking it by arbitrary number.
 Let's  $Q= 9699691$ be the  prime number because as stated above   none of $p_j$ (i.e $2$ to $19$ prime numbers) divides $Q$ which can be represented as 
$Q= 2\times3\times5\times7\times11\times13\times17\times19+1$ which must be prime number.
But $Q = 9699691=347\times27953$ which is a composite number. 
The thing is that the statement "$Q$ is prime or else it can be written as a product of two or more primes" implies there exists a prime dividing $Q$ but it ain't at our list. Since none of the $p_j$ have this property, there has to be another prime not in the list which divides $Q$.. which is why 
 $347$, $27953$ does... both are primes.

Hope, this clarifies everything.