Finite Set of Primes, n is not prime

There is a finite set $$\{p_1, \ldots, p_r\}$$ of all primes and $$n = p_1 p_2 \cdots p_r + 1 $$ is not prime $$\Rightarrow$$ Contradiction.

Explanation

n has a prime divisor p. (*)

Is p in the set $$\{p_1, \ldots, p_r\}$$ ?


 * Yes, $$p \in \{p_1, \ldots, p_r\}\Rightarrow$$ Contradiction (more)
 * No, $$p \notin \{p_1, \ldots, p_r\}\Rightarrow$$ Contradiction (*)