N has a Prime Divisor in the Set of Primes

$$n = p_1 p_2 \cdots p_r + 1 $$ has a prime divisor $$p \in \{p_1, \ldots, p_r\} \Rightarrow$$ Contradiction

Explanation

p divides n

p divides $$p_1 p_2 \cdots p_r$$

p divides $$n - p_1 p_2 \cdots p_r = 1$$ (more)

p divides 1 $$\Rightarrow$$ Contradiction