RO-TST 2003 Problem 10

Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$ , having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $-1+\prod_{p\in A}p$ are found in $M$ . Prove that $M= \mathcal{P}$ .

Valentin Vornicu