p is any of the primes, x is the result of multiplying all the other primes in the list.
If px+1 was divisible by p, then px+1 = py, for some whole number y
dividing by p, y = x + 1/p. This cannot be a whole number as p >= 2 and x is a whole number.
I'm sure there's a better proof, but that's just off the top of my head.