Both proofs are wrong, second one is closest. Second one should not claim that N is a prime (it likely isn't). It should say N is not divisible by any of p_i, and since due to Fun. Theo. of Arith. it is such that N = Sum {c_i q_i} where q_i are prime, and none of q_i in {p_i} which shows a finite list of primes is not possible construct.