In proof of infinitude of primes, it is stated that $p_1p_2...p_n +1$ has no factor in the list of product (i.e., $p_1$ to $p_n$) due to addition with 1. However, the resulting value can be a prime or composite. If it is a prime, it is no issue as a new prime has been found. But, if a composite has been found then it has to be a product of at least 2 primes. I want proof that in the case of composite number, the prime factor is a new one and cannot be from the list.
My proof approach would state that it is obvious that none of the given primes would divide the composite number, and hence the prime factorization would have new primes. Stated differently, any prime in the list would not divide any part (factor) of the new number.
But, it is non-rigorous and if some help be provided to make it look rigorous, or else I should accept that it is an axiom and no rigor is possible.