A Mathematical Conjecture, Part II

I recently devised a theorem that I conjectured required higher set theory to prove. It doesn't:

Let the index of the ith prime be i, i.e., the index of 2 is 1, the index of 3 is 2, etc.

If we take the prime factorization of an integer n greater than 1, the index of at least one of the primes will be relatively prime to n.

This can be easily proved by considering the smallest prime factor of n. Its index will either be 1 or it will be the product of primes smaller than any that divide n. In other words, that index will be relatively prime to n.