A Mathematical Conjecture

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 using ZFC set theory. (Hint: Consider the enumeration of the ordinals less than ε₀ discussed here and then recall the Axiom of Foundation.) I'm not sure if it can be proved from the Peano postulates. An enumeration of ε₀ cannot be derived from the Peano postulates, but there might be other proofs.

Addendum: There is another proof.