Sure. Sorry, I didn't take the time to work this out on paper before posting or I would have realized that the condition itself is wrong. The condition lcm(|X|,|Y|) != |X| instead should be that |Y| has some prime factor that |X| does not.

Here is an explanation with the new condition:

Let p be any prime factor of |Y| that |X| does not have. It follows from Euclid's First Theorem[1] that p cannot divide |X|^n for any n [2]. Since every integer (> 1) has a unique prime factorization, it follows that |X|^n can't divide |Y|, because the prime p divides |Y| but not |X|^n.

[1] http://mathworld.wolfram.com/EuclidsTheorems.html [2] We are given that p does not divide |X|^1. Suppose that p also does not divide |X|^(n-1) for some n > 1. |X|^n = |X|^(n-1) * |X|^1, so by Euclid's First Theorem, if p divides |X|^n it must divide either |X|^(n-1) or |X|^1. We know it divides neither, so p does not divide |X|^n. By induction, this is true for all n > 0.