I know all subrings of $\mathbb{Q}$ is ｛$\mathbb{Zs}$｜$\mathbb{S}$ is complement of union of $\mathbb{pZ}$ where $p$ is a prime｝.
If the number of prime numbers is finite, the number of distinct subring of $\mathbb{Q}$ is also finite. If we could prove infinity of the number of distinct subrings of $\mathbb{Q}$ without using the fact that there are infinitely many primes, we can recover the proof of infiniteness of prime numbers.
So, I am seeking the proof of the title.Thank you for your help.