A set $\mathcal{H}=\{h_1,h_2,...,h_{k_0}\}$ is admissible if it fails to contain a member of at least one residue class modulo $p$ for any prime $p$. According to Zhang, the set $\mathcal{H}$ being composed of $k_0$ distinct primes, each of which is larger than $k_0$, implies that $\mathcal{H}$ is admissible.
It's clear to me that $\mathcal{H}$ cannot "fail" admissibility with respect to any prime $p$ larger than $k_0$, since $\mathcal{H}$ contains fewer than $p$ elements and therefore cannot contain $p$ distinct residue classes modulo $p$. But how do we know that $\mathcal{H}$ can't contain all the residue classes for a prime $p$ with $p\leq k_0$?
For every prime $p\leq k_0$, $\mathcal H$ fails to contain an element from the residue class $0\pmod p$, since $\mathcal H$ consists only of primes which strictly exceed $p$.