We have an elliptic curve $E/K$ and $\mathfrak{p}$ is a finite prime of $K$. If $\mathfrak{p}$ is a prime of good reudction for $E$, and $m$ is prime to the characteristic of the residue field of $\mathfrak{p}$, then the reduction map
$$ E(K)[m] \to \tilde{E}$$
is injective.
In Silverman's AEC he proves this for local fields VII.3.1 but his proof uses formal groups. Perhaps because the prior arguments want to include the bad reduction case. Is there a more elementary proof of this for when $K$ is a number field?