Let $G$ be an elementary abelian pro-$p$ group. Then we have that
$$G=\prod\limits_{\mathfrak{m}}C_p$$
where $\mathfrak{m}$ is a cardinal. We have this as a direct consequence of Theorem 4.3.8. from Profinite Groups, Luis Ribes Pavel Zalesskii:
We also know that $G$ is a profinite group. So it can be described as an inverse limit of finite groups: $$G=\lim\limits_{\substack{\leftarrow\\ i\in I}} G_i$$
I want to show that $\mathfrak{m}=|I|$.
I was thinking about using that: When $G$ is an infinite profinite group, it follows from Theorem 2.1.3 (from Profinite Groups, Luis Ribes Pavel Zalesskii) that the cardinal of any fundamental system of neighborhoods of 1 consisting of open subgroups is the same. But I am struggling to reach a conclusion.