elementary set theory - Question about cardinal arithmetic

Let $|I|\leq \aleph _\alpha$, and let $\left\{ \beta _i \right\} _{i\in I}$ be cardinals with $\beta _i \leq \aleph _{\alpha}$.

Is it true that $\aleph_{\alpha +1} > \sum _{i\in I}\beta _i$?

Using choice, I think it's possible to write $\aleph _{\alpha+1}=\sum _{i\in I}\aleph_{\alpha +1}$, and I think
$$\sum _{i\in I}\aleph_{\alpha +1} > \sum _{i\in I}\beta _i$$ holds because $|I|\leq \aleph _\alpha$, but I'm not sure whether this is true nor how to prove it.