I am reading Jech's book Introduction to Set Theory.
As an introduction to Silver's theorem, it is stated that:
If $\aleph_\alpha$ is a strong limit cardinal and if $k < \aleph_\alpha$ and $\lambda < \aleph_\alpha$ then $k^\lambda < \aleph_\alpha$.
It is stated, that using the above claim, one can prove that:
If for any $\alpha < \omega_1$, $2^\aleph_\alpha = \aleph_{\alpha + 1}$, then, ${\aleph_\alpha}^{\aleph_1} < \aleph_{\omega_1}$?
How can I prove that?
Thank you!
Edit: Now that I think of it, Maybe this is because:
If $2^{\aleph_\alpha} < \aleph_{\alpha + 1}$ implies $\aleph_{\omega_1}$ is strong limit? Am I in the right direction?