Via PCF theory, we can prove in $\mathsf{ZFC}$ alone that if $\aleph_\omega$ is a strong limit cardinal then $2^{\aleph_\omega}<\aleph_{\omega_4}$. However, PCF theory is rather complicated. I'm curious whether there is a "toy" version of PCF theory - or an entirely different argument - which can prove any bound at all on $2^{\aleph_\omega}$ under the assumption that $\aleph_\omega$ is a strong limit cardinal.
To make this more precise, is there a simpler way than PCF theory to prove the following in $\mathsf{ZFC}$ alone?
If $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega}$ is less than the least weakly inaccessible.
(Recall that a weakly inaccessible cardinal is a regular limit cardinal - in particular, in contrast with strong accessibility if $\kappa$ is weakly inaccessible we may still have $\lambda<\kappa\le 2^\lambda$. So the fact above isn't trivial.)