Prove that $\displaystyle H(\beth_{\omega})$ has cardinality $\displaystyle \beth_{\omega}$. Which axioms of ZFC are satified by $\displaystyle H(\beth_{\omega})$?

Here, for a given cardinal $\displaystyle \kappa$, $\displaystyle H(\kappa)$ denotes the collection of sets whose transitive closure has cardinality less than $\displaystyle \kappa$. Also, $\displaystyle \beth_{\alpha}$ is defined by $\displaystyle \beth_{0}= \aleph_0=|\mathbb{N}|=\omega$, $\displaystyle \beth_{\beta +1}=2^{\beth_{\beta}}$, $\displaystyle \beth_{\lambda} = \text{sup} \{ \beth_{\beta} | \beta < \lambda \}$, if $\displaystyle \text{Limit}(\lambda)$.

I do not have any good ideas on how to prove this. I would appreciate a few hints. Thanks.

Link:

Zermelo?Fraenkel set theory - Wikipedia, the free encyclopedia