cardinality, ZFC

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

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

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