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.

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