Suppose that $\displaystyle \kappa$ is a regular cardinal such that $\displaystyle 2^{\lambda} < \kappa$ for all $\displaystyle \lambda < \kappa$. Prove that $\displaystyle H(\kappa)=\mathcal{V}_{\kappa}$. Which axioms of ZFC does $\displaystyle \mathcal{V}_{\kappa}$ satisfy?

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, a well orderable infinite cardinal $\displaystyle \kappa$ is regular if $\displaystyle \text{cf}(\kappa) =_c \kappa$, where $\displaystyle \text{cf}$ denotes the cofinality. The set $\displaystyle \mathcal{V}_{\alpha}$ is defined by $\displaystyle \mathcal{V}_{0}= \emptyset$, $\displaystyle \mathcal{V}_{\alpha+1}=\mathcal{P}(\mathcal{V}_{\a lpha})$, and $\displaystyle \mathcal{V}_{\lambda}= \bigcup_{\alpha< \lambda} \mathcal{V}_{\alpha}$ if $\displaystyle \text{Limit}(\lambda)$.

I am not sure how to prove this. I would appreciate a few hints or suggestions. Thank you.

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