# regular cardinal. ZFC

Suppose that $\kappa$ is a regular cardinal such that $2^{\lambda} < \kappa$ for all $\lambda < \kappa$. Prove that $H(\kappa)=\mathcal{V}_{\kappa}$. Which axioms of ZFC does $\mathcal{V}_{\kappa}$ satisfy?
Here, for a given cardinal $\kappa$, $H(\kappa)$ denotes the collection of sets whose transitive closure has cardinality less than $\kappa$. Also, a well orderable infinite cardinal $\kappa$ is regular if $\text{cf}(\kappa) =_c \kappa$, where $\text{cf}$ denotes the cofinality. The set $\mathcal{V}_{\alpha}$ is defined by $\mathcal{V}_{0}= \emptyset$, $\mathcal{V}_{\alpha+1}=\mathcal{P}(\mathcal{V}_{\a lpha})$, and $\mathcal{V}_{\lambda}= \bigcup_{\alpha< \lambda} \mathcal{V}_{\alpha}$ if $\text{Limit}(\lambda)$.