= Solution
The structure $H_\kappa$ of <hereditarily small sets> is transitive. If $\kappa\le\omega$, all its sets are hereditarily finite, so it cannot satisfy Infinity. Therefore the assumed model has $\kappa>\omega$.
Let $\mu<\kappa$ be a <cardinal>. The <ordinal> $\mu$ belongs to $H_\kappa$, and every actual <subset> of $\mu$ also belongs to $H_\kappa$: its <transitive closure> has size at most $\max(\mu,\omega)<\kappa$. The <Power set> axiom inside $H_\kappa$ therefore produces the actual $\mathcal P(\mu)$, since all <subsets> relevant to the internal definition are present. This <power set> itself belongs to $H_\kappa$, so
$$
2^\mu=|\mathcal P(\mu)|<\kappa.
$$
Thus $\kappa$ is a strong limit. Its regularity was assumed, and we have proved uncountability. Hence \b[$\kappa$ is strongly inaccessible].
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