= Solution
Consider the structure $(M,\in)$.
For extensionality, let $x,y\in M$ have the same members in $M$. Since $M$ is transitive, every actual member of $x$ or $y$ lies in $M$. Thus $x$ and $y$ have the same members in $V$, and ambient extensionality gives $x=y$.
By hypothesis $\varnothing\in M$, and it has no members in the induced structure, so the empty-set axiom holds. If $x,y\in M$, then $\{x,y\}\in M$ by closure, and its internal members are exactly $x$ and $y$, proving pairing.
Finally, if $x\in M$, then $\bigcup x\in M$. For $z\in M$,
$$
z\in\bigcup x
\quad\Longleftrightarrow\quad
\exists y\in x\ (z\in y).
$$
Every such $y$ lies in $M$ by transitivity, so the same equivalence holds internally. Hence union is satisfied. This is the <basic set-theoretic axioms inherited by a transitive class> argument.
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