= Solution
The collection of <hereditarily finite sets> is
$$
V_\omega=\bigcup_{n<\omega}V_n,
$$
where $V_0=\varnothing$ and $V_{n+1}=\mathcal P(V_n)$. Every $V_n$ is finite, and the axiom of infinity together with replacement and union makes $V_\omega$ a set. Hence this collection is not a proper class.
It is transitive: if $x\in V_n$, then $x\subset V_{n-1}$, so every $y\in x$ belongs to $V_{n-1}\subset V_\omega$. It is also $\varphi$-closed. If $x\in V_n$ and
$$
s=\{z\in x:\varphi(z)\},
$$
then $s\subset x\subset V_{n-1}$, so $s\in V_n\subset V_\omega$.
Back to article page
