= Solution
The <cumulative hierarchy> is defined by <transfinite recursion> on the <ordinals>:
$$
\boxed{V_0=\varnothing,\qquad V_{\alpha+1}=\mathcal P(V_\alpha),\qquad V_\lambda=\bigcup_{\beta<\lambda}V_\beta\quad(\lambda\text{ a nonzero limit}).}
$$
The recursion theorem of <ZF> supplies each <set> stage. The whole universe is $V=\bigcup_{\alpha\in\mathrm{Ord}}V_\alpha$, a proper-class <union>. Equivalently, $x\in V_\alpha$ exactly when the rank of $x$ is less than $\alpha$, where $\operatorname{rank}(x)=\sup\{\operatorname{rank}(z)+1:z\in x\}$.
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