Rank bounding for sets in the cumulative hierarchy ($a\subseteq V⟹a\in V$)

If every member of a set $a$ belongs to some cumulative-hierarchy stage, replacement collects the least such stages. Their ordinal supremum gives a common stage $V_{\beta }$ containing every element of $a$, so $a\in V_{\beta +1}$. The argument also works without foundation and bounds a set of well-founded elements inside the hierarchy.

Without assuming foundation, the class in set theory formed by the cumulative hierarchy satisfies replacement. Relativize a functional logical formula to that definable class in set theory and use ambient replacement to collect its range. Every output belongs to the hierarchy class in set theory, and the rank-bounding result places the range set itself in that class in set theory.