Rank of Set Determined by Members

Theorem
Let $S$ be a set.

Let $\operatorname{rank} \left({ S }\right)$ denote the rank of $S$.

Then:


 * $\operatorname{rank} \left({ S }\right) = \bigcap \left\{ x \in \operatorname{On} : \forall y \in S: \operatorname{rank} \left({ y }\right) < x \right\}$

Proof
Let:
 * $T = \bigcap \left\{ x \in \operatorname{On} : \forall y \in S: \operatorname{rank} \left({ y }\right) < x \right\}$

Let $y \in S$.

Then by Membership Rank Inequality:
 * $\operatorname{rank} \left({ x }\right) < \operatorname{rank} \left({ S }\right)$

Therefore:
 * $T \subseteq \operatorname{rank} \left({ S }\right)$

Conversely, take any $x \in T$.

By the definition of $T$:
 * $\forall y \in S: \operatorname{rank} \left({ y }\right) < x$

From Ordinal is Subset of Rank of Small Class iff Not in Von Neumann Hierarchy:
 * $\forall y \in S: y \in V \left({ x }\right)$

where $V \left({x}\right)$ denotes the von Neumann hierarchy.

Therefore by the definition of subset:
 * $S \subseteq V \left({x}\right)$

By the definition of rank.
 * $\operatorname{rank} \left({ S }\right) \le x$

Therefore:
 * $\operatorname{rank} \left({ S }\right) = T$