Definition:Rank (Set Theory)

Definition
Let $A$ be a class.

Let $V$ denote the von Neumann hierarchy.

Then, the rank of $A$ is the smallest ordinal $x$ such that $A \in V\left({x+1}\right)$, given that $x$ exists.

Notation
The rank of the class $A$ is sometimes denoted as $\operatorname{rank} \left({ A }\right)$.