Rank of Ordinal

Theorem
Let $x$ be an ordinal.

Let $\map {\operatorname {rank} } x$ denote the rank of $x$.

Then:


 * $\map {\operatorname {rank} } x = x$

Proof
The proof shall proceed by Transfinite Induction (Strong Induction) on $x$.

Suppose $\forall y \in x: \map {\operatorname {rank} } y = y$.

Then: