Order Isomorphism between Ordinals and Proper Class

Theorem
Suppose $(A,\prec)$ is a strict well-ordering

Suppose $A$ is a proper class

Suppose, for every $x \in A$, that the initial segment of $x$ is a set.

Then, we may make the following definitions:

Set $G$ equal to the collection of ordered pairs $(x,y)$ such that:


 * $y \in (A \setminus \operatorname{Im}(x) )$
 * $( A \setminus \operatorname{Im}(x) ) \cap A_y = \varnothing$

Use transfinite recursion to construct a mapping $F$ such that:


 * The domain of $F$ is $\operatorname{On}$.
 * For all ordinals $x$, $F(x) = G(F \restriction x)$.

Then, $F : \operatorname{On} \to A$ is an order isomorphism between $( \operatorname{On}, \in )$ and $( A , \prec )$.

Also see

 * Transfinite Recursion
 * Condition for Injective Mapping on Ordinals
 * Maximal Injective Mapping from Ordinals to a Set