Fundamental Theorem of Well-Ordering

Theorem
Let $\struct {A, \preccurlyeq_A}$ and $\struct {B, \preccurlyeq_B}$ be well-ordered classes.

Then either:
 * $\struct {A, \preccurlyeq_A}$ is order isomorphic to a lower section of $\struct {B, \preccurlyeq_B}$, or perhaps all of $\struct {B, \preccurlyeq_B}$

or:
 * $\struct {B, \preccurlyeq_B}$ is order isomorphic to a lower section of $\struct {A, \preccurlyeq_A}$, or perhaps all of $\struct {A, \preccurlyeq_A}$.

Proof
Let $N$ be the class of all order isomorphisms from lower sections of $\struct {A, \preccurlyeq_A}$ which are sets to lower sections of $\struct {B, \preccurlyeq_B}$ which are sets.

Each such order isomorphism is itself a set.

By Isomorphisms between Lower Sections of Well-Ordered Classes are Nested, $N$ is a nest.

$N$ may or may not contain a maximal element the subset relation.

However, note that no order isomorphism from a proper lower section of $\struct {A, \preccurlyeq_A}$ onto a proper lower section of $\struct {B, \preccurlyeq_B}$ can be a maximal element of $N$.

Indeed, suppose $\psi$ is an order isomorphism from a proper lower section $a^\prec$ of $\struct {A, \preccurlyeq_A}$ onto a proper lower section $b^\prec$ of $\struct {B, \preccurlyeq_B}$.

Then $\psi \cup \tuple {a, b}$ is an element of $N$ which is a proper extension of $\psi$.

Let $I = \bigcup N$ be the union of $N$.

Because $N$ is a nest, $I$ must be an order isomorphism from a lower section of $\struct {A, \preccurlyeq_A}$ (not necessarily proper) onto a lower section of $\struct {B, \preccurlyeq_B}$ (not necessarily proper).

It remains to be shown that either:
 * the domain of $I$ is the whole of $A$

or:
 * the image of $I$ is the whole of $B$.

Equivalently, we can instead show that:
 * the domain of $I$ cannot be a proper lower section of $\struct {A, \preccurlyeq_A}$

and:
 * the image of $I$ cannot be a proper lower section of $\struct {B, \preccurlyeq_B}$.

, the above were the case.

Because $I = \bigcup N$, $I$ would then be a maximal element of $N$.

But we have shown that no order isomorphism from a proper lower section of $\struct {A, \preccurlyeq_A}$ onto a proper lower section of $\struct {B, \preccurlyeq_B}$ can be a maximal element of $N$.

Therefore $I$ cannot be an order isomorphism from a proper lower section of $\struct {A, \preccurlyeq_A}$ onto a proper lower section of $\struct {B, \preccurlyeq_B}$.

The result follows.

Also presented as
This can also be presented as:

Let $\struct {A, \preccurlyeq_A}$ and $\struct {B, \preccurlyeq_B}$ be well-ordered classes.

Then either:
 * $\struct {A, \preccurlyeq_A}$ is order isomorphic to a lower section of $\struct {B, \preccurlyeq_B}$

or:
 * $\struct {B, \preccurlyeq_B}$ is order isomorphic to a lower section of $\struct {A, \preccurlyeq_A}$

or:
 * $\struct {A, \preccurlyeq_A}$ is order isomorphic to $\struct {B, \preccurlyeq_B}$.

Also known as
The Fundamental Theorem of Well-Ordering is also known as the comparability theorem for well-orderings, or simply the comparability theorem.