Cofinal Ordinal Relation is Reflexive

Theorem
Let $x$ be an ordinal.

Then $x$ is cofinal to itself.

That is:


 * $\operatorname{cof} \left({ x,x }\right)$

Proof
Each of the conditions for cofinal ordinals shall be verified:


 * $x \le x$ follows by Set is Subset of Itself.

The mapping $f : x \to x$ can simply be the identity mapping $I_x$.


 * $I_x : x \to x$

Moreover, $a < b \implies I_x \left({ a }\right) < I_x \left({b}\right)$ by the definition of the identity mapping.

Therefore:


 * $I_x$ is strictly increasing.

Finally, $I_x \left({a}\right) \ge a$ by the definition of identity mapping.

It follows that for all $a \in x$, there is a $b \in x$ such that $I_x\left({b}\right) \ge a$, since we may set $b = a$.