Cofinal Ordinal Relation is Reflexive

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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 definition of identity mapping.

It follows by Existential Generalisation that:

$\forall a \in x: \exists b \in x: I_x \left({b}\right) \ge a$

$\blacksquare$


Sources