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

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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

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.52 \ (1)$