Class of All Cardinals is Subclass of Class of All Ordinals

Theorem

Let $\NN$ denote the class of all cardinals.

Let $\On$ denote the class of all ordinals.

Then:

$\NN \subseteq \On$

Proof

By definition of the class of all cardinals:

$\NN = \set {x \in \On: \exists y: x = \card y}$

Every element of $\NN$ is thus an element of $\On$.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.37$