Definition 1. An ordinal $\alpha$ is a cardinal number or cardinal if
\[\forall \beta < \alpha : \beta \lnequ \alpha.\]The class of all cardinals is denoted by $\Card$.
Definition 2. Every natural number is a cardinal, called a finite cardinal. A cardinal is infinite if it is not finite.
Proposition 1. $\Card$ is a proper class.
Proof. Show that
\[\Ord \subseteq \bigcup \Card.\]
Proposition 2. Every infinite cardinal is a limit ordinal.
Proof. By SUC > Proposition 2.