Minimally Inductive Set is Infinite Cardinal

Theorem

$\omega$, the minimally inductive set, is an element of the class of infinite cardinals $\NN'$.

Proof

By Cardinal Number Less than Ordinal: Corollary:

$\card \omega \le \omega$

Moreover, for any $n \in \omega$, by Cardinal of Finite Ordinal:

$\card n < \card {n + 1} \le \card \omega$

Thus by Cardinal of Finite Ordinal:

$n \in \card \omega$

Therefore:

$\omega = \card \omega$

By Cardinal of Cardinal Equal to Cardinal: Corollary:

$\omega \in \NN'$

$\blacksquare$