User:Dfeuer/Existence of Inductive Set implies Axiom of Infinity

Theorem
Suppose that there is an inductive set $a$.

Then the axiom of infinity holds. That is, the class $\omega$ of all natural numbers is a set.

Proof
Let $x \in \omega$.

Then by the definition of natural number, $x$ is in every inductive set, so in particular $x \in a$.

Thus by the definition of subclass, $\omega \subseteq a$.

By the subset axiom, $\omega$ is a set.