User:Dfeuer/Existence of Set that is not a Natural Number implies Axiom of Infinity

Theorem

Suppose there is a set $x$ such that $x$ is not a User:Dfeuer/Definition:Natural Number.

Then the User:Dfeuer/Axiom of Infinity holds. That is, the class $\omega$ of natural numbers is a set.

Proof

Since $x$ is not a natural number, there is an inductive set $a$ such that $x \notin a$.

Thus by User:Dfeuer/Existence of Inductive Set implies Axiom of Infinity, the User:Dfeuer/Axiom of Infinity holds.

$\blacksquare$