User:Dfeuer/Successor of Natural Number is Natural Number

Theorem

Let $n$ be a natural number.

Then $n^+$ is also a natural number.

Proof

Let $a$ be an User:Dfeuer/Definition:Inductive Set.

Since $n$ is a natural number, $n \in a$.

Thus by the definition of inductive set, $n^+ \in a$.

Since this holds for all inductive sets $a$, $n^+$ is a natural number.

$\blacksquare$