Minimally Inductive Set forms Peano Structure

Theorem
Let $\omega$ be the minimal infinite successor set.

Then $\omega$ fulfils Peano's axioms, and hence $\omega$ is a Peano structure.

Proof
From the definition:

P1
$\omega \ne \varnothing$ by definition.

P2
Let $s: \omega \to \omega$ be defined as:
 * $\forall X \in \omega: s \left({X}\right) = X^+$

where $X^+$ is the successor of $X$.

It will be shown later that $s$ as defined here has the properties required for $\omega$ to be a Peano structure.

P3

 * P3: $\forall m, n \in P: s \left({m}\right) = s \left({n}\right) \implies m = n$