Principle of Mathematical Induction/Proof

Proof
Let $\Z_{\ge n_0}$ denote the set:
 * $S = \set {n \in \Z: n \ge n_0}$

Let $S$ be the set of integers defined as:
 * $S = \set {n \in \Z_{\ge n_0}: \map P n}$

That is, the set of all integers for which $n \ge n_0$ and for which $\map P n$ holds.

From Subset of Set with Propositional Function we have that:
 * $S \subseteq \Z_{\ge n_0}$

From $(1)$ we have that $\map P {n_0}$.

Hence $n_0 \in S$.

Let $k \in S$.

Then $\map P k$ holds.

But by $(2)$, $\map P {k + 1}$ also holds.

This implies $k + 1 \in S$.

So as:
 * $S \subseteq \Z_{\ge n_0}$

and:
 * $S$ satisfies $(1)$ and $(2)$

it follows by the Principle of Finite Induction that $S = \Z_{\ge n_0}$.

Hence for all $n \ge n_0$, $\map P n$ holds.