Natural Numbers form Inductive Set/Proof 1

Proof
By definition of the natural numbers:


 * $\N = \ds \bigcap \II$

where $\II$ is the collection of all inductive sets.

Suppose that $n \in \N$.

Then by definition of intersection:


 * $\forall I \in \II: n \in I$

Because all these $I$ are inductive:


 * $\forall I \in \II: n + 1 \in I$

Again by definition of intersection:


 * $n + 1 \in \N$