Well-Ordering Principle/Corollary

Corollary to Well-Ordering Principle

Let $\le_1$ denote the restriction of the usual ordering $\le$ to the set $\N_{\ne 0}$ of natural numbers without zero.

The relational structure $\struct {\N_{\ne 0}, \le_1}$ forms a well-ordered set.

Proof

We have by the Well-Ordering Principle that $\struct {\N, \le}$ forms a well-ordered set.

We also have that $\N_{\ne 0}$ is a subset of $\N$.

The result follows from Subset of Well-Ordered Set is Well-Ordered.

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations