Natural Number m is Less than n implies n is not Greater than Successor of n/Proof using Von Neumann Construction

Theorem

Let $\N$ be the natural numbers.

Let $m, n \in \N$.

Then:

$m < n \implies m + 1 \le n$

Proof

Let $\N$ be defined as the von Neumann construction $\omega$.

By definition of the ordering on von Neumann construction:

$m \le n \iff m \subseteq n$

From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is minimally inductive class under the successor mapping.

The result is then a direct application of Characteristics of Minimally Inductive Class under Progressing Mapping: Image of Proper Subset is Subset:

$m \subset n \implies m^+ \subseteq n$

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 5$ Applications to natural numbers: Theorem $5.3 \ (2)$