Natural Number Less than or Equal to Successor of Another
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Theorem
Let $\N$ be the natural numbers.
Let $m, n \in \N$ such that $m \le n^+$.
Then either:
- $(1): \quad m \le n$
or:
- $(2): \quad m = n^+$
Proof
Let $m \le n^+$.
Suppose $m \le n$ is false.
Then:
- $n^+ \le m$
and because $m \le n^+$:
- $m = n^+$
$\blacksquare$
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 $5.8$