Problem 16 Proof Recitation

Proof by Joelle Walker


Statement: For all natural numbers x,y, if $y<Succ(x),$ then $y≤x$
Proof: Assume that for natural numbers, x,y that $y<Succ(x),$. $y<Succ(x),$$y∈Succ(x),$→ ($y∈$xU{x}). So 1.$y∈x$ or 2. $y∈${x} by the properties proved in 15a.
1. ($y∈x$) → ($y<x$)
2. ($y∈${x}) → ($y=x$)
And by trichotomy proved by 15b, (($y<x$) or ($y=x$)) → $y≤x$ QED.

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