Ordering of Integers is Reversed by Negation

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Theorem

Let $x, y \in \Z$ such that $x > y$.

Then:

$-x < -y$

Proof

From the formal definition of integers, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.

Let $x = \eqclass {a, b} {}$ and $y = \eqclass {c, d} {}$ for some $x, y \in \Z$.

We have:

 $\ds x$ $>$ $\ds y$ $\ds \leadsto \ \$ $\ds \eqclass {a, b} {}$ $>$ $\ds \eqclass {c, d} {}$ Definition of Integer $\ds \leadsto \ \$ $\ds a + d$ $>$ $\ds b + c$ Definition of Strict Ordering on Integers $\ds \leadsto \ \$ $\ds b + c$ $<$ $\ds a + d$ $\ds \leadsto \ \$ $\ds \eqclass {b, a} {}$ $<$ $\ds \eqclass {d, c} {}$ $\ds \leadsto \ \$ $\ds -\eqclass {a, b} {}$ $<$ $\ds -\eqclass {c, d} {}$ Negative of Integer $\ds \leadsto \ \$ $\ds -x$ $<$ $\ds -y$ Definition of Integer

$\blacksquare$