# Real Number is Greater than Zero iff its Negative is Less than Zero

## Theorem

$\forall x \in \R: x > 0 \iff \paren {-x} < 0$

## Proof

Let $x > 0$.

 $\displaystyle x$ $>$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle x + \paren {-x}$ $>$ $\displaystyle 0 + \paren {-x}$ Real Number Axioms: $\R \text O 2$: compatibility with addition $\displaystyle \leadsto \ \$ $\displaystyle 0$ $>$ $\displaystyle 0 + \paren {-x}$ Real Number Axioms: $\R \text A 4$: Inverses $\displaystyle \leadsto \ \$ $\displaystyle 0$ $>$ $\displaystyle \paren {-x}$ Real Number Axioms: $\R \text A 3$: Identity

$\Box$

Let $\paren {-x} < 0$.

 $\displaystyle \paren {-x}$ $<$ $\displaystyle 0$ $\displaystyle \leadsto \ \$ $\displaystyle x + \paren {-x}$ $<$ $\displaystyle x + 0$ Real Number Axioms: $\R \text O 2$: compatibility with addition $\displaystyle \leadsto \ \$ $\displaystyle 0$ $<$ $\displaystyle x + 0$ Real Number Axioms: $\R \text A 4$: Inverses $\displaystyle \leadsto \ \$ $\displaystyle 0$ $<$ $\displaystyle x$ Real Number Axioms: $\R \text A 3$: Identity

$\blacksquare$