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

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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

$\blacksquare$


Sources