Reciprocal of Real Number is Non-Zero

Theorem

 * $\forall x \in \R: x \ne 0 \implies \dfrac 1 x \ne 0$

Proof
that:
 * $\exists x \in \R_{\ne 0}: \dfrac 1 x = 0$

From Real Zero is Zero Element
 * $\dfrac 1 x \times x = 0$

But from Real Number Axioms: $\R \text M 4$: Inverses:
 * $\dfrac 1 x \times x = 1$

The result follows by Proof by Contradiction.