Reciprocal of Real Number is Non-Zero

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


Aiming for a contradiction, suppose 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.