Theorem rerecne0d

Description: The reciprocal of a nonzero number is nonzero. (Contributed by SN, 4-Apr-2026)

	Ref	Expression
Hypotheses	sn-rereccld.a	$$⊢\phi \to A\in ℝ$$
	sn-rereccld.z	$$⊢\phi \to A\ne 0$$
Assertion	rerecne0d	$$⊢\phi \to 1{/}_{ℝ}A\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		sn-rereccld.a	$$⊢\phi \to A\in ℝ$$
2		sn-rereccld.z	$$⊢\phi \to A\ne 0$$
3		ax-1ne0	$$⊢1\ne 0$$
4		1red	$$⊢\phi \to 1\in ℝ$$
5	4 1 2	redivne0bd	$$⊢\phi \to \left(1\ne 0\leftrightarrow 1{/}_{ℝ}A\ne 0\right)$$
6	3 5	mpbii	$$⊢\phi \to 1{/}_{ℝ}A\ne 0$$