Theorem recne0d

Description: The reciprocal of a nonzero number is nonzero. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	div1d.1	$$⊢\phi \to A\in ℂ$$
	reccld.2	$$⊢\phi \to A\ne 0$$
Assertion	recne0d	$$⊢\phi \to \frac{1}{A}\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		div1d.1	$$⊢\phi \to A\in ℂ$$
2		reccld.2	$$⊢\phi \to A\ne 0$$
3		recne0	$$⊢\left(A\in ℂ\wedge A\ne 0\right)\to \frac{1}{A}\ne 0$$
4	1 2 3	syl2anc	$$⊢\phi \to \frac{1}{A}\ne 0$$