Theorem gt0ne0d

Description: Positive implies nonzero. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypothesis	gt0ne0d.1	$$⊢\phi \to 0<A$$
Assertion	gt0ne0d	$$⊢\phi \to A\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		gt0ne0d.1	$$⊢\phi \to 0<A$$
2		0re	$$⊢0\in ℝ$$
3		ltne	$$⊢\left(0\in ℝ\wedge 0<A\right)\to A\ne 0$$
4	2 1 3	sylancr	$$⊢\phi \to A\ne 0$$