Theorem gt0ne0

Description: Positive implies nonzero. (Contributed by NM, 3-Oct-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	gt0ne0	$$⊢\left(A\in ℝ\wedge 0<A\right)\to A\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		0red	$$⊢A\in ℝ\to 0\in ℝ$$
2		ltne	$$⊢\left(0\in ℝ\wedge 0<A\right)\to A\ne 0$$
3	1 2	sylan	$$⊢\left(A\in ℝ\wedge 0<A\right)\to A\ne 0$$