Theorem ne0gt0d

Description: A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	$$⊢\phi \to A\in ℝ$$
	ne0gt0d.2	$$⊢\phi \to 0\le A$$
	ne0gt0d.3	$$⊢\phi \to A\ne 0$$
Assertion	ne0gt0d	$$⊢\phi \to 0<A$$

Proof

Step	Hyp	Ref	Expression
1		ltd.1	$$⊢\phi \to A\in ℝ$$
2		ne0gt0d.2	$$⊢\phi \to 0\le A$$
3		ne0gt0d.3	$$⊢\phi \to A\ne 0$$
4		ne0gt0	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left(A\ne 0\leftrightarrow 0<A\right)$$
5	1 2 4	syl2anc	$$⊢\phi \to \left(A\ne 0\leftrightarrow 0<A\right)$$
6	3 5	mpbid	$$⊢\phi \to 0<A$$