Theorem lt0ne0d

Description: Something less than zero is not zero. Deduction form. (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypothesis	lt0ne0d.1	$$⊢\phi \to A<0$$
Assertion	lt0ne0d	$$⊢\phi \to A\ne 0$$

Proof

Step	Hyp	Ref	Expression
1		lt0ne0d.1	$$⊢\phi \to A<0$$
2		0re	$$⊢0\in ℝ$$
3	2	ltnri	$$⊢\neg 0<0$$
4		breq1	$$⊢A=0\to \left(A<0\leftrightarrow 0<0\right)$$
5	3 4	mtbiri	$$⊢A=0\to \neg A<0$$
6	5	necon2ai	$$⊢A<0\to A\ne 0$$
7	1 6	syl	$$⊢\phi \to A\ne 0$$