Theorem lt0ne0d

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

	Ref	Expression
Hypothesis	lt0ne0d.1	⊢ ( 𝜑 → 𝐴 < 0 )
Assertion	lt0ne0d	⊢ ( 𝜑 → 𝐴 ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		lt0ne0d.1	⊢ ( 𝜑 → 𝐴 < 0 )
2		0re	⊢ 0 ∈ ℝ
3	2	ltnri	⊢ ¬ 0 < 0
4		breq1	⊢ ( 𝐴 = 0 → ( 𝐴 < 0 ↔ 0 < 0 ) )
5	3 4	mtbiri	⊢ ( 𝐴 = 0 → ¬ 𝐴 < 0 )
6	5	necon2ai	⊢ ( 𝐴 < 0 → 𝐴 ≠ 0 )
7	1 6	syl	⊢ ( 𝜑 → 𝐴 ≠ 0 )