Theorem ne0gt0

Description: A nonzero nonnegative number is positive. (Contributed by NM, 20-Nov-2007)

	Ref	Expression
Assertion	ne0gt0	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 𝐴 ≠ 0 ↔ 0 < 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		lttri2	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐴 ≠ 0 ↔ ( 𝐴 < 0 ∨ 0 < 𝐴 ) ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ≠ 0 ↔ ( 𝐴 < 0 ∨ 0 < 𝐴 ) ) )
4	3	adantr	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 𝐴 ≠ 0 ↔ ( 𝐴 < 0 ∨ 0 < 𝐴 ) ) )
5		lenlt	⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 ≤ 𝐴 ↔ ¬ 𝐴 < 0 ) )
6	1 5	mpan	⊢ ( 𝐴 ∈ ℝ → ( 0 ≤ 𝐴 ↔ ¬ 𝐴 < 0 ) )
7	6	biimpa	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ¬ 𝐴 < 0 )
8		biorf	⊢ ( ¬ 𝐴 < 0 → ( 0 < 𝐴 ↔ ( 𝐴 < 0 ∨ 0 < 𝐴 ) ) )
9	7 8	syl	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 0 < 𝐴 ↔ ( 𝐴 < 0 ∨ 0 < 𝐴 ) ) )
10	4 9	bitr4d	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( 𝐴 ≠ 0 ↔ 0 < 𝐴 ) )