This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )