Description: A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | ge0nemnf | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) → 𝐴 ≠ -∞ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ge0gtmnf | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) → -∞ < 𝐴 ) | |
2 | ngtmnft | ⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 = -∞ ↔ ¬ -∞ < 𝐴 ) ) | |
3 | 2 | adantr | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) → ( 𝐴 = -∞ ↔ ¬ -∞ < 𝐴 ) ) |
4 | 3 | necon2abid | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) → ( -∞ < 𝐴 ↔ 𝐴 ≠ -∞ ) ) |
5 | 1 4 | mpbid | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ) → 𝐴 ≠ -∞ ) |