Theorem ge0nemnf

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 ≠ -∞ )

Proof

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 ≠ -∞ )