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Metamath Proof Explorer

Theorem ge0gtmnf

Description: A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Assertion ge0gtmnf ⊢ A ∈ ℝ * ∧ 0 ≤ A → −∞ < A

Proof

Step Hyp Ref Expression
1 mnflt0 ⊢ −∞ < 0
2 mnfxr ⊢ −∞ ∈ ℝ *
3 0xr ⊢ 0 ∈ ℝ *
4 xrltletr ⊢ −∞ ∈ ℝ * ∧ 0 ∈ ℝ * ∧ A ∈ ℝ * → −∞ < 0 ∧ 0 ≤ A → −∞ < A
5 2 3 4 mp3an12 ⊢ A ∈ ℝ * → −∞ < 0 ∧ 0 ≤ A → −∞ < A
6 5 imp ⊢ A ∈ ℝ * ∧ −∞ < 0 ∧ 0 ≤ A → −∞ < A
7 1 6 mpanr1 ⊢ A ∈ ℝ * ∧ 0 ≤ A → −∞ < A