ge0gtmnf - Metamath Proof Explorer

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Theorem ge0gtmnf 13214

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

**Assertion**
Ref	Expression
ge0gtmnf	⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)

**Proof of Theorem ge0gtmnf**
Step	Hyp	Ref	Expression
1		mnflt0 13167	. 2 ⊢ -∞ < 0
2		mnfxr 11318	. . . 4 ⊢ -∞ ∈ ℝ^*
3		0xr 11308	. . . 4 ⊢ 0 ∈ ℝ^*
4		xrltletr 13199	. . . 4 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((-∞ < 0 ∧ 0 ≤ 𝐴) → -∞ < 𝐴))
5	2, 3, 4	mp3an12 1453	. . 3 ⊢ (𝐴 ∈ ℝ^* → ((-∞ < 0 ∧ 0 ≤ 𝐴) → -∞ < 𝐴))
6	5	imp 406	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ (-∞ < 0 ∧ 0 ≤ 𝐴)) → -∞ < 𝐴)
7	1, 6	mpanr1 703	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 0cc0 11155 -∞cmnf 11293 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301

This theorem is referenced by: ge0nemnf 13215 xrrege0 13216 pcgcd1 16915 pnfnei 23228 isnghm3 24746 ovolunnul 25535 radcnvle 26463 psercnlem1 26469