ge0gtmnf - Metamath Proof Explorer

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Theorem ge0gtmnf 11877

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 11835	. 2 ⊢ -∞ < 0
2		mnfxr 9975	. . . 4 ⊢ -∞ ∈ ℝ^*
3		0xr 9965	. . . 4 ⊢ 0 ∈ ℝ^*
4		xrltletr 11864	. . . 4 ⊢ ((-∞ ∈ ℝ^* ∧ 0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((-∞ < 0 ∧ 0 ≤ 𝐴) → -∞ < 𝐴))
5	2, 3, 4	mp3an12 1406	. . 3 ⊢ (𝐴 ∈ ℝ^* → ((-∞ < 0 ∧ 0 ≤ 𝐴) → -∞ < 𝐴))
6	5	imp 444	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ (-∞ < 0 ∧ 0 ≤ 𝐴)) → -∞ < 𝐴)
7	1, 6	mpanr1 715	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 0 ≤ 𝐴) → -∞ < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 class class class wbr 4583 0cc0 9815 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959

This theorem is referenced by: ge0nemnf 11878 xrrege0 11879 pcgcd1 15419 pnfnei 20834 isnghm3 22339 ovolunnul 23075 radcnvle 23978 psercnlem1 23983