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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0nemnfd | Structured version Visualization version GIF version |
Description: A nonnegative extended real is not minus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
xrge0nemnfd.1 | ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
xrge0nemnfd | ⊢ (𝜑 → 𝐴 ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 11267 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → -∞ ∈ ℝ*) |
3 | iccssxr 13403 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
4 | xrge0nemnfd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) | |
5 | 3, 4 | sselid 3979 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
6 | 0xr 11257 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
8 | mnflt0 13101 | . . . 4 ⊢ -∞ < 0 | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ < 0) |
10 | pnfxr 11264 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → +∞ ∈ ℝ*) |
12 | iccgelb 13376 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴) | |
13 | 7, 11, 4, 12 | syl3anc 1372 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) |
14 | 2, 7, 5, 9, 13 | xrltletrd 13136 | . 2 ⊢ (𝜑 → -∞ < 𝐴) |
15 | 2, 5, 14 | xrgtned 43967 | 1 ⊢ (𝜑 → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5147 (class class class)co 7404 0cc0 11106 +∞cpnf 11241 -∞cmnf 11242 ℝ*cxr 11243 < clt 11244 ≤ cle 11245 [,]cicc 13323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-addrcl 11167 ax-rnegex 11177 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-icc 13327 |
This theorem is referenced by: ovolsplit 44639 caragenuncllem 45163 |
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