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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 11343 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → -∞ ∈ ℝ*) |
3 | iccssxr 13486 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
4 | xrge0nemnfd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) | |
5 | 3, 4 | sselid 4000 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
6 | 0xr 11333 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
8 | mnflt0 13184 | . . . 4 ⊢ -∞ < 0 | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ < 0) |
10 | pnfxr 11340 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → +∞ ∈ ℝ*) |
12 | iccgelb 13459 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴) | |
13 | 7, 11, 4, 12 | syl3anc 1371 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) |
14 | 2, 7, 5, 9, 13 | xrltletrd 13219 | . 2 ⊢ (𝜑 → -∞ < 𝐴) |
15 | 2, 5, 14 | xrgtned 45172 | 1 ⊢ (𝜑 → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2103 ≠ wne 2942 class class class wbr 5169 (class class class)co 7445 0cc0 11180 +∞cpnf 11317 -∞cmnf 11318 ℝ*cxr 11319 < clt 11320 ≤ cle 11321 [,]cicc 13406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-addrcl 11241 ax-rnegex 11251 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-po 5611 df-so 5612 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-1st 8026 df-2nd 8027 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-icc 13410 |
This theorem is referenced by: ovolsplit 45844 caragenuncllem 46368 |
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