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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 11349 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → -∞ ∈ ℝ*) |
3 | iccssxr 13492 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
4 | xrge0nemnfd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) | |
5 | 3, 4 | sselid 4006 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
6 | 0xr 11339 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
8 | mnflt0 13190 | . . . 4 ⊢ -∞ < 0 | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ < 0) |
10 | pnfxr 11346 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → +∞ ∈ ℝ*) |
12 | iccgelb 13465 | . . . 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 13225 | . 2 ⊢ (𝜑 → -∞ < 𝐴) |
15 | 2, 5, 14 | xrgtned 45239 | 1 ⊢ (𝜑 → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 (class class class)co 7450 0cc0 11186 +∞cpnf 11323 -∞cmnf 11324 ℝ*cxr 11325 < clt 11326 ≤ cle 11327 [,]cicc 13412 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-addrcl 11247 ax-rnegex 11257 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-ov 7453 df-oprab 7454 df-mpo 7455 df-1st 8032 df-2nd 8033 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-icc 13416 |
This theorem is referenced by: ovolsplit 45911 caragenuncllem 46435 |
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