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Mirrors > Home > MPE Home > Th. List > xrge0neqmnf | Structured version Visualization version GIF version |
Description: A nonnegative extended real is not equal to minus infinity. (Contributed by Thierry Arnoux, 9-Jun-2017.) (Proof shortened by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
xrge0neqmnf | ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliccxr 12572 | . 2 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ∈ ℝ*) | |
2 | 0xr 10423 | . . 3 ⊢ 0 ∈ ℝ* | |
3 | pnfxr 10430 | . . 3 ⊢ +∞ ∈ ℝ* | |
4 | iccgelb 12542 | . . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴) | |
5 | 2, 3, 4 | mp3an12 1524 | . 2 ⊢ (𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴) |
6 | ge0nemnf 12316 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞) | |
7 | 1, 5, 6 | syl2anc 579 | 1 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4886 (class class class)co 6922 0cc0 10272 +∞cpnf 10408 -∞cmnf 10409 ℝ*cxr 10410 ≤ cle 10412 [,]cicc 12490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-addrcl 10333 ax-rnegex 10343 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-icc 12494 |
This theorem is referenced by: xrge0nre 12591 xrge0adddir 30254 xrge0npcan 30256 hasheuni 30745 esumcvgre 30751 carsgclctunlem2 30979 sge0split 41554 sge0nemnf 41565 |
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