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Mirrors > Home > MPE Home > Th. List > mnfle | Structured version Visualization version GIF version |
Description: Minus infinity is less than or equal to any extended real. (Contributed by NM, 19-Jan-2006.) |
Ref | Expression |
---|---|
mnfle | ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nltmnf 12274 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) | |
2 | mnfxr 10434 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | xrlenlt 10442 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ ≤ 𝐴 ↔ ¬ 𝐴 < -∞)) | |
4 | 2, 3 | mpan 680 | . 2 ⊢ (𝐴 ∈ ℝ* → (-∞ ≤ 𝐴 ↔ ¬ 𝐴 < -∞)) |
5 | 1, 4 | mpbird 249 | 1 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∈ wcel 2107 class class class wbr 4886 -∞cmnf 10409 ℝ*cxr 10410 < clt 10411 ≤ cle 10412 |
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 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 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-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 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-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 |
This theorem is referenced by: ngtmnft 12309 xrre2 12313 xleadd1a 12395 xlt2add 12402 xsubge0 12403 xlesubadd 12405 xlemul1a 12430 supxrmnf 12459 elioc2 12548 iccmax 12561 xrsdsreclblem 20188 leordtvallem2 21423 lecldbas 21431 tgioo 23007 xrtgioo 23017 ioombl 23769 ismbfd 23843 degltlem1 24269 ply1rem 24360 xrdifh 30106 tpr2rico 30556 itg2gt0cn 34090 hbtlem2 38653 supxrgelem 40461 supxrge 40462 suplesup 40463 xrlexaddrp 40476 infxr 40491 infleinf 40496 mnfled 40517 eliocre 40644 fouriersw 41375 |
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