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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 13169 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) | |
2 | mnfxr 11316 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | xrlenlt 11324 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ ≤ 𝐴 ↔ ¬ 𝐴 < -∞)) | |
4 | 2, 3 | mpan 690 | . 2 ⊢ (𝐴 ∈ ℝ* → (-∞ ≤ 𝐴 ↔ ¬ 𝐴 < -∞)) |
5 | 1, 4 | mpbird 257 | 1 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∈ wcel 2106 class class class wbr 5148 -∞cmnf 11291 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 |
This theorem is referenced by: mnfled 13175 ngtmnft 13205 xrre2 13209 xleadd1a 13292 xlt2add 13299 xsubge0 13300 xlesubadd 13302 xlemul1a 13327 supxrmnf 13356 elioc2 13447 iccmax 13460 xrsdsreclblem 21448 leordtvallem2 23235 lecldbas 23243 tgioo 24832 xrtgioo 24842 ioombl 25614 ismbfd 25688 degltlem1 26126 ply1rem 26220 xrdifh 32789 tpr2rico 33873 itg2gt0cn 37662 hbtlem2 43113 supxrgelem 45287 supxrge 45288 suplesup 45289 xrlexaddrp 45302 infxr 45317 infleinf 45322 eliocre 45462 fouriersw 46187 |
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