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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 12686 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) | |
2 | mnfxr 10855 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | xrlenlt 10863 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ ≤ 𝐴 ↔ ¬ 𝐴 < -∞)) | |
4 | 2, 3 | mpan 690 | . 2 ⊢ (𝐴 ∈ ℝ* → (-∞ ≤ 𝐴 ↔ ¬ 𝐴 < -∞)) |
5 | 1, 4 | mpbird 260 | 1 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∈ wcel 2112 class class class wbr 5039 -∞cmnf 10830 ℝ*cxr 10831 < clt 10832 ≤ cle 10833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 |
This theorem is referenced by: ngtmnft 12721 xrre2 12725 xleadd1a 12808 xlt2add 12815 xsubge0 12816 xlesubadd 12818 xlemul1a 12843 supxrmnf 12872 elioc2 12963 iccmax 12976 xrsdsreclblem 20363 leordtvallem2 22062 lecldbas 22070 tgioo 23647 xrtgioo 23657 ioombl 24416 ismbfd 24490 degltlem1 24924 ply1rem 25015 xrdifh 30775 tpr2rico 31530 itg2gt0cn 35518 hbtlem2 40593 supxrgelem 42490 supxrge 42491 suplesup 42492 xrlexaddrp 42505 infxr 42520 infleinf 42525 mnfled 42542 eliocre 42663 fouriersw 43390 |
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