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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 11839 | . 2 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < -∞) | |
2 | mnfxr 9975 | . . 3 ⊢ -∞ ∈ ℝ* | |
3 | xrlenlt 9982 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ ≤ 𝐴 ↔ ¬ 𝐴 < -∞)) | |
4 | 2, 3 | mpan 702 | . 2 ⊢ (𝐴 ∈ ℝ* → (-∞ ≤ 𝐴 ↔ ¬ 𝐴 < -∞)) |
5 | 1, 4 | mpbird 246 | 1 ⊢ (𝐴 ∈ ℝ* → -∞ ≤ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∈ wcel 1977 class class class wbr 4583 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 |
This theorem is referenced by: ngtmnft 11872 xrre2 11875 xleadd1a 11955 xlt2add 11962 xsubge0 11963 xlesubadd 11965 xlemul1a 11990 supxrmnf 12019 elioc2 12107 iccmax 12120 xrsdsreclblem 19611 leordtvallem2 20825 lecldbas 20833 tgioo 22407 xrtgioo 22417 ioombl 23140 ismbfd 23213 degltlem1 23636 ply1rem 23727 xrdifh 28932 tpr2rico 29286 itg2gt0cn 32635 hbtlem2 36713 supxrgelem 38494 supxrge 38495 suplesup 38496 xrlexaddrp 38509 infxr 38524 infleinf 38529 mnfled 38550 eliocre 38581 fouriersw 39124 |
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