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Mirrors > Home > MPE Home > Th. List > mnfltd | Structured version Visualization version GIF version |
Description: Minus infinity is less than any (finite) real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
mnfltd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
mnfltd | ⊢ (𝜑 → -∞ < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfltd.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | mnflt 13136 | . 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → -∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 class class class wbr 5148 ℝcr 11138 -∞cmnf 11277 < clt 11279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5684 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 |
This theorem is referenced by: qbtwnxr 13212 xltnegi 13228 supxrre 13339 infxrre 13348 caucvgrlem 15652 tgioo 24725 reconnlem1 24755 reconnlem2 24756 ovoliunlem1 25444 ovoliun 25447 ioombl1lem2 25501 ismbf3d 25596 dvferm1lem 25929 dvferm2lem 25931 degltlem1 26021 ply1divex 26085 dvdsq1p 26110 logdmnrp 26588 atans2 26876 ply1degltel 33265 ply1degleel 33266 ply1degltlss 33267 ply1degltdimlem 33320 areacirclem5 37185 aks6d1c5lem3 41608 infleinflem2 44753 xrralrecnnge 44772 icoopn 44910 icomnfinre 44937 ressiocsup 44939 ressioosup 44940 preimaiocmnf 44946 limciccioolb 45009 limsupre 45029 limcresioolb 45031 limcleqr 45032 xlimmnfvlem1 45220 fourierdlem32 45527 fourierdlem46 45540 fourierdlem48 45542 fourierdlem49 45543 fourierdlem74 45568 fourierdlem88 45582 fourierdlem95 45589 fourierdlem103 45597 fourierdlem104 45598 fouriersw 45619 ioorrnopnxrlem 45694 hspdifhsp 46004 hspmbllem2 46015 pimgtmnf2 46102 smfsuplem1 46199 |
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