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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 13165 | . 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → -∞ < 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 -∞cmnf 11293 < clt 11295 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 |
| This theorem is referenced by: qbtwnxr 13242 xltnegi 13258 supxrre 13369 infxrre 13378 caucvgrlem 15709 tgioo 24817 reconnlem1 24848 reconnlem2 24849 ovoliunlem1 25537 ovoliun 25540 ioombl1lem2 25594 ismbf3d 25689 dvferm1lem 26022 dvferm2lem 26024 degltlem1 26111 ply1divex 26176 dvdsq1p 26202 logdmnrp 26683 atans2 26974 ply1degltel 33615 ply1degleel 33616 ply1degltlss 33617 ply1degltdimlem 33673 areacirclem5 37719 aks6d1c5lem3 42138 infleinflem2 45382 xrralrecnnge 45401 icoopn 45538 icomnfinre 45565 ressiocsup 45567 ressioosup 45568 preimaiocmnf 45574 limciccioolb 45636 limsupre 45656 limcresioolb 45658 limcleqr 45659 xlimmnfvlem1 45847 fourierdlem32 46154 fourierdlem46 46167 fourierdlem48 46169 fourierdlem49 46170 fourierdlem74 46195 fourierdlem88 46209 fourierdlem95 46216 fourierdlem103 46224 fourierdlem104 46225 fouriersw 46246 ioorrnopnxrlem 46321 hspdifhsp 46631 hspmbllem2 46642 pimgtmnf2 46729 smfsuplem1 46826 |
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