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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 12739 | . 2 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → -∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 class class class wbr 5067 ℝcr 10752 -∞cmnf 10889 < clt 10891 |
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 2016 ax-8 2113 ax-9 2121 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 |
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-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-xp 5571 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 |
This theorem is referenced by: qbtwnxr 12814 xltnegi 12830 supxrre 12941 infxrre 12950 caucvgrlem 15260 tgioo 23717 reconnlem1 23747 reconnlem2 23748 ovoliunlem1 24423 ovoliun 24426 ioombl1lem2 24480 ismbf3d 24575 dvferm1lem 24905 dvferm2lem 24907 degltlem1 24994 ply1divex 25058 dvdsq1p 25082 logdmnrp 25553 atans2 25838 areacirclem5 35632 infleinflem2 42611 xrralrecnnge 42631 icoopn 42766 icomnfinre 42793 ressiocsup 42795 ressioosup 42796 preimaiocmnf 42802 limciccioolb 42865 limsupre 42885 limcresioolb 42887 limcleqr 42888 xlimmnfvlem1 43076 fourierdlem32 43383 fourierdlem46 43396 fourierdlem48 43398 fourierdlem49 43399 fourierdlem74 43424 fourierdlem88 43438 fourierdlem95 43445 fourierdlem103 43453 fourierdlem104 43454 fouriersw 43475 ioorrnopnxrlem 43550 hspdifhsp 43857 hspmbllem2 43868 pimltmnf2 43938 pimgtmnf2 43951 smfsuplem1 44044 |
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