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| Mirrors > Home > MPE Home > Th. List > mnflt | Structured version Visualization version GIF version | ||
| Description: Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.) |
| Ref | Expression |
|---|---|
| mnflt | ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ -∞ = -∞ | |
| 2 | olc 869 | . . . 4 ⊢ ((-∞ = -∞ ∧ 𝐴 ∈ ℝ) → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))) | |
| 3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐴 ∈ ℝ → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
| 4 | 3 | olcd 875 | . 2 ⊢ (𝐴 ∈ ℝ → ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))) |
| 5 | mnfxr 11318 | . . 3 ⊢ -∞ ∈ ℝ* | |
| 6 | rexr 11307 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
| 7 | ltxr 13157 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
| 8 | 5, 6, 7 | sylancr 587 | . 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))) |
| 9 | 4, 8 | mpbird 257 | 1 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 <ℝ cltrr 11159 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 < 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: mnfltd 13166 mnflt0 13167 mnfltxr 13169 xrlttri 13181 xrlttr 13182 xrrebnd 13210 xrre3 13213 qbtwnxr 13242 xrsupsslem 13349 xrub 13354 elico2 13451 elicc2 13452 ioomax 13462 elioomnf 13484 difreicc 13524 icopnfcld 24788 iocmnfcld 24789 xrtgioo 24828 bndth 24990 mbfmax 25684 itg2seq 25777 ellogdm 26681 esumcvgsum 34089 dya2iocbrsiga 34277 dya2icobrsiga 34278 orvclteel 34475 iooelexlt 37363 itg2addnclem 37678 asindmre 37710 dvasin 37711 dvacos 37712 rfcnpre4 45039 infrpge 45362 infxr 45378 infxrunb2 45379 infleinflem2 45382 icccncfext 45902 fourierdlem113 46234 fouriersw 46246 pimgtmnff 46737 iccpartigtl 47410 |
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