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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 867 | . . . 4 ⊢ ((-∞ = -∞ ∧ 𝐴 ∈ ℝ) → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))) | |
3 | 1, 2 | mpan 689 | . . 3 ⊢ (𝐴 ∈ ℝ → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
4 | 3 | olcd 873 | . 2 ⊢ (𝐴 ∈ ℝ → ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))) |
5 | mnfxr 11213 | . . 3 ⊢ -∞ ∈ ℝ* | |
6 | rexr 11202 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
7 | ltxr 13037 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
8 | 5, 6, 7 | sylancr 588 | . 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 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 ℝcr 11051 <ℝ cltrr 11056 +∞cpnf 11187 -∞cmnf 11188 ℝ*cxr 11189 < clt 11190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 |
This theorem is referenced by: mnfltd 13046 mnflt0 13047 mnfltxr 13049 xrlttri 13059 xrlttr 13060 xrrebnd 13088 xrre3 13091 qbtwnxr 13120 xrsupsslem 13227 xrub 13232 elico2 13329 elicc2 13330 ioomax 13340 elioomnf 13362 difreicc 13402 icopnfcld 24134 iocmnfcld 24135 xrtgioo 24172 bndth 24324 mbfmax 25016 itg2seq 25110 ellogdm 25997 esumcvgsum 32690 dya2iocbrsiga 32878 dya2icobrsiga 32879 orvclteel 33075 iooelexlt 35836 itg2addnclem 36132 asindmre 36164 dvasin 36165 dvacos 36166 rfcnpre4 43246 infrpge 43592 infxr 43608 infxrunb2 43609 infleinflem2 43612 icccncfext 44135 fourierdlem113 44467 fouriersw 44479 pimgtmnff 44970 iccpartigtl 45622 |
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