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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 2726 | . . . 4 ⊢ -∞ = -∞ | |
2 | olc 866 | . . . 4 ⊢ ((-∞ = -∞ ∧ 𝐴 ∈ ℝ) → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ ℝ → ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))) |
4 | 3 | olcd 872 | . 2 ⊢ (𝐴 ∈ ℝ → ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ)))) |
5 | mnfxr 11321 | . . 3 ⊢ -∞ ∈ ℝ* | |
6 | rexr 11310 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
7 | ltxr 13149 | . . 3 ⊢ ((-∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))) | |
8 | 5, 6, 7 | sylancr 585 | . 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ↔ ((((-∞ ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ -∞ <ℝ 𝐴) ∨ (-∞ = -∞ ∧ 𝐴 = +∞)) ∨ ((-∞ ∈ ℝ ∧ 𝐴 = +∞) ∨ (-∞ = -∞ ∧ 𝐴 ∈ ℝ))))) |
9 | 4, 8 | mpbird 256 | 1 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 class class class wbr 5153 ℝcr 11157 <ℝ cltrr 11162 +∞cpnf 11295 -∞cmnf 11296 ℝ*cxr 11297 < clt 11298 |
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 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-xp 5688 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 |
This theorem is referenced by: mnfltd 13158 mnflt0 13159 mnfltxr 13161 xrlttri 13172 xrlttr 13173 xrrebnd 13201 xrre3 13204 qbtwnxr 13233 xrsupsslem 13340 xrub 13345 elico2 13442 elicc2 13443 ioomax 13453 elioomnf 13475 difreicc 13515 icopnfcld 24775 iocmnfcld 24776 xrtgioo 24813 bndth 24975 mbfmax 25669 itg2seq 25763 ellogdm 26666 esumcvgsum 33921 dya2iocbrsiga 34109 dya2icobrsiga 34110 orvclteel 34306 iooelexlt 37069 itg2addnclem 37372 asindmre 37404 dvasin 37405 dvacos 37406 rfcnpre4 44633 infrpge 44966 infxr 44982 infxrunb2 44983 infleinflem2 44986 icccncfext 45508 fourierdlem113 45840 fouriersw 45852 pimgtmnff 46343 iccpartigtl 46995 |
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