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Mirrors > Home > MPE Home > Th. List > eliooord | Structured version Visualization version GIF version |
Description: Ordering implied by a member of an open interval of reals. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 9-May-2014.) |
Ref | Expression |
---|---|
eliooord | ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliooxr 12785 | . . . 4 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) | |
2 | elioo2 12769 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐴 ∈ (𝐵(,)𝐶) ↔ (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) |
4 | 3 | ibi 268 | . 2 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) |
5 | 3simpc 1142 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶) → (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) | |
6 | 4, 5 | syl 17 | 1 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 ∈ wcel 2105 class class class wbr 5058 (class class class)co 7145 ℝcr 10525 ℝ*cxr 10663 < clt 10664 (,)cioo 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7680 df-2nd 7681 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ioo 12732 |
This theorem is referenced by: elioo4g 12787 iccssioo2 12799 qdensere 23307 zcld 23350 reconnlem2 23364 xrge0tsms 23371 ovolioo 24098 ioorcl2 24102 itgsplitioo 24367 dvferm1lem 24510 dvferm2lem 24512 dvferm 24514 dvlt0 24531 dvivthlem1 24534 lhop1lem 24539 lhop1 24540 lhop2 24541 dvcvx 24546 ftc1lem4 24565 itgsubstlem 24574 itgsubst 24575 pilem2 24969 pilem3 24970 pigt2lt4 24971 tangtx 25020 tanabsge 25021 cosne0 25041 tanord 25049 tanregt0 25050 argimlt0 25123 logneg2 25125 divlogrlim 25145 logno1 25146 logcnlem3 25154 dvloglem 25158 logf1o2 25160 loglesqrt 25266 asinsin 25397 acoscos 25398 atanlogaddlem 25418 atanlogsub 25421 atantan 25428 atanbndlem 25430 scvxcvx 25491 lgamgulmlem2 25535 basellem8 25593 vmalogdivsum2 26042 vmalogdivsum 26043 2vmadivsumlem 26044 chpdifbndlem1 26057 selberg3lem1 26061 selberg3 26063 selberg4lem1 26064 selberg4 26065 selberg3r 26073 selberg4r 26074 selberg34r 26075 pntrlog2bndlem1 26081 pntrlog2bndlem2 26082 pntrlog2bndlem3 26083 pntrlog2bndlem4 26084 pntrlog2bndlem5 26085 pntrlog2bndlem6a 26086 pntrlog2bndlem6 26087 pntrlog2bnd 26088 pntpbnd1a 26089 pntpbnd1 26090 pntpbnd2 26091 pntpbnd 26092 pntibndlem2 26095 pntibndlem3 26096 pntibnd 26097 pntlemd 26098 pntlemb 26101 pntlemr 26106 pnt 26118 padicabv 26134 xrge0tsmsd 30620 fct2relem 31768 logdivsqrle 31821 knoppndvlem3 33751 iooelexlt 34526 relowlssretop 34527 poimir 34807 itg2gt0cn 34829 ftc1cnnclem 34847 radcnvrat 40526 cncfiooicclem1 42056 itgioocnicc 42142 iblcncfioo 42143 amgmwlem 44801 |
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