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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 13436 | . . . 4 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) | |
2 | elioo2 13419 | . . . 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 266 | . 2 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) |
5 | 3simpc 1147 | . 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 205 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 class class class wbr 5153 (class class class)co 7424 ℝcr 11157 ℝ*cxr 11297 < clt 11298 (,)cioo 13378 |
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-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-pre-lttri 11232 ax-pre-lttrn 11233 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 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-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-ioo 13382 |
This theorem is referenced by: elioo4g 13438 iccssioo2 13451 qdensere 24777 zcld 24820 reconnlem2 24834 xrge0tsms 24841 ovolioo 25588 ioorcl2 25592 itgsplitioo 25858 dvferm1lem 26007 dvferm2lem 26009 dvferm 26011 dvlt0 26029 dvivthlem1 26032 lhop1lem 26037 lhop1 26038 lhop2 26039 dvcvx 26044 ftc1lem4 26065 itgsubstlem 26074 itgsubst 26075 pilem2 26482 pilem3 26483 pigt2lt4 26484 tangtx 26533 tanabsge 26534 cosne0 26556 cos0pilt1 26559 tanord 26565 tanregt0 26566 argimlt0 26640 logneg2 26642 divlogrlim 26662 logno1 26663 logcnlem3 26671 dvloglem 26675 logf1o2 26677 loglesqrt 26789 asinsin 26920 acoscos 26921 atanlogaddlem 26941 atanlogsub 26944 atantan 26951 atanbndlem 26953 scvxcvx 27014 lgamgulmlem2 27058 basellem8 27116 vmalogdivsum2 27567 vmalogdivsum 27568 2vmadivsumlem 27569 chpdifbndlem1 27582 selberg3lem1 27586 selberg3 27588 selberg4lem1 27589 selberg4 27590 selberg3r 27598 selberg4r 27599 selberg34r 27600 pntrlog2bndlem1 27606 pntrlog2bndlem2 27607 pntrlog2bndlem3 27608 pntrlog2bndlem4 27609 pntrlog2bndlem5 27610 pntrlog2bndlem6a 27611 pntrlog2bndlem6 27612 pntrlog2bnd 27613 pntpbnd1a 27614 pntpbnd1 27615 pntpbnd2 27616 pntpbnd 27617 pntibndlem2 27620 pntibndlem3 27621 pntibnd 27622 pntlemd 27623 pntlemb 27626 pntlemr 27631 pnt 27643 padicabv 27659 xrge0tsmsd 32926 fct2relem 34443 logdivsqrle 34496 knoppndvlem3 36217 iooelexlt 37069 relowlssretop 37070 poimir 37354 itg2gt0cn 37376 ftc1cnnclem 37392 aks4d1p1p5 41774 radcnvrat 43988 cncfiooicclem1 45514 itgioocnicc 45598 iblcncfioo 45599 amgmwlem 48550 |
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