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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 13445 | . . . 4 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) | |
| 2 | elioo2 13428 | . . . 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 267 | . 2 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐴 ∈ ℝ ∧ 𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) |
| 5 | 3simpc 1151 | . 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 206 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 ℝ*cxr 11294 < clt 11295 (,)cioo 13387 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 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-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 |
| This theorem is referenced by: elioo4g 13447 iccssioo2 13460 qdensere 24790 zcld 24835 reconnlem2 24849 xrge0tsms 24856 ovolioo 25603 ioorcl2 25607 itgsplitioo 25873 dvferm1lem 26022 dvferm2lem 26024 dvferm 26026 dvlt0 26044 dvivthlem1 26047 lhop1lem 26052 lhop1 26053 lhop2 26054 dvcvx 26059 ftc1lem4 26080 itgsubstlem 26089 itgsubst 26090 pilem2 26496 pilem3 26497 pigt2lt4 26498 tangtx 26547 tanabsge 26548 cosne0 26571 cos0pilt1 26574 tanord 26580 tanregt0 26581 argimlt0 26655 logneg2 26657 divlogrlim 26677 logno1 26678 logcnlem3 26686 dvloglem 26690 logf1o2 26692 loglesqrt 26804 asinsin 26935 acoscos 26936 atanlogaddlem 26956 atanlogsub 26959 atantan 26966 atanbndlem 26968 scvxcvx 27029 lgamgulmlem2 27073 basellem8 27131 vmalogdivsum2 27582 vmalogdivsum 27583 2vmadivsumlem 27584 chpdifbndlem1 27597 selberg3lem1 27601 selberg3 27603 selberg4lem1 27604 selberg4 27605 selberg3r 27613 selberg4r 27614 selberg34r 27615 pntrlog2bndlem1 27621 pntrlog2bndlem2 27622 pntrlog2bndlem3 27623 pntrlog2bndlem4 27624 pntrlog2bndlem5 27625 pntrlog2bndlem6a 27626 pntrlog2bndlem6 27627 pntrlog2bnd 27628 pntpbnd1a 27629 pntpbnd1 27630 pntpbnd2 27631 pntpbnd 27632 pntibndlem2 27635 pntibndlem3 27636 pntibnd 27637 pntlemd 27638 pntlemb 27641 pntlemr 27646 pnt 27658 padicabv 27674 xrge0tsmsd 33065 fct2relem 34612 logdivsqrle 34665 knoppndvlem3 36515 iooelexlt 37363 relowlssretop 37364 poimir 37660 itg2gt0cn 37682 ftc1cnnclem 37698 aks4d1p1p5 42076 radcnvrat 44333 cncfiooicclem1 45908 itgioocnicc 45992 iblcncfioo 45993 amgmwlem 49321 |
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