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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 13328 | . . . 4 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → (𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) | |
2 | elioo2 13311 | . . . 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 205 ∧ wa 397 ∧ w3a 1088 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 ℝcr 11055 ℝ*cxr 11193 < clt 11194 (,)cioo 13270 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 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-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-ioo 13274 |
This theorem is referenced by: elioo4g 13330 iccssioo2 13343 qdensere 24149 zcld 24192 reconnlem2 24206 xrge0tsms 24213 ovolioo 24948 ioorcl2 24952 itgsplitioo 25218 dvferm1lem 25364 dvferm2lem 25366 dvferm 25368 dvlt0 25385 dvivthlem1 25388 lhop1lem 25393 lhop1 25394 lhop2 25395 dvcvx 25400 ftc1lem4 25419 itgsubstlem 25428 itgsubst 25429 pilem2 25827 pilem3 25828 pigt2lt4 25829 tangtx 25878 tanabsge 25879 cosne0 25901 cos0pilt1 25904 tanord 25910 tanregt0 25911 argimlt0 25984 logneg2 25986 divlogrlim 26006 logno1 26007 logcnlem3 26015 dvloglem 26019 logf1o2 26021 loglesqrt 26127 asinsin 26258 acoscos 26259 atanlogaddlem 26279 atanlogsub 26282 atantan 26289 atanbndlem 26291 scvxcvx 26351 lgamgulmlem2 26395 basellem8 26453 vmalogdivsum2 26902 vmalogdivsum 26903 2vmadivsumlem 26904 chpdifbndlem1 26917 selberg3lem1 26921 selberg3 26923 selberg4lem1 26924 selberg4 26925 selberg3r 26933 selberg4r 26934 selberg34r 26935 pntrlog2bndlem1 26941 pntrlog2bndlem2 26942 pntrlog2bndlem3 26943 pntrlog2bndlem4 26944 pntrlog2bndlem5 26945 pntrlog2bndlem6a 26946 pntrlog2bndlem6 26947 pntrlog2bnd 26948 pntpbnd1a 26949 pntpbnd1 26950 pntpbnd2 26951 pntpbnd 26952 pntibndlem2 26955 pntibndlem3 26956 pntibnd 26957 pntlemd 26958 pntlemb 26961 pntlemr 26966 pnt 26978 padicabv 26994 xrge0tsmsd 31948 fct2relem 33267 logdivsqrle 33320 knoppndvlem3 35023 iooelexlt 35879 relowlssretop 35880 poimir 36157 itg2gt0cn 36179 ftc1cnnclem 36195 aks4d1p1p5 40578 radcnvrat 42682 cncfiooicclem1 44220 itgioocnicc 44304 iblcncfioo 44305 amgmwlem 47335 |
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