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Mirrors > Home > MPE Home > Th. List > elioo2 | Structured version Visualization version GIF version |
Description: Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.) |
Ref | Expression |
---|---|
elioo2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooval2 12761 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}) | |
2 | 1 | eleq2d 2898 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ 𝐶 ∈ {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})) |
3 | breq2 5062 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 < 𝑥 ↔ 𝐴 < 𝐶)) | |
4 | breq1 5061 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 < 𝐵 ↔ 𝐶 < 𝐵)) | |
5 | 3, 4 | anbi12d 630 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
6 | 5 | elrab 3679 | . . 3 ⊢ (𝐶 ∈ {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ↔ (𝐶 ∈ ℝ ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
7 | 3anass 1087 | . . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
8 | 6, 7 | bitr4i 279 | . 2 ⊢ (𝐶 ∈ {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) |
9 | 2, 8 | syl6bb 288 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {crab 3142 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: eliooord 12786 elioopnf 12821 elioomnf 12822 difreicc 12860 xov1plusxeqvd 12874 tanhbnd 15504 bl2ioo 23329 xrtgioo 23343 zcld 23350 iccntr 23358 icccmplem2 23360 reconnlem1 23363 reconnlem2 23364 icoopnst 23472 iocopnst 23473 ivthlem3 23983 ovolicc2lem1 24047 ovolicc2lem5 24051 ioombl1lem4 24091 mbfmax 24179 itg2monolem1 24280 itg2monolem3 24282 dvferm1lem 24510 dvferm2lem 24512 dvlip2 24521 dvivthlem1 24534 lhop1lem 24539 lhop 24542 dvcnvrelem1 24543 dvcnvre 24545 itgsubst 24575 sincosq1sgn 25013 sincosq2sgn 25014 sincosq3sgn 25015 sincosq4sgn 25016 coseq00topi 25017 tanabsge 25021 sinq12gt0 25022 sinq12ge0 25023 cosq14gt0 25025 sincos6thpi 25030 sineq0 25038 cosordlem 25042 tanord1 25048 tanord 25049 argregt0 25120 argimgt0 25122 argimlt0 25123 dvloglem 25158 logf1o2 25160 efopnlem2 25167 asinsinlem 25396 acoscos 25398 atanlogsublem 25420 atantan 25428 atanbndlem 25430 atanbnd 25431 atan1 25433 scvxcvx 25491 basellem1 25586 pntibndlem1 26093 pntibnd 26097 pntlemc 26099 padicabvf 26135 padicabvcxp 26136 dfrp2 30418 cnre2csqlem 31053 ivthALT 33581 iooelexlt 34526 itg2gt0cn 34829 iblabsnclem 34837 dvasin 34860 areacirclem1 34864 areacirc 34869 cvgdvgrat 40525 radcnvrat 40526 sineq0ALT 41151 ioogtlb 41650 eliood 41653 eliooshift 41662 iooltub 41666 limciccioolb 41782 limcicciooub 41798 cncfioobdlem 42059 ditgeqiooicc 42125 dirkercncflem1 42269 dirkercncflem4 42272 fourierdlem10 42283 fourierdlem32 42305 fourierdlem62 42334 fourierdlem81 42353 fourierdlem82 42354 fourierdlem93 42365 fourierdlem104 42376 fourierdlem111 42383 |
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