elioo2 - Metamath Proof Explorer

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Theorem elioo2 13419

Description: Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)

**Assertion**
Ref	Expression
elioo2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))

Proof of Theorem elioo2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iooval2 13411	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
2	1	eleq2d 2812	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ 𝐶 ∈ {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}))
3		breq2 5157	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 < 𝑥 ↔ 𝐴 < 𝐶))
4		breq1 5156	. . . . 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 3681	. . 3 ⊢ (𝐶 ∈ {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ↔ (𝐶 ∈ ℝ ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
7		3anass 1092	. . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
8	6, 7	bitr4i 277	. 2 ⊢ (𝐶 ∈ {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
9	2, 8	bitrdi 286	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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 {crab 3419 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: dfrp2 13427 eliooord 13437 elioopnf 13474 elioomnf 13475 difreicc 13515 xov1plusxeqvd 13529 tanhbnd 16163 bl2ioo 24799 xrtgioo 24813 zcld 24820 iccntr 24828 icccmplem2 24830 reconnlem1 24833 reconnlem2 24834 icoopnst 24954 iocopnst 24955 ivthlem3 25473 ovolicc2lem1 25537 ovolicc2lem5 25541 ioombl1lem4 25581 mbfmax 25669 itg2monolem1 25771 itg2monolem3 25773 dvferm1lem 26007 dvferm2lem 26009 dvlip2 26019 dvivthlem1 26032 lhop1lem 26037 lhop 26040 dvcnvrelem1 26041 dvcnvre 26043 itgsubst 26075 sincosq1sgn 26526 sincosq2sgn 26527 sincosq3sgn 26528 sincosq4sgn 26529 coseq00topi 26530 tanabsge 26534 sinq12gt0 26535 sinq12ge0 26536 cosq14gt0 26538 sincos6thpi 26543 sineq0 26551 cos02pilt1 26553 cosq34lt1 26554 cosordlem 26557 cos0pilt1 26559 tanord1 26564 tanord 26565 argregt0 26637 argimgt0 26639 argimlt0 26640 dvloglem 26675 logf1o2 26677 efopnlem2 26684 asinsinlem 26919 acoscos 26921 atanlogsublem 26943 atantan 26951 atanbndlem 26953 atanbnd 26954 atan1 26956 scvxcvx 27014 basellem1 27109 pntibndlem1 27618 pntibnd 27622 pntlemc 27624 padicabvf 27660 padicabvcxp 27661 cnre2csqlem 33725 ivthALT 36047 iooelexlt 37069 itg2gt0cn 37376 iblabsnclem 37384 dvasin 37405 areacirclem1 37409 areacirc 37414 dvrelog3 41764 0nonelalab 41766 cvgdvgrat 43987 radcnvrat 43988 sineq0ALT 44613 ioogtlb 45113 eliood 45116 eliooshift 45124 iooltub 45128 limciccioolb 45242 limcicciooub 45258 cncfioobdlem 45517 ditgeqiooicc 45581 dirkercncflem1 45724 dirkercncflem4 45727 fourierdlem10 45738 fourierdlem32 45760 fourierdlem62 45789 fourierdlem81 45808 fourierdlem82 45809 fourierdlem93 45820 fourierdlem104 45831 fourierdlem111 45838

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