elioo2 - Metamath Proof Explorer

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Theorem elioo2 13428

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 13420	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)})
2	1	eleq2d 2827	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ 𝐶 ∈ {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)}))
3		breq2 5147	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐴 < 𝑥 ↔ 𝐴 < 𝐶))
4		breq1 5146	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 < 𝐵 ↔ 𝐶 < 𝐵))
5	3, 4	anbi12d 632	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝐴 < 𝑥 ∧ 𝑥 < 𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
6	5	elrab 3692	. . 3 ⊢ (𝐶 ∈ {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ↔ (𝐶 ∈ ℝ ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
7		3anass 1095	. . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
8	6, 7	bitr4i 278	. 2 ⊢ (𝐶 ∈ {𝑥 ∈ ℝ ∣ (𝐴 < 𝑥 ∧ 𝑥 < 𝐵)} ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
9	2, 8	bitrdi 287	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {crab 3436 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: dfrp2 13436 eliooord 13446 elioopnf 13483 elioomnf 13484 difreicc 13524 xov1plusxeqvd 13538 tanhbnd 16197 bl2ioo 24813 xrtgioo 24828 zcld 24835 iccntr 24843 icccmplem2 24845 reconnlem1 24848 reconnlem2 24849 icoopnst 24969 iocopnst 24970 ivthlem3 25488 ovolicc2lem1 25552 ovolicc2lem5 25556 ioombl1lem4 25596 mbfmax 25684 itg2monolem1 25785 itg2monolem3 25787 dvferm1lem 26022 dvferm2lem 26024 dvlip2 26034 dvivthlem1 26047 lhop1lem 26052 lhop 26055 dvcnvrelem1 26056 dvcnvre 26058 itgsubst 26090 sincosq1sgn 26540 sincosq2sgn 26541 sincosq3sgn 26542 sincosq4sgn 26543 coseq00topi 26544 tanabsge 26548 sinq12gt0 26549 sinq12ge0 26550 cosq14gt0 26552 sincos6thpi 26558 sineq0 26566 cos02pilt1 26568 cosq34lt1 26569 cosordlem 26572 cos0pilt1 26574 tanord1 26579 tanord 26580 argregt0 26652 argimgt0 26654 argimlt0 26655 dvloglem 26690 logf1o2 26692 efopnlem2 26699 asinsinlem 26934 acoscos 26936 atanlogsublem 26958 atantan 26966 atanbndlem 26968 atanbnd 26969 atan1 26971 scvxcvx 27029 basellem1 27124 pntibndlem1 27633 pntibnd 27637 pntlemc 27639 padicabvf 27675 padicabvcxp 27676 cnre2csqlem 33909 ivthALT 36336 iooelexlt 37363 itg2gt0cn 37682 iblabsnclem 37690 dvasin 37711 areacirclem1 37715 areacirc 37720 dvrelog3 42066 0nonelalab 42068 cvgdvgrat 44332 radcnvrat 44333 sineq0ALT 44957 ioogtlb 45508 eliood 45511 eliooshift 45519 iooltub 45523 limciccioolb 45636 limcicciooub 45652 cncfioobdlem 45911 ditgeqiooicc 45975 dirkercncflem1 46118 dirkercncflem4 46121 fourierdlem10 46132 fourierdlem32 46154 fourierdlem62 46183 fourierdlem81 46202 fourierdlem82 46203 fourierdlem93 46214 fourierdlem104 46225 fourierdlem111 46232

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