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| Description: The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| lbicc2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simp1 1136 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ*) | |
| 2 | xrleid 13194 | . . 3 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) | |
| 3 | 2 | 3ad2ant1 1133 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴) | 
| 4 | simp3 1138 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
| 5 | elicc1 13432 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) | |
| 6 | 5 | 3adant3 1132 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))) | 
| 7 | 1, 3, 4, 6 | mpbir3and 1342 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 ℝ*cxr 11295 ≤ cle 11297 [,]cicc 13391 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-pre-lttri 11230 ax-pre-lttrn 11231 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-icc 13395 | 
| This theorem is referenced by: icccmplem1 24845 reconnlem2 24850 oprpiece1res1 24983 pcoass 25058 ivthlem1 25487 ivth2 25491 ivthle 25492 ivthle2 25493 evthicc 25495 ovolicc2lem5 25557 dyadmaxlem 25633 rolle 26029 cmvth 26030 cmvthOLD 26031 mvth 26032 dvlip 26033 c1liplem1 26036 dveq0 26040 dvgt0lem1 26042 lhop1lem 26053 dvcnvrelem1 26057 dvcvx 26060 dvfsumle 26061 dvfsumleOLD 26062 dvfsumge 26063 dvfsumabs 26064 dvfsumlem2 26068 dvfsumlem2OLD 26069 ftc2 26086 ftc2ditglem 26087 itgparts 26089 itgsubstlem 26090 itgpowd 26092 taylfval 26401 tayl0 26404 efcvx 26494 pige3ALT 26563 logccv 26706 loglesqrt 26805 eliccioo 32914 ftc2re 34614 cvmliftlem6 35296 cvmliftlem8 35298 cvmliftlem9 35299 cvmliftlem10 35300 cvmliftlem13 35302 ivthALT 36337 ftc2nc 37710 areacirc 37721 iccintsng 45541 icccncfext 45907 cncfiooicclem1 45913 dvbdfbdioolem1 45948 itgsin0pilem1 45970 itgcoscmulx 45989 itgsincmulx 45994 fourierdlem20 46147 fourierdlem51 46177 fourierdlem54 46180 fourierdlem64 46190 fourierdlem73 46199 fourierdlem81 46207 fourierdlem102 46228 fourierdlem103 46229 fourierdlem104 46230 fourierdlem114 46240 etransclem46 46300 hoidmv1lelem1 46611 | 
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