lbicc2 - Metamath Proof Explorer

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Theorem lbicc2 13392

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.)

**Assertion**
Ref	Expression
lbicc2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))

**Proof of Theorem lbicc2**
Step	Hyp	Ref	Expression
1		simp1 1137	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ^*)
2		xrleid 13077	. . 3 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
3	2	3ad2ant1 1134	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐴)
4		simp3 1139	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
5		elicc1 13317	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ^ ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
6	5	3adant3 1133	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
7	1, 3, 4, 6	mpbir3and 1344	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 ∈ wcel 2114 class class class wbr 5100 (class class class)co 7368 ℝ^*cxr 11177 ≤ cle 11179 [,]cicc 13276

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-icc 13280

This theorem is referenced by: icccmplem1 24782 reconnlem2 24787 oprpiece1res1 24920 pcoass 24995 ivthlem1 25423 ivth2 25427 ivthle 25428 ivthle2 25429 evthicc 25431 ovolicc2lem5 25493 dyadmaxlem 25569 rolle 25965 cmvth 25966 cmvthOLD 25967 mvth 25968 dvlip 25969 c1liplem1 25972 dveq0 25976 dvgt0lem1 25978 lhop1lem 25989 dvcnvrelem1 25993 dvcvx 25996 dvfsumle 25997 dvfsumleOLD 25998 dvfsumge 25999 dvfsumabs 26000 dvfsumlem2 26004 dvfsumlem2OLD 26005 ftc2 26022 ftc2ditglem 26023 itgparts 26025 itgsubstlem 26026 itgpowd 26028 taylfval 26337 tayl0 26340 efcvx 26430 pige3ALT 26500 logccv 26643 loglesqrt 26742 eliccioo 33027 ftc2re 34780 cvmliftlem6 35510 cvmliftlem8 35512 cvmliftlem9 35513 cvmliftlem10 35514 cvmliftlem13 35516 ivthALT 36555 ftc2nc 37957 areacirc 37968 iccintsng 45887 icccncfext 46249 cncfiooicclem1 46255 dvbdfbdioolem1 46290 itgsin0pilem1 46312 itgcoscmulx 46331 itgsincmulx 46336 fourierdlem20 46489 fourierdlem51 46519 fourierdlem54 46522 fourierdlem64 46532 fourierdlem73 46541 fourierdlem81 46549 fourierdlem102 46570 fourierdlem103 46571 fourierdlem104 46572 fourierdlem114 46582 etransclem46 46642 hoidmv1lelem1 46953

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