ubicc2 - Metamath Proof Explorer

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Theorem ubicc2 13490

Description: The upper bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.)

**Assertion**
Ref	Expression
ubicc2	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))

**Proof of Theorem ubicc2**
Step	Hyp	Ref	Expression
1		simp2 1134	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
2		simp3 1135	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
3		xrleid 13178	. . 3 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
4	3	3ad2ant2 1131	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
5		elicc1 13416	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)))
6	5	3adant3 1129	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)))
7	1, 2, 4, 6	mpbir3and 1339	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 ∈ wcel 2099 class class class wbr 5145 (class class class)co 7416 ℝ^*cxr 11288 ≤ cle 11290 [,]cicc 13375

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 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-pre-lttri 11223 ax-pre-lttrn 11224

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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-icc 13379

This theorem is referenced by: xnn0xrge0 13531 iccpnfcnv 24957 oprpiece1res2 24965 ivthlem2 25469 ivth2 25472 ivthle 25473 ivthle2 25474 dyadmaxlem 25614 cmvth 26011 cmvthOLD 26012 mvth 26013 dvlip 26014 c1liplem1 26017 dvgt0lem1 26023 lhop1lem 26034 dvcnvrelem1 26038 dvcvx 26041 dvfsumle 26042 dvfsumleOLD 26043 dvfsumge 26044 dvfsumabs 26045 dvfsumlem2 26049 dvfsumlem2OLD 26050 ftc2 26067 ftc2ditglem 26068 itgparts 26070 itgsubstlem 26071 itgpowd 26073 efcvx 26476 pige3ALT 26544 cos0pilt1 26556 logccv 26687 loglesqrt 26786 pntlem3 27635 eliccioo 32795 xrge0iifcnv 33761 lmxrge0 33780 esumpinfval 33919 hashf2 33930 esumcvg 33932 ftc2re 34457 cvmliftlem7 35132 cvmliftlem10 35135 ivthALT 36060 ftc2nc 37416 areacirc 37427 iccintsng 45177 pnfel0pnf 45182 limcicciooub 45294 icccncfext 45544 dvbdfbdioolem1 45585 itgsin0pilem1 45607 itgcoscmulx 45626 itgsincmulx 45631 itgsubsticc 45633 fourierdlem20 45784 fourierdlem54 45817 fourierdlem64 45827 fourierdlem81 45844 fourierdlem102 45865 fourierdlem103 45866 fourierdlem104 45867 fourierdlem114 45877 etransclem46 45937

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