ubicc2 - Metamath Proof Explorer

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Theorem ubicc2 13379

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 1137	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
2		simp3 1138	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
3		xrleid 13063	. . 3 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
4	3	3ad2ant2 1134	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
5		elicc1 13303	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)))
6	5	3adant3 1132	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)))
7	1, 2, 4, 6	mpbir3and 1343	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 ∈ wcel 2113 class class class wbr 5096 (class class class)co 7356 ℝ^*cxr 11163 ≤ cle 11165 [,]cicc 13262

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-pre-lttri 11098 ax-pre-lttrn 11099

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-po 5530 df-so 5531 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-icc 13266

This theorem is referenced by: xnn0xrge0 13420 iccpnfcnv 24896 oprpiece1res2 24904 ivthlem2 25407 ivth2 25410 ivthle 25411 ivthle2 25412 dyadmaxlem 25552 cmvth 25949 cmvthOLD 25950 mvth 25951 dvlip 25952 c1liplem1 25955 dvgt0lem1 25961 lhop1lem 25972 dvcnvrelem1 25976 dvcvx 25979 dvfsumle 25980 dvfsumleOLD 25981 dvfsumge 25982 dvfsumabs 25983 dvfsumlem2 25987 dvfsumlem2OLD 25988 ftc2 26005 ftc2ditglem 26006 itgparts 26008 itgsubstlem 26009 itgpowd 26011 efcvx 26413 pige3ALT 26483 cos0pilt1 26495 logccv 26626 loglesqrt 26725 pntlem3 27574 eliccioo 32961 xrge0iifcnv 34039 lmxrge0 34058 esumpinfval 34179 hashf2 34190 esumcvg 34192 ftc2re 34704 cvmliftlem7 35434 cvmliftlem10 35437 ivthALT 36478 ftc2nc 37842 areacirc 37853 iccintsng 45711 pnfel0pnf 45716 limcicciooub 45823 icccncfext 46073 dvbdfbdioolem1 46114 itgsin0pilem1 46136 itgcoscmulx 46155 itgsincmulx 46160 itgsubsticc 46162 fourierdlem20 46313 fourierdlem54 46346 fourierdlem64 46356 fourierdlem81 46373 fourierdlem102 46394 fourierdlem103 46395 fourierdlem104 46396 fourierdlem114 46406 etransclem46 46466

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