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Mirrors > Home > MPE Home > Th. List > ubicc2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ubicc2 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1136 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ*) | |
2 | simp3 1137 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵) | |
3 | xrleid 12885 | . . 3 ⊢ (𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵) | |
4 | 3 | 3ad2ant2 1133 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵) |
5 | elicc1 13123 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))) | |
6 | 5 | 3adant3 1131 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))) |
7 | 1, 2, 4, 6 | mpbir3and 1341 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝ*cxr 11008 ≤ cle 11010 [,]cicc 13082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-icc 13086 |
This theorem is referenced by: xnn0xrge0 13238 iccpnfcnv 24107 oprpiece1res2 24115 ivthlem2 24616 ivth2 24619 ivthle 24620 ivthle2 24621 dyadmaxlem 24761 cmvth 25155 mvth 25156 dvlip 25157 c1liplem1 25160 dvgt0lem1 25166 lhop1lem 25177 dvcnvrelem1 25181 dvcvx 25184 dvfsumle 25185 dvfsumge 25186 dvfsumabs 25187 dvfsumlem2 25191 ftc2 25208 ftc2ditglem 25209 itgparts 25211 itgsubstlem 25212 itgpowd 25214 efcvx 25608 pige3ALT 25676 cos0pilt1 25688 logccv 25818 loglesqrt 25911 pntlem3 26757 eliccioo 31205 xrge0iifcnv 31883 lmxrge0 31902 esumpinfval 32041 hashf2 32052 esumcvg 32054 ftc2re 32578 cvmliftlem7 33253 cvmliftlem10 33256 ivthALT 34524 ftc2nc 35859 areacirc 35870 iccintsng 43061 pnfel0pnf 43066 limcicciooub 43178 icccncfext 43428 dvbdfbdioolem1 43469 itgsin0pilem1 43491 itgcoscmulx 43510 itgsincmulx 43515 itgsubsticc 43517 fourierdlem20 43668 fourierdlem54 43701 fourierdlem64 43711 fourierdlem81 43728 fourierdlem102 43749 fourierdlem103 43750 fourierdlem104 43751 fourierdlem114 43761 etransclem46 43821 |
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