ubicc2 - Metamath Proof Explorer

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Theorem ubicc2 13393

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 1138	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
2		simp3 1139	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
3		xrleid 13077	. . 3 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
4	3	3ad2ant2 1135	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
5		elicc1 13317	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)))
6	5	3adant3 1133	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)))
7	1, 2, 4, 6	mpbir3and 1344	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (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: xnn0xrge0 13434 iccpnfcnv 24910 oprpiece1res2 24918 ivthlem2 25421 ivth2 25424 ivthle 25425 ivthle2 25426 dyadmaxlem 25566 cmvth 25963 cmvthOLD 25964 mvth 25965 dvlip 25966 c1liplem1 25969 dvgt0lem1 25975 lhop1lem 25986 dvcnvrelem1 25990 dvcvx 25993 dvfsumle 25994 dvfsumleOLD 25995 dvfsumge 25996 dvfsumabs 25997 dvfsumlem2 26001 dvfsumlem2OLD 26002 ftc2 26019 ftc2ditglem 26020 itgparts 26022 itgsubstlem 26023 itgpowd 26025 efcvx 26427 pige3ALT 26497 cos0pilt1 26509 logccv 26640 loglesqrt 26739 pntlem3 27588 eliccioo 33023 xrge0iifcnv 34111 lmxrge0 34130 esumpinfval 34251 hashf2 34262 esumcvg 34264 ftc2re 34776 cvmliftlem7 35507 cvmliftlem10 35510 ivthALT 36551 ftc2nc 37953 areacirc 37964 iccintsng 45883 pnfel0pnf 45888 limcicciooub 45995 icccncfext 46245 dvbdfbdioolem1 46286 itgsin0pilem1 46308 itgcoscmulx 46327 itgsincmulx 46332 itgsubsticc 46334 fourierdlem20 46485 fourierdlem54 46518 fourierdlem64 46528 fourierdlem81 46545 fourierdlem102 46566 fourierdlem103 46567 fourierdlem104 46568 fourierdlem114 46578 etransclem46 46638

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