ubicc2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ubicc2		Structured version Visualization version GIF version

Theorem ubicc2 13502

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 1136	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ^*)
2		simp3 1137	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐴 ≤ 𝐵)
3		xrleid 13190	. . 3 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
4	3	3ad2ant2 1133	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐵)
5		elicc1 13428	. . 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 206 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝ^*cxr 11292 ≤ cle 11294 [,]cicc 13387

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-icc 13391

This theorem is referenced by: xnn0xrge0 13543 iccpnfcnv 24989 oprpiece1res2 24997 ivthlem2 25501 ivth2 25504 ivthle 25505 ivthle2 25506 dyadmaxlem 25646 cmvth 26044 cmvthOLD 26045 mvth 26046 dvlip 26047 c1liplem1 26050 dvgt0lem1 26056 lhop1lem 26067 dvcnvrelem1 26071 dvcvx 26074 dvfsumle 26075 dvfsumleOLD 26076 dvfsumge 26077 dvfsumabs 26078 dvfsumlem2 26082 dvfsumlem2OLD 26083 ftc2 26100 ftc2ditglem 26101 itgparts 26103 itgsubstlem 26104 itgpowd 26106 efcvx 26508 pige3ALT 26577 cos0pilt1 26589 logccv 26720 loglesqrt 26819 pntlem3 27668 eliccioo 32898 xrge0iifcnv 33894 lmxrge0 33913 esumpinfval 34054 hashf2 34065 esumcvg 34067 ftc2re 34592 cvmliftlem7 35276 cvmliftlem10 35279 ivthALT 36318 ftc2nc 37689 areacirc 37700 iccintsng 45476 pnfel0pnf 45481 limcicciooub 45593 icccncfext 45843 dvbdfbdioolem1 45884 itgsin0pilem1 45906 itgcoscmulx 45925 itgsincmulx 45930 itgsubsticc 45932 fourierdlem20 46083 fourierdlem54 46116 fourierdlem64 46126 fourierdlem81 46143 fourierdlem102 46164 fourierdlem103 46165 fourierdlem104 46166 fourierdlem114 46176 etransclem46 46236

W3C validator