elicc2 - Metamath Proof Explorer

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Theorem elicc2 13311

Description: Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.)

**Assertion**
Ref	Expression
elicc2	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))

**Proof of Theorem elicc2**
Step	Hyp	Ref	Expression
1		rexr 11158	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
2		rexr 11158	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
3		elicc1 13289	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
5		mnfxr 11169	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
6	5	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ ∈ ℝ^*)
7	1	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 ∈ ℝ^*)
8		simpr1 1195	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ^*)
9		mnflt 13022	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
10	9	ad2antrr 726	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ < 𝐴)
11		simpr2 1196	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 ≤ 𝐶)
12	6, 7, 8, 10, 11	xrltletrd 13060	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ < 𝐶)
13	2	ad2antlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 ∈ ℝ^*)
14		pnfxr 11166	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
15	14	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → +∞ ∈ ℝ^*)
16		simpr3 1197	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ≤ 𝐵)
17		ltpnf 13019	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
18	17	ad2antlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 < +∞)
19	8, 13, 15, 16, 18	xrlelttrd 13059	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 < +∞)
20		xrrebnd 13067	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞)))
21	8, 20	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞)))
22	12, 19, 21	mpbir2and 713	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ)
23	22, 11, 16	3jca 1128	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
24	23	ex 412	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
25		rexr 11158	. . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
26	25	3anim1i 1152	. . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
27	24, 26	impbid1 225	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
28	4, 27	bitrd 279	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2111 class class class wbr 5089 (class class class)co 7346 ℝcr 11005 +∞cpnf 11143 -∞cmnf 11144 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 [,]cicc 13248

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-pre-lttri 11080 ax-pre-lttrn 11081

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-po 5522 df-so 5523 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-icc 13252

This theorem is referenced by: elicc2i 13312 iccssre 13329 iccsupr 13342 iccneg 13372 iccsplit 13385 iccshftr 13386 iccshftl 13388 iccdil 13390 icccntr 13392 iccf1o 13396 supicc 13401 icco1 15447 iccntr 24737 icccmplem1 24738 icccmplem2 24739 icccmplem3 24740 reconnlem1 24742 reconnlem2 24743 cnmpopc 24849 icoopnst 24863 iocopnst 24864 cnheiborlem 24880 ivthlem2 25380 ivthlem3 25381 ivthicc 25386 evthicc2 25388 ovolficc 25396 ovolicc1 25444 ovolicc2lem2 25446 ovolicc2lem5 25449 ovolicopnf 25452 dyadmaxlem 25525 opnmbllem 25529 volsup2 25533 volcn 25534 mbfi1fseqlem6 25648 itgspliticc 25765 itgsplitioo 25766 ditgcl 25786 ditgswap 25787 ditgsplitlem 25788 ditgsplit 25789 dvlip 25925 dvlip2 25927 dveq0 25932 dvgt0lem1 25934 dvivthlem1 25940 dvne0 25943 dvcnvrelem1 25949 dvcnvrelem2 25950 dvcnvre 25951 dvfsumlem2 25960 dvfsumlem2OLD 25961 ftc1lem1 25969 ftc1lem2 25970 ftc1a 25971 ftc1lem4 25973 ftc2 25978 ftc2ditglem 25979 itgsubstlem 25982 pserulm 26358 loglesqrt 26698 log2tlbnd 26882 ppisval 27041 chtleppi 27148 fsumvma2 27152 chpchtsum 27157 chpub 27158 rplogsumlem2 27423 chpdifbndlem1 27491 pntibndlem2a 27528 pntibndlem2 27529 pntlemj 27541 pntlem3 27547 pntleml 27549 resconn 35290 cvmliftlem10 35338 opnmbllem0 37706 ftc2nc 37752 areacirclem2 37759 areacirclem4 37761 areacirc 37763 isbnd3 37834 isbnd3b 37835 prdsbnd 37843 iccbnd 37890 intlewftc 42164 dvrelog2 42167 aks4d1p1p5 42178 eliccd 45614 eliccre 45615 iccshift 45628 iccsuble 45629 limcicciooub 45745 icccncfext 45995 itgsubsticc 46084 iblcncfioo 46086 itgiccshift 46088 itgperiod 46089 itgsbtaddcnst 46090 fourierdlem42 46257 fourierdlem54 46268 fourierdlem63 46277 fourierdlem65 46279 fourierdlem74 46288 fourierdlem75 46289 fourierdlem82 46296 fourierdlem93 46307 fourierdlem101 46315 fourierdlem104 46318 fourierdlem111 46325 reorelicc 48821

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