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Theorem elicc2 12795

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 10681	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
2		rexr 10681	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
3		elicc1 12776	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
4	1, 2, 3	syl2an 597	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
5		mnfxr 10692	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
6	5	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ ∈ ℝ^*)
7	1	ad2antrr 724	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 ∈ ℝ^*)
8		simpr1 1190	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ^*)
9		mnflt 12512	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
10	9	ad2antrr 724	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ < 𝐴)
11		simpr2 1191	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 ≤ 𝐶)
12	6, 7, 8, 10, 11	xrltletrd 12548	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ < 𝐶)
13	2	ad2antlr 725	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 ∈ ℝ^*)
14		pnfxr 10689	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
15	14	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → +∞ ∈ ℝ^*)
16		simpr3 1192	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ≤ 𝐵)
17		ltpnf 12509	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
18	17	ad2antlr 725	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 < +∞)
19	8, 13, 15, 16, 18	xrlelttrd 12547	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 < +∞)
20		xrrebnd 12555	. . . . . . 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 711	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ)
23	22, 11, 16	3jca 1124	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
24	23	ex 415	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
25		rexr 10681	. . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
26	25	3anim1i 1148	. . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
27	24, 26	impbid1 227	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
28	4, 27	bitrd 281	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 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 +∞cpnf 10666 -∞cmnf 10667 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 [,]cicc 12735

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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-icc 12739

This theorem is referenced by: elicc2i 12796 iccssre 12812 iccsupr 12824 iccneg 12852 iccsplit 12865 iccshftr 12866 iccshftl 12868 iccdil 12870 icccntr 12872 iccf1o 12876 supicc 12880 icco1 14891 iccntr 23423 icccmplem1 23424 icccmplem2 23425 icccmplem3 23426 reconnlem1 23428 reconnlem2 23429 cnmpopc 23526 icoopnst 23537 iocopnst 23538 cnheiborlem 23552 ivthlem2 24047 ivthlem3 24048 ivthicc 24053 evthicc2 24055 ovolficc 24063 ovolicc1 24111 ovolicc2lem2 24113 ovolicc2lem5 24116 ovolicopnf 24119 dyadmaxlem 24192 opnmbllem 24196 volsup2 24200 volcn 24201 mbfi1fseqlem6 24315 itgspliticc 24431 itgsplitioo 24432 ditgcl 24450 ditgswap 24451 ditgsplitlem 24452 ditgsplit 24453 dvlip 24584 dvlip2 24586 dveq0 24591 dvgt0lem1 24593 dvivthlem1 24599 dvne0 24602 dvcnvrelem1 24608 dvcnvrelem2 24609 dvcnvre 24610 dvfsumlem2 24618 ftc1lem1 24626 ftc1lem2 24627 ftc1a 24628 ftc1lem4 24630 ftc2 24635 ftc2ditglem 24636 itgsubstlem 24639 pserulm 25004 loglesqrt 25333 log2tlbnd 25517 ppisval 25675 chtleppi 25780 fsumvma2 25784 chpchtsum 25789 chpub 25790 rplogsumlem2 26055 chpdifbndlem1 26123 pntibndlem2a 26160 pntibndlem2 26161 pntlemj 26173 pntlem3 26179 pntleml 26181 resconn 32488 cvmliftlem10 32536 opnmbllem0 34922 ftc2nc 34970 areacirclem2 34977 areacirclem4 34979 areacirc 34981 isbnd3 35056 isbnd3b 35057 prdsbnd 35065 iccbnd 35112 eliccd 41772 eliccre 41774 iccshift 41787 iccsuble 41788 limcicciooub 41911 icccncfext 42163 itgsubsticc 42254 iblcncfioo 42256 itgiccshift 42258 itgperiod 42259 itgsbtaddcnst 42260 fourierdlem42 42428 fourierdlem54 42439 fourierdlem63 42448 fourierdlem65 42450 fourierdlem74 42459 fourierdlem75 42460 fourierdlem82 42467 fourierdlem93 42478 fourierdlem101 42486 fourierdlem104 42489 fourierdlem111 42496 reorelicc 44691

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