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Theorem elicc2 13467

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 11282	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
2		rexr 11282	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
3		elicc1 13445	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
4	1, 2, 3	syl2an 608	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
5		mnfxr 11293	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
6	5	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ ∈ ℝ^*)
7	1	ad2antrr 739	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 ∈ ℝ^*)
8		simpr1 1213	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ^*)
9		mnflt 13177	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
10	9	ad2antrr 739	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ < 𝐴)
11		simpr2 1214	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐴 ≤ 𝐶)
12	6, 7, 8, 10, 11	xrltletrd 13215	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → -∞ < 𝐶)
13	2	ad2antlr 740	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 ∈ ℝ^*)
14		pnfxr 11290	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
15	14	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → +∞ ∈ ℝ^*)
16		simpr3 1215	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ≤ 𝐵)
17		ltpnf 13174	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
18	17	ad2antlr 740	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐵 < +∞)
19	8, 13, 15, 16, 18	xrlelttrd 13214	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 < +∞)
20		xrrebnd 13223	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞)))
21	8, 20	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞)))
22	12, 19, 21	mpbir2and 726	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝐶 ∈ ℝ)
23	22, 11, 16	3jca 1146	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
24	23	ex 418	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
25		rexr 11282	. . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
26	25	3anim1i 1170	. . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) → (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
27	24, 26	impbid1 228	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
28	4, 27	bitrd 282	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∈ wcel 2145 class class class wbr 5103 (class class class)co 7414 ℝcr 11126 +∞cpnf 11267 -∞cmnf 11268 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 [,]cicc 13404

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-pre-lttri 11201 ax-pre-lttrn 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-icc 13408

This theorem is used by: elicc2i 13468 iccssre 13485 iccsupr 13498 iccneg 13528 iccsplit 13541 iccshftr 13542 iccshftl 13544 iccdil 13546 icccntr 13548 iccf1o 13552 supicc 13557 icco1 15630 iccntr 25051 icccmplem1 25052 icccmplem2 25053 icccmplem3 25054 reconnlem1 25056 reconnlem2 25057 cnmpopc 25159 icoopnst 25170 iocopnst 25171 cnheiborlem 25185 ivthlem2 25683 ivthlem3 25684 ivthicc 25689 evthicc2 25691 ovolficc 25699 ovolicc1 25747 ovolicc2lem2 25749 ovolicc2lem5 25752 ovolicopnf 25755 dyadmaxlem 25828 opnmbllem 25832 volsup2 25836 volcn 25837 mbfi1fseqlem6 25951 itgspliticc 26067 itgsplitioo 26068 ditgcl 26088 ditgswap 26089 ditgsplitlem 26090 ditgsplit 26091 dvlip 26223 dvlip2 26225 dveq0 26230 dvgt0lem1 26232 dvivthlem1 26238 dvne0 26241 dvcnvrelem1 26247 dvcnvrelem2 26248 dvcnvre 26249 dvfsumlem2 26257 ftc1lem1 26265 ftc1lem2 26266 ftc1a 26267 ftc1lem4 26269 ftc2 26274 ftc2ditglem 26275 itgsubstlem 26278 pserulm 26661 loglesqrt 27001 log2tlbnd 27185 ppisval 27343 chtleppi 27449 fsumvma2 27453 chpchtsum 27458 chpub 27459 rplogsumlem2 27724 chpdifbndlem1 27792 pntibndlem2a 27829 pntibndlem2 27830 pntlemj 27842 pntlem3 27848 pntleml 27850 resconn 35828 cvmliftlem10 35876 opnmbllem0 38408 ftc2nc 38454 areacirclem2 38461 areacirclem4 38463 areacirc 38465 isbnd3 38537 isbnd3b 38538 prdsbnd 38546 iccbnd 38593 intlewftc 42930 dvrelog2 42933 aks4d1p1p5 42944 eliccd 46337 eliccre 46338 iccshift 46351 iccsuble 46352 limcicciooub 46468 icccncfext 46718 itgsubsticc 46807 iblcncfioo 46809 itgiccshift 46811 itgperiod 46812 itgsbtaddcnst 46813 fourierdlem42 46980 fourierdlem54 46991 fourierdlem63 47000 fourierdlem65 47002 fourierdlem74 47011 fourierdlem75 47012 fourierdlem82 47019 fourierdlem93 47030 fourierdlem101 47038 fourierdlem104 47041 fourierdlem111 47048 reorelicc 49643

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