elicoelioo - Mathbox for Thierry Arnoux

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Theorem elicoelioo 29540

Description: Relate elementhood to a closed-below, open-above interval with elementhood to the same open interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.)

**Assertion**
Ref	Expression
elicoelioo	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵))))

**Proof of Theorem elicoelioo**
Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐴 ∈ ℝ^*)
2		simpl2 1065	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐵 ∈ ℝ^*)
3		simprl 794	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 ∈ (𝐴[,)𝐵))
4		elico1 12218	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
5	4	biimpa 501	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
6	5	simp1d 1073	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ^)
7	1, 2, 3, 6	syl21anc 1325	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 ∈ ℝ^*)
8	5	simp2d 1074	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶)
9	1, 2, 3, 8	syl21anc 1325	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐴 ≤ 𝐶)
10	1, 2	jca 554	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
11		simprr 796	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → ¬ 𝐶 ∈ (𝐴(,)𝐵))
12	5	simp3d 1075	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
13	10, 3, 12	syl2anc 693	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 < 𝐵)
14		elioo1 12215	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
15	14	notbid 308	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ 𝐶 ∈ (𝐴(,)𝐵) ↔ ¬ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
16	15	biimpa 501	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → ¬ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
17		3anan32 1050	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) ∧ 𝐴 < 𝐶))
18	17	notbii 310	. . . . . . . . . . 11 ⊢ (¬ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ ¬ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) ∧ 𝐴 < 𝐶))
19		imnan 438	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) → ¬ 𝐴 < 𝐶) ↔ ¬ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) ∧ 𝐴 < 𝐶))
20	18, 19	bitr4i 267	. . . . . . . . . 10 ⊢ (¬ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) → ¬ 𝐴 < 𝐶))
21	16, 20	sylib 208	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → ((𝐶 ∈ ℝ^ ∧ 𝐶 < 𝐵) → ¬ 𝐴 < 𝐶))
22	21	imp 445	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) ∧ (𝐶 ∈ ℝ^ ∧ 𝐶 < 𝐵)) → ¬ 𝐴 < 𝐶)
23	10, 11, 7, 13, 22	syl22anc 1327	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → ¬ 𝐴 < 𝐶)
24		xeqlelt 29538	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 = 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ ¬ 𝐴 < 𝐶)))
25	24	biimpar 502	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ ¬ 𝐴 < 𝐶)) → 𝐴 = 𝐶)
26	1, 7, 9, 23, 25	syl22anc 1327	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐴 = 𝐶)
27	26	ex 450	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 = 𝐶))
28		eqcom 2629	. . . . 5 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴)
29	27, 28	syl6ib 241	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 = 𝐴))
30		pm5.6 951	. . . 4 ⊢ (((𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 = 𝐴) ↔ (𝐶 ∈ (𝐴[,)𝐵) → (𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴)))
31	29, 30	sylib 208	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴)))
32		orcom 402	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵)))
33	31, 32	syl6ib 241	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵))))
34		simpr 477	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
35		simpl1 1064	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ∈ ℝ^*)
36	34, 35	eqeltrd 2701	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 ∈ ℝ^*)
37		xrleid 11983	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
38	35, 37	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ≤ 𝐴)
39	38, 34	breqtrrd 4681	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ≤ 𝐶)
40		simpl3 1066	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 < 𝐵)
41	34, 40	eqbrtrd 4675	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 < 𝐵)
42		simpl2 1065	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐵 ∈ ℝ^*)
43	35, 42, 4	syl2anc 693	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
44	36, 39, 41, 43	mpbir3and 1245	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 ∈ (𝐴[,)𝐵))
45		ioossico 12262	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵)
46		simpr 477	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴(,)𝐵))
47	45, 46	sseldi 3601	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴[,)𝐵))
48	44, 47	jaodan 826	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 ∈ (𝐴[,)𝐵))
49	48	ex 450	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴[,)𝐵)))
50	33, 49	impbid 202	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 (,)cioo 12175 [,)cico 12177

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioo 12179 df-ico 12181

This theorem is referenced by: xrge0mulc1cn 29987

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