elicoelioo - Mathbox for Thierry Arnoux

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Theorem elicoelioo 31989

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 1192	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐴 ∈ ℝ^*)
2		simpl2 1193	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐵 ∈ ℝ^*)
3		simprl 770	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 ∈ (𝐴[,)𝐵))
4		elico1 13367	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
5	4	biimpa 478	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
6	5	simp1d 1143	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ^)
7	1, 2, 3, 6	syl21anc 837	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 ∈ ℝ^*)
8	5	simp2d 1144	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶)
9	1, 2, 3, 8	syl21anc 837	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐴 ≤ 𝐶)
10	1, 2	jca 513	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
11		simprr 772	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → ¬ 𝐶 ∈ (𝐴(,)𝐵))
12	5	simp3d 1145	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
13	10, 3, 12	syl2anc 585	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 < 𝐵)
14		elioo1 13364	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
15	14	notbid 318	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ 𝐶 ∈ (𝐴(,)𝐵) ↔ ¬ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
16	15	biimpa 478	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → ¬ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
17		3anan32 1098	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) ∧ 𝐴 < 𝐶))
18	17	notbii 320	. . . . . . . . . . 11 ⊢ (¬ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ ¬ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) ∧ 𝐴 < 𝐶))
19		imnan 401	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) → ¬ 𝐴 < 𝐶) ↔ ¬ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) ∧ 𝐴 < 𝐶))
20	18, 19	bitr4i 278	. . . . . . . . . 10 ⊢ (¬ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) → ¬ 𝐴 < 𝐶))
21	16, 20	sylib 217	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → ((𝐶 ∈ ℝ^ ∧ 𝐶 < 𝐵) → ¬ 𝐴 < 𝐶))
22	21	imp 408	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) ∧ (𝐶 ∈ ℝ^ ∧ 𝐶 < 𝐵)) → ¬ 𝐴 < 𝐶)
23	10, 11, 7, 13, 22	syl22anc 838	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → ¬ 𝐴 < 𝐶)
24		xeqlelt 31987	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 = 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ ¬ 𝐴 < 𝐶)))
25	24	biimpar 479	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ ¬ 𝐴 < 𝐶)) → 𝐴 = 𝐶)
26	1, 7, 9, 23, 25	syl22anc 838	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐴 = 𝐶)
27	26	ex 414	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 = 𝐶))
28		eqcom 2740	. . . . 5 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴)
29	27, 28	imbitrdi 250	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 = 𝐴))
30		pm5.6 1001	. . . 4 ⊢ (((𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 = 𝐴) ↔ (𝐶 ∈ (𝐴[,)𝐵) → (𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴)))
31	29, 30	sylib 217	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴)))
32		orcom 869	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵)))
33	31, 32	imbitrdi 250	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵))))
34		simpr 486	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
35		simpl1 1192	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ∈ ℝ^*)
36	34, 35	eqeltrd 2834	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 ∈ ℝ^*)
37	35	xrleidd 13131	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ≤ 𝐴)
38	37, 34	breqtrrd 5177	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ≤ 𝐶)
39		simpl3 1194	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 < 𝐵)
40	34, 39	eqbrtrd 5171	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 < 𝐵)
41		simpl2 1193	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐵 ∈ ℝ^*)
42	35, 41, 4	syl2anc 585	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
43	36, 38, 40, 42	mpbir3and 1343	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 ∈ (𝐴[,)𝐵))
44		ioossico 13415	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵)
45		simpr 486	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴(,)𝐵))
46	44, 45	sselid 3981	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴[,)𝐵))
47	43, 46	jaodan 957	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 ∈ (𝐴[,)𝐵))
48	47	ex 414	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴[,)𝐵)))
49	33, 48	impbid 211	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 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 class class class wbr 5149 (class class class)co 7409 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 (,)cioo 13324 [,)cico 13326

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-ioo 13328 df-ico 13330

This theorem is referenced by: xrge0mulc1cn 32921

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