elicoelioo - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > elicoelioo		Structured version Visualization version GIF version

Theorem elicoelioo 30482

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 1187	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐴 ∈ ℝ^*)
2		simpl2 1188	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐵 ∈ ℝ^*)
3		simprl 769	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 ∈ (𝐴[,)𝐵))
4		elico1 12763	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
5	4	biimpa 479	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
6	5	simp1d 1138	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ^)
7	1, 2, 3, 6	syl21anc 835	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 ∈ ℝ^*)
8	5	simp2d 1139	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶)
9	1, 2, 3, 8	syl21anc 835	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐴 ≤ 𝐶)
10	1, 2	jca 514	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
11		simprr 771	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → ¬ 𝐶 ∈ (𝐴(,)𝐵))
12	5	simp3d 1140	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 < 𝐵)
13	10, 3, 12	syl2anc 586	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 < 𝐵)
14		elioo1 12760	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
15	14	notbid 320	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ 𝐶 ∈ (𝐴(,)𝐵) ↔ ¬ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
16	15	biimpa 479	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → ¬ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
17		3anan32 1093	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) ∧ 𝐴 < 𝐶))
18	17	notbii 322	. . . . . . . . . . 11 ⊢ (¬ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ ¬ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) ∧ 𝐴 < 𝐶))
19		imnan 402	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) → ¬ 𝐴 < 𝐶) ↔ ¬ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) ∧ 𝐴 < 𝐶))
20	18, 19	bitr4i 280	. . . . . . . . . 10 ⊢ (¬ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ ((𝐶 ∈ ℝ^* ∧ 𝐶 < 𝐵) → ¬ 𝐴 < 𝐶))
21	16, 20	sylib 220	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → ((𝐶 ∈ ℝ^ ∧ 𝐶 < 𝐵) → ¬ 𝐴 < 𝐶))
22	21	imp 409	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) ∧ (𝐶 ∈ ℝ^ ∧ 𝐶 < 𝐵)) → ¬ 𝐴 < 𝐶)
23	10, 11, 7, 13, 22	syl22anc 836	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → ¬ 𝐴 < 𝐶)
24		xeqlelt 30480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 = 𝐶 ↔ (𝐴 ≤ 𝐶 ∧ ¬ 𝐴 < 𝐶)))
25	24	biimpar 480	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ ¬ 𝐴 < 𝐶)) → 𝐴 = 𝐶)
26	1, 7, 9, 23, 25	syl22anc 836	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐴 = 𝐶)
27	26	ex 415	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐴 = 𝐶))
28		eqcom 2827	. . . . 5 ⊢ (𝐴 = 𝐶 ↔ 𝐶 = 𝐴)
29	27, 28	syl6ib 253	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 = 𝐴))
30		pm5.6 998	. . . 4 ⊢ (((𝐶 ∈ (𝐴[,)𝐵) ∧ ¬ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 = 𝐴) ↔ (𝐶 ∈ (𝐴[,)𝐵) → (𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴)))
31	29, 30	sylib 220	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴)))
32		orcom 866	. . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵)))
33	31, 32	syl6ib 253	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵))))
34		simpr 487	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
35		simpl1 1187	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ∈ ℝ^*)
36	34, 35	eqeltrd 2911	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 ∈ ℝ^*)
37	35	xrleidd 12527	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ≤ 𝐴)
38	37, 34	breqtrrd 5075	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ≤ 𝐶)
39		simpl3 1189	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 < 𝐵)
40	34, 39	eqbrtrd 5069	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 < 𝐵)
41		simpl2 1188	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐵 ∈ ℝ^*)
42	35, 41, 4	syl2anc 586	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
43	36, 38, 40, 42	mpbir3and 1338	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 ∈ (𝐴[,)𝐵))
44		ioossico 12808	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵)
45		simpr 487	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴(,)𝐵))
46	44, 45	sseldi 3948	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴[,)𝐵))
47	43, 46	jaodan 954	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ∧ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵))) → 𝐶 ∈ (𝐴[,)𝐵))
48	47	ex 415	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ((𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴[,)𝐵)))
49	33, 48	impbid 214	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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5047 (class class class)co 7137 ℝ^*cxr 10655 < clt 10656 ≤ cle 10657 (,)cioo 12720 [,)cico 12722

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-pre-lttri 10592 ax-pre-lttrn 10593

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-br 5048 df-opab 5110 df-mpt 5128 df-id 5441 df-po 5455 df-so 5456 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-ov 7140 df-oprab 7141 df-mpo 7142 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-ioo 12724 df-ico 12726

This theorem is referenced by: xrge0mulc1cn 31186

W3C validator