Theorem eliccelico 30615

Description: Relate elementhood to a closed interval with elementhood to the same closed-below, open-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.)

Assertion

Ref	Expression
eliccelico	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵)))

Proof of Theorem eliccelico

Step	Hyp	Ref	Expression
1		simpl1 1189	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵))) → 𝐴 ∈ ℝ*)
2		simpl2 1190	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵))) → 𝐵 ∈ ℝ*)
3		simprl 771	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵))) → 𝐶 ∈ (𝐴[,]𝐵))
4		elicc1 12816	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
5	4	biimpa 481	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
6	5	simp1d 1140	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ℝ*)
7	1, 2, 3, 6	syl21anc 837	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵))) → 𝐶 ∈ ℝ*)
8	5	simp3d 1142	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
9	1, 2, 3, 8	syl21anc 837	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵))) → 𝐶 ≤ 𝐵)
10	1, 2	jca 516	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵))) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*))
11		simprr 773	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵))) → ¬ 𝐶 ∈ (𝐴[,)𝐵))
12	5	simp2d 1141	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
13	10, 3, 12	syl2anc 588	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵))) → 𝐴 ≤ 𝐶)
14		elico1 12815	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
15	14	notbid 322	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (¬ 𝐶 ∈ (𝐴[,)𝐵) ↔ ¬ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
16	15	biimpa 481	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵)) → ¬ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
17		df-3an 1087	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) ↔ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶) ∧ 𝐶 < 𝐵))
18	17	notbii 324	. . . . . . . . 9 ⊢ (¬ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) ↔ ¬ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶) ∧ 𝐶 < 𝐵))
19		imnan 404	. . . . . . . . 9 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶) → ¬ 𝐶 < 𝐵) ↔ ¬ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶) ∧ 𝐶 < 𝐵))
20	18, 19	bitr4i 281	. . . . . . . 8 ⊢ (¬ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) ↔ ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶) → ¬ 𝐶 < 𝐵))
21	16, 20	sylib 221	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶) → ¬ 𝐶 < 𝐵))
22	21	imp 411	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵)) ∧ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶)) → ¬ 𝐶 < 𝐵)
23	10, 11, 7, 13, 22	syl22anc 838	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵))) → ¬ 𝐶 < 𝐵)
24		xeqlelt 30614	. . . . . 6 ⊢ ((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 = 𝐵 ↔ (𝐶 ≤ 𝐵 ∧ ¬ 𝐶 < 𝐵)))
25	24	biimpar 482	. . . . 5 ⊢ (((𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ≤ 𝐵 ∧ ¬ 𝐶 < 𝐵)) → 𝐶 = 𝐵)
26	7, 2, 9, 23, 25	syl22anc 838	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵))) → 𝐶 = 𝐵)
27	26	ex 417	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 = 𝐵))
28		pm5.6 1000	. . 3 ⊢ (((𝐶 ∈ (𝐴[,]𝐵) ∧ ¬ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 = 𝐵) ↔ (𝐶 ∈ (𝐴[,]𝐵) → (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵)))
29	27, 28	sylib 221	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) → (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵)))
30		icossicc 12861	. . . . 5 ⊢ (𝐴[,)𝐵) ⊆ (𝐴[,]𝐵)
31		simpr 489	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ (𝐴[,)𝐵))
32	30, 31	sseldi 3891	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ (𝐴[,]𝐵))
33		simpr 489	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐶 = 𝐵)
34		simpl2 1190	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐵 ∈ ℝ*)
35	33, 34	eqeltrd 2853	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐶 ∈ ℝ*)
36		simpl3 1191	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐴 ≤ 𝐵)
37	36, 33	breqtrrd 5061	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐴 ≤ 𝐶)
38	34	xrleidd 12579	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐵 ≤ 𝐵)
39	33, 38	eqbrtrd 5055	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐶 ≤ 𝐵)
40		simpl1 1189	. . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐴 ∈ ℝ*)
41	40, 34, 4	syl2anc 588	. . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
42	35, 37, 39, 41	mpbir3and 1340	. . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐶 ∈ (𝐴[,]𝐵))
43	32, 42	jaodan 956	. . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ∈ (𝐴[,]𝐵))
44	43	ex 417	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵) → 𝐶 ∈ (𝐴[,]𝐵)))
45	29, 44	impbid 215	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 ∈ (𝐴[,)𝐵) ∨ 𝐶 = 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   class class class wbr 5033  (class class class)co 7151  ℝ^*cxr 10705   < clt 10706   ≤ cle 10707  [,)cico 12774  [,]cicc 12775

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10624  ax-resscn 10625  ax-pre-lttri 10642  ax-pre-lttrn 10643

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-po 5444  df-so 5445  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  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 7154  df-oprab 7155  df-mpo 7156  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-pnf 10708  df-mnf 10709  df-xr 10710  df-ltxr 10711  df-le 10712  df-ico 12778  df-icc 12779

This theorem is referenced by:  xrge0adddir  30820  esumcvg  31566