iocinico - Mathbox for Jon Pennant

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Theorem iocinico 43658

Description: The intersection of two sets that meet at a point is that point. (Contributed by Jon Pennant, 12-Jun-2019.)

**Assertion**
Ref	Expression
iocinico	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝐵})

Proof of Theorem iocinico Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-in 3897	. . . . . 6 ⊢ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝑥 ∣ (𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶))}
2		elioc1 13338	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
3	2	3adant3 1138	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
4		3simpb 1155	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵) → (𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵))
5	3, 4	biimtrdi 254	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) → (𝑥 ∈ ℝ^ ∧ 𝑥 ≤ 𝐵)))
6		elico1 13339	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶)))
7	6	3adant1 1136	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶)))
8		3simpa 1154	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶) → (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥))
9	7, 8	biimtrdi 254	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) → (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥)))
10	5, 9	anim12d 615	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → ((𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶)) → ((𝑥 ∈ ℝ^ ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥))))
11		simpll 772	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
12		simprr 778	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝐵 ≤ 𝑥)
13		simplr 774	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝑥 ≤ 𝐵)
14	11, 12, 13	3jca 1134	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵))
15	10, 14	syl6 35	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → ((𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶)) → (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
16		elicc1 13340	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
17	16	anidms 571	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
18	17	3ad2ant2 1140	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
19	15, 18	sylibrd 260	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ (𝐵[,]𝐵)))
20	19	ss2abdv 4003	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → {𝑥 ∣ (𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶))} ⊆ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)})
21	1, 20	eqsstrid 3960	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)})
22		abid2 2877	. . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)} = (𝐵[,]𝐵)
23	21, 22	sseqtrdi 3962	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ (𝐵[,]𝐵))
24	23	adantr 481	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ (𝐵[,]𝐵))
25		iccid 13341	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵[,]𝐵) = {𝐵})
26	25	3ad2ant2 1140	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵[,]𝐵) = {𝐵})
27	26	adantr 481	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵[,]𝐵) = {𝐵})
28	24, 27	sseqtrd 3958	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ {𝐵})
29		simpl2 1199	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ^)
30		simprl 776	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 < 𝐵)
31	29	xrleidd 13101	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ≤ 𝐵)
32		elioc1 13338	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
33	32	3adant3 1138	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
34	33	adantr 481	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
35	29, 30, 31, 34	mpbir3and 1349	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ (𝐴(,]𝐵))
36		simprr 778	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 < 𝐶)
37		elico1 13339	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ∈ (𝐵[,)𝐶) ↔ (𝐵 ∈ ℝ^ ∧ 𝐵 ≤ 𝐵 ∧ 𝐵 < 𝐶)))
38	37	3adant1 1136	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ∈ (𝐵[,)𝐶) ↔ (𝐵 ∈ ℝ^ ∧ 𝐵 ≤ 𝐵 ∧ 𝐵 < 𝐶)))
39	38	adantr 481	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵 ∈ (𝐵[,)𝐶) ↔ (𝐵 ∈ ℝ^ ∧ 𝐵 ≤ 𝐵 ∧ 𝐵 < 𝐶)))
40	29, 31, 36, 39	mpbir3and 1349	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ (𝐵[,)𝐶))
41	35, 40	elind 4136	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)))
42	41	snssd 4725	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → {𝐵} ⊆ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)))
43	28, 42	eqssd 3939	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝐵})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 {cab 2718 ∩ cin 3889 ⊆ wss 3890 {csn 4562 class class class wbr 5079 (class class class)co 7363 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 (,]cioc 13297 [,)cico 13298 [,]cicc 13299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-pre-lttri 11110 ax-pre-lttrn 11111

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-ioc 13301 df-ico 13302 df-icc 13303

This theorem is referenced by: (None)

W3C validator