iocinico - Mathbox for Jon Pennant

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Theorem iocinico 43737

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 3906	. . . . . 6 ⊢ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝑥 ∣ (𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶))}
2		elioc1 13381	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
3	2	3adant3 1141	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
4		3simpb 1158	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵) → (𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵))
5	3, 4	biimtrdi 255	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) → (𝑥 ∈ ℝ^ ∧ 𝑥 ≤ 𝐵)))
6		elico1 13382	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶)))
7	6	3adant1 1139	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶)))
8		3simpa 1157	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶) → (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥))
9	7, 8	biimtrdi 255	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) → (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥)))
10	5, 9	anim12d 617	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → ((𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶)) → ((𝑥 ∈ ℝ^ ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥))))
11		simpll 774	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
12		simprr 780	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝐵 ≤ 𝑥)
13		simplr 776	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝑥 ≤ 𝐵)
14	11, 12, 13	3jca 1137	. . . . . . . . 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 13383	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
17	16	anidms 573	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
18	17	3ad2ant2 1143	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
19	15, 18	sylibrd 261	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ (𝐵[,]𝐵)))
20	19	ss2abdv 4013	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → {𝑥 ∣ (𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶))} ⊆ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)})
21	1, 20	eqsstrid 3969	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)})
22		abid2 2893	. . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)} = (𝐵[,]𝐵)
23	21, 22	sseqtrdi 3971	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ (𝐵[,]𝐵))
24	23	adantr 483	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ (𝐵[,]𝐵))
25		iccid 13384	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵[,]𝐵) = {𝐵})
26	25	3ad2ant2 1143	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵[,]𝐵) = {𝐵})
27	26	adantr 483	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵[,]𝐵) = {𝐵})
28	24, 27	sseqtrd 3967	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ {𝐵})
29		simpl2 1202	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ^)
30		simprl 778	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 < 𝐵)
31	29	xrleidd 13144	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ≤ 𝐵)
32		elioc1 13381	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
33	32	3adant3 1141	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
34	33	adantr 483	. . . . 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 1352	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ (𝐴(,]𝐵))
36		simprr 780	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 < 𝐶)
37		elico1 13382	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ∈ (𝐵[,)𝐶) ↔ (𝐵 ∈ ℝ^ ∧ 𝐵 ≤ 𝐵 ∧ 𝐵 < 𝐶)))
38	37	3adant1 1139	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ∈ (𝐵[,)𝐶) ↔ (𝐵 ∈ ℝ^ ∧ 𝐵 ≤ 𝐵 ∧ 𝐵 < 𝐶)))
39	38	adantr 483	. . . . 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 1352	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ (𝐵[,)𝐶))
41	35, 40	elind 4147	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)))
42	41	snssd 4739	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → {𝐵} ⊆ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)))
43	28, 42	eqssd 3948	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 208 ∧ wa 398 ∧ w3a 1095 = wceq 1554 ∈ wcel 2136 {cab 2734 ∩ cin 3898 ⊆ wss 3899 {csn 4576 class class class wbr 5094 (class class class)co 7385 ℝ^*cxr 11205 < clt 11206 ≤ cle 11207 (,]cioc 13340 [,)cico 13341 [,]cicc 13342

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-cnex 11119 ax-resscn 11120 ax-pre-lttri 11137 ax-pre-lttrn 11138

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5095 df-opab 5157 df-mpt 5176 df-id 5535 df-po 5548 df-so 5549 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-ov 7388 df-oprab 7389 df-mpo 7390 df-er 8666 df-en 8917 df-dom 8918 df-sdom 8919 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-ioc 13344 df-ico 13345 df-icc 13346

This theorem is referenced by: (None)

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