iocinico - Mathbox for Jon Pennant

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Theorem iocinico 43558

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 3910	. . . . . 6 ⊢ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝑥 ∣ (𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶))}
2		elioc1 13315	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
3	2	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
4		3simpb 1150	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵) → (𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵))
5	3, 4	biimtrdi 253	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) → (𝑥 ∈ ℝ^ ∧ 𝑥 ≤ 𝐵)))
6		elico1 13316	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶)))
7	6	3adant1 1131	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶)))
8		3simpa 1149	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶) → (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥))
9	7, 8	biimtrdi 253	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) → (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥)))
10	5, 9	anim12d 610	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → ((𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶)) → ((𝑥 ∈ ℝ^ ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥))))
11		simpll 767	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
12		simprr 773	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝐵 ≤ 𝑥)
13		simplr 769	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝑥 ≤ 𝐵)
14	11, 12, 13	3jca 1129	. . . . . . . . 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 13317	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
17	16	anidms 566	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
18	17	3ad2ant2 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
19	15, 18	sylibrd 259	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ (𝐵[,]𝐵)))
20	19	ss2abdv 4019	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → {𝑥 ∣ (𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶))} ⊆ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)})
21	1, 20	eqsstrid 3974	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)})
22		abid2 2874	. . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)} = (𝐵[,]𝐵)
23	21, 22	sseqtrdi 3976	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ (𝐵[,]𝐵))
24	23	adantr 480	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ (𝐵[,]𝐵))
25		iccid 13318	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵[,]𝐵) = {𝐵})
26	25	3ad2ant2 1135	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵[,]𝐵) = {𝐵})
27	26	adantr 480	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → (𝐵[,]𝐵) = {𝐵})
28	24, 27	sseqtrd 3972	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ {𝐵})
29		simpl2 1194	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ^)
30		simprl 771	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 < 𝐵)
31	29	xrleidd 13078	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ≤ 𝐵)
32		elioc1 13315	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
33	32	3adant3 1133	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
34	33	adantr 480	. . . . 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 1344	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ (𝐴(,]𝐵))
36		simprr 773	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 < 𝐶)
37		elico1 13316	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ∈ (𝐵[,)𝐶) ↔ (𝐵 ∈ ℝ^ ∧ 𝐵 ≤ 𝐵 ∧ 𝐵 < 𝐶)))
38	37	3adant1 1131	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ∈ (𝐵[,)𝐶) ↔ (𝐵 ∈ ℝ^ ∧ 𝐵 ≤ 𝐵 ∧ 𝐵 < 𝐶)))
39	38	adantr 480	. . . . 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 1344	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ (𝐵[,)𝐶))
41	35, 40	elind 4154	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)))
42	41	snssd 4767	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → {𝐵} ⊆ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)))
43	28, 42	eqssd 3953	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {cab 2715 ∩ cin 3902 ⊆ wss 3903 {csn 4582 class class class wbr 5100 (class class class)co 7368 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 (,]cioc 13274 [,)cico 13275 [,]cicc 13276

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-ioc 13278 df-ico 13279 df-icc 13280

This theorem is referenced by: (None)

W3C validator