iocinico - Mathbox for Jon Pennant

	Mathbox for Jon Pennant		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > iocinico		Structured version Visualization version GIF version

Theorem iocinico 41951

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 3955	. . . . . 6 ⊢ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) = {𝑥 ∣ (𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶))}
2		elioc1 13365	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
3	2	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵)))
4		3simpb 1149	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐴 < 𝑥 ∧ 𝑥 ≤ 𝐵) → (𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵))
5	3, 4	syl6bi 252	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐴(,]𝐵) → (𝑥 ∈ ℝ^ ∧ 𝑥 ≤ 𝐵)))
6		elico1 13366	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶)))
7	6	3adant1 1130	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶)))
8		3simpa 1148	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 < 𝐶) → (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥))
9	7, 8	syl6bi 252	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,)𝐶) → (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥)))
10	5, 9	anim12d 609	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → ((𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶)) → ((𝑥 ∈ ℝ^ ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥))))
11		simpll 765	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝑥 ∈ ℝ^*)
12		simprr 771	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝐵 ≤ 𝑥)
13		simplr 767	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ^* ∧ 𝑥 ≤ 𝐵) ∧ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥)) → 𝑥 ≤ 𝐵)
14	11, 12, 13	3jca 1128	. . . . . . . . 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 13367	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
17	16	anidms 567	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^* ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
18	17	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑥 ∈ (𝐵[,]𝐵) ↔ (𝑥 ∈ ℝ^ ∧ 𝐵 ≤ 𝑥 ∧ 𝑥 ≤ 𝐵)))
19	15, 18	sylibrd 258	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ (𝐵[,]𝐵)))
20	19	ss2abdv 4060	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → {𝑥 ∣ (𝑥 ∈ (𝐴(,]𝐵) ∧ 𝑥 ∈ (𝐵[,)𝐶))} ⊆ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)})
21	1, 20	eqsstrid 4030	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)})
22		abid2 2871	. . . . 5 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐵[,]𝐵)} = (𝐵[,]𝐵)
23	21, 22	sseqtrdi 4032	. . . 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 13368	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (𝐵[,]𝐵) = {𝐵})
26	25	3ad2ant2 1134	. . . 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 4022	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)) ⊆ {𝐵})
29		simpl2 1192	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ^)
30		simprl 769	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐴 < 𝐵)
31	29	xrleidd 13130	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ≤ 𝐵)
32		elioc1 13365	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
33	32	3adant3 1132	. . . . . 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 1342	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ (𝐴(,]𝐵))
36		simprr 771	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 < 𝐶)
37		elico1 13366	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝐵 ∈ (𝐵[,)𝐶) ↔ (𝐵 ∈ ℝ^ ∧ 𝐵 ≤ 𝐵 ∧ 𝐵 < 𝐶)))
38	37	3adant1 1130	. . . . . 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 1342	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ (𝐵[,)𝐶))
41	35, 40	elind 4194	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)))
42	41	snssd 4812	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → {𝐵} ⊆ ((𝐴(,]𝐵) ∩ (𝐵[,)𝐶)))
43	28, 42	eqssd 3999	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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {cab 2709 ∩ cin 3947 ⊆ wss 3948 {csn 4628 class class class wbr 5148 (class class class)co 7408 ℝ^*cxr 11246 < clt 11247 ≤ cle 11248 (,]cioc 13324 [,)cico 13325 [,]cicc 13326

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-ioc 13328 df-ico 13329 df-icc 13330

This theorem is referenced by: (None)

W3C validator