iccintsng - Mathbox for Glauco Siliprandi

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Theorem iccintsng 45877

Description: Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

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

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

Step	Hyp	Ref	Expression
1		simpl1 1193	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → 𝐴 ∈ ℝ^)
2		simpl2 1194	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → 𝐵 ∈ ℝ^)
3		simprl 771	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → 𝑥 ∈ (𝐴[,]𝐵))
4		iccleub 13329	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑥 ∈ (𝐴[,]𝐵)) → 𝑥 ≤ 𝐵)
5	1, 2, 3, 4	syl3anc 1374	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → 𝑥 ≤ 𝐵)
6		simpl3 1195	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → 𝐶 ∈ ℝ^)
7		simprr 773	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → 𝑥 ∈ (𝐵[,]𝐶))
8		iccgelb 13330	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝐵 ≤ 𝑥)
9	2, 6, 7, 8	syl3anc 1374	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → 𝐵 ≤ 𝑥)
10		eliccxr 13363	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → 𝑥 ∈ ℝ^*)
11	3, 10	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → 𝑥 ∈ ℝ^)
12	11, 2	jca 511	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → (𝑥 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*))
13		xrletri3 13080	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑥 = 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥)))
14	12, 13	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → (𝑥 = 𝐵 ↔ (𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥)))
15	5, 9, 14	mpbir2and 714	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))) → 𝑥 = 𝐵)
16	15	ex 412	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝑥 = 𝐵))
17	16	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶)) → 𝑥 = 𝐵))
18		simpll1 1214	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) ∧ 𝑥 = 𝐵) → 𝐴 ∈ ℝ^)
19		simpll2 1215	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) ∧ 𝑥 = 𝐵) → 𝐵 ∈ ℝ^)
20		simplrl 777	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) ∧ 𝑥 = 𝐵) → 𝐴 ≤ 𝐵)
21		simpr 484	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
22		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
23		ubicc2 13393	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
24	23	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 = 𝐵) → 𝐵 ∈ (𝐴[,]𝐵))
25	22, 24	eqeltrd 2837	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≤ 𝐵) ∧ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
26	18, 19, 20, 21, 25	syl31anc 1376	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) ∧ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
27		simpll3 1216	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) ∧ 𝑥 = 𝐵) → 𝐶 ∈ ℝ^)
28		simplrr 778	. . . . . . 7 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) ∧ 𝑥 = 𝐵) → 𝐵 ≤ 𝐶)
29		simpr 484	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵)
30		lbicc2 13392	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) → 𝐵 ∈ (𝐵[,]𝐶))
31	30	adantr 480	. . . . . . . 8 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) ∧ 𝑥 = 𝐵) → 𝐵 ∈ (𝐵[,]𝐶))
32	29, 31	eqeltrd 2837	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐵 ≤ 𝐶) ∧ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,]𝐶))
33	19, 27, 28, 21, 32	syl31anc 1376	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) ∧ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,]𝐶))
34	26, 33	jca 511	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) ∧ 𝑥 = 𝐵) → (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶)))
35	34	ex 412	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → (𝑥 = 𝐵 → (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶))))
36	17, 35	impbid 212	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → ((𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶)) ↔ 𝑥 = 𝐵))
37		elin 3919	. . 3 ⊢ (𝑥 ∈ ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) ↔ (𝑥 ∈ (𝐴[,]𝐵) ∧ 𝑥 ∈ (𝐵[,]𝐶)))
38		velsn 4598	. . 3 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
39	36, 37, 38	3bitr4g 314	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶)) → (𝑥 ∈ ((𝐴[,]𝐵) ∩ (𝐵[,]𝐶)) ↔ 𝑥 ∈ {𝐵}))
40	39	eqrdv 2735	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 ∩ cin 3902 {csn 4582 class class class wbr 5100 (class class class)co 7368 ℝ^*cxr 11177 ≤ cle 11179 [,]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-iun 4950 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-1st 7943 df-2nd 7944 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-icc 13280

This theorem is referenced by: iblspltprt 46325

W3C validator