difioo - Mathbox for Thierry Arnoux

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Theorem difioo 32712

Description: The difference between two open intervals sharing the same lower bound. (Contributed by Thierry Arnoux, 26-Sep-2017.)

**Assertion**
Ref	Expression
difioo	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶))

**Proof of Theorem difioo**
Step	Hyp	Ref	Expression
1		incom 4175	. . . 4 ⊢ ((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ((𝐵[,)𝐶) ∩ (𝐴(,)𝐵))
2		joiniooico 32704	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ ∧ ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)))
3	2	anassrs 467	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → (((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ ∧ ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)))
4	3	simpld 494	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ∅)
5	1, 4	eqtr3id 2779	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → ((𝐵[,)𝐶) ∩ (𝐴(,)𝐵)) = ∅)
6	3	simprd 495	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶))
7		uncom 4124	. . . . 5 ⊢ ((𝐵[,)𝐶) ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶))
8	7	a1i 11	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → ((𝐵[,)𝐶) ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)))
9		simpll1 1213	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → 𝐴 ∈ ℝ^)
10		simpl3 1194	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐶 ∈ ℝ^)
11	10	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → 𝐶 ∈ ℝ^)
12	9	xrleidd 13119	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → 𝐴 ≤ 𝐴)
13		simpr 484	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → 𝐵 ≤ 𝐶)
14		ioossioo 13409	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐵 ≤ 𝐶)) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
15	9, 11, 12, 13, 14	syl22anc 838	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐶))
16		ssequn2 4155	. . . . 5 ⊢ ((𝐴(,)𝐵) ⊆ (𝐴(,)𝐶) ↔ ((𝐴(,)𝐶) ∪ (𝐴(,)𝐵)) = (𝐴(,)𝐶))
17	15, 16	sylib 218	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐶) ∪ (𝐴(,)𝐵)) = (𝐴(,)𝐶))
18	6, 8, 17	3eqtr4d 2775	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → ((𝐵[,)𝐶) ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐶) ∪ (𝐴(,)𝐵)))
19		difeq 32454	. . 3 ⊢ (((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶) ↔ (((𝐵[,)𝐶) ∩ (𝐴(,)𝐵)) = ∅ ∧ ((𝐵[,)𝐶) ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐶) ∪ (𝐴(,)𝐵))))
20	5, 18, 19	sylanbrc 583	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶))
21		simpll1 1213	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → 𝐴 ∈ ℝ^)
22		simpl2 1193	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^)
23	22	adantr 480	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → 𝐵 ∈ ℝ^)
24	21	xrleidd 13119	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → 𝐴 ≤ 𝐴)
25	10	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → 𝐶 ∈ ℝ^)
26		simpr 484	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → 𝐶 < 𝐵)
27	25, 23, 26	xrltled 13117	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵)
28		ioossioo 13409	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐶 ≤ 𝐵)) → (𝐴(,)𝐶) ⊆ (𝐴(,)𝐵))
29	21, 23, 24, 27, 28	syl22anc 838	. . . 4 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → (𝐴(,)𝐶) ⊆ (𝐴(,)𝐵))
30		ssdif0 4332	. . . 4 ⊢ ((𝐴(,)𝐶) ⊆ (𝐴(,)𝐵) ↔ ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = ∅)
31	29, 30	sylib 218	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = ∅)
32		ico0 13359	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐵[,)𝐶) = ∅ ↔ 𝐶 ≤ 𝐵))
33	32	biimpar 477	. . . 4 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐶 ≤ 𝐵) → (𝐵[,)𝐶) = ∅)
34	23, 25, 27, 33	syl21anc 837	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → (𝐵[,)𝐶) = ∅)
35	31, 34	eqtr4d 2768	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶))
36		xrlelttric 32682	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ≤ 𝐶 ∨ 𝐶 < 𝐵))
37	22, 10, 36	syl2anc 584	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ≤ 𝐶 ∨ 𝐶 < 𝐵))
38	20, 35, 37	mpjaodan 960	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∖ cdif 3914 ∪ cun 3915 ∩ cin 3916 ⊆ wss 3917 ∅c0 4299 class class class wbr 5110 (class class class)co 7390 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 (,)cioo 13313 [,)cico 13315

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-n0 12450 df-z 12537 df-uz 12801 df-q 12915 df-ioo 13317 df-ico 13319

This theorem is referenced by: dya2iocbrsiga 34273 dya2icobrsiga 34274

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