difioo - Mathbox for Thierry Arnoux

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Theorem difioo 32763

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 4159	. . . 4 ⊢ ((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ((𝐵[,)𝐶) ∩ (𝐴(,)𝐵))
2		joiniooico 32755	. . . . . 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 2780	. . 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 4108	. . . . 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 13051	. . . . . 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 13341	. . . . . 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 4139	. . . . 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 2776	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → ((𝐵[,)𝐶) ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐶) ∪ (𝐴(,)𝐵)))
19		difeq 32496	. . 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 13051	. . . . 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 13049	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵)
28		ioossioo 13341	. . . . 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 4316	. . . 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 13291	. . . . 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 2769	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶))
36		xrlelttric 32733	. . 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 1541 ∈ wcel 2111 ∖ cdif 3899 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 ∅c0 4283 class class class wbr 5091 (class class class)co 7346 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 (,)cioo 13245 [,)cico 13247

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-ioo 13249 df-ico 13251

This theorem is referenced by: dya2iocbrsiga 34286 dya2icobrsiga 34287

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