difioo - Mathbox for Thierry Arnoux

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Theorem difioo 32738

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 4162	. . . 4 ⊢ ((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ((𝐵[,)𝐶) ∩ (𝐴(,)𝐵))
2		joiniooico 32730	. . . . . 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 2778	. . 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 4111	. . . . 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 13072	. . . . . 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 13362	. . . . . 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 4142	. . . . 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 2774	. . 3 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐵 ≤ 𝐶) → ((𝐵[,)𝐶) ∪ (𝐴(,)𝐵)) = ((𝐴(,)𝐶) ∪ (𝐴(,)𝐵)))
19		difeq 32480	. . 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 13072	. . . . 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 13070	. . . . 5 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → 𝐶 ≤ 𝐵)
28		ioossioo 13362	. . . . 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 4319	. . . 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 13312	. . . . 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 2767	. 2 ⊢ ((((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 < 𝐵) ∧ 𝐶 < 𝐵) → ((𝐴(,)𝐶) ∖ (𝐴(,)𝐵)) = (𝐵[,)𝐶))
36		xrlelttric 32708	. . 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 3902 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 class class class wbr 5095 (class class class)co 7353 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 (,)cioo 13266 [,)cico 13268

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 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-n0 12403 df-z 12490 df-uz 12754 df-q 12868 df-ioo 13270 df-ico 13272

This theorem is referenced by: dya2iocbrsiga 34245 dya2icobrsiga 34246

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