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Theorem ioombl 25866

Description: An open real interval is measurable. (Contributed by Mario Carneiro, 16-Jun-2014.)

**Assertion**
Ref	Expression
ioombl	⊢ (𝐴(,)𝐵) ∈ dom vol

Proof of Theorem ioombl Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snunioo 13590	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
2	1	3expa 1136	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
3	2	adantrr 730	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
4		lbico1 13512	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
5	4	3expa 1136	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
6	5	adantrr 730	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ (𝐴[,)𝐵))
7	6	snssd 4747	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ⊆ (𝐴[,)𝐵))
8		iccid 13502	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → (𝐴[,]𝐴) = {𝐴})
9	8	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴[,]𝐴) = {𝐴})
10	9	ineq1d 4165	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ({𝐴} ∩ (𝐴(,)𝐵)))
11		simpll 779	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ ℝ^)
12		simplr 781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐵 ∈ ℝ^)
13		df-icc 13464	. . . . . . . . . . 11 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
14		df-ioo 13461	. . . . . . . . . . 11 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
15		xrltnle 11357	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴))
16	13, 14, 15	ixxdisj 13472	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ∅)
17	11, 11, 12, 16	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ∅)
18	10, 17	eqtr3d 2798	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ({𝐴} ∩ (𝐴(,)𝐵)) = ∅)
19		uneqdifeq 4448	. . . . . . . 8 ⊢ (({𝐴} ⊆ (𝐴[,)𝐵) ∧ ({𝐴} ∩ (𝐴(,)𝐵)) = ∅) → (({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵)))
20	7, 18, 19	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵)))
21	3, 20	mpbid 235	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵))
22		mnfxr 11347	. . . . . . . . . 10 ⊢ -∞ ∈ ℝ^*
23	22	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → -∞ ∈ ℝ^)
24		simprr 785	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → -∞ < 𝐴)
25		simprl 783	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 < 𝐵)
26		xrre2 13281	. . . . . . . . 9 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (-∞ < 𝐴 ∧ 𝐴 < 𝐵)) → 𝐴 ∈ ℝ)
27	23, 11, 12, 24, 25, 26	syl32anc 1405	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ ℝ)
28		icombl 25865	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)
29	27, 12, 28	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴[,)𝐵) ∈ dom vol)
30	27	snssd 4747	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ⊆ ℝ)
31		ovolsn 25796	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)
32	27, 31	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (vol‘{𝐴}) = 0)
33		nulmbl 25836	. . . . . . . 8 ⊢ (({𝐴} ⊆ ℝ ∧ (vol*‘{𝐴}) = 0) → {𝐴} ∈ dom vol)
34	30, 32, 33	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ∈ dom vol)
35		difmbl 25844	. . . . . . 7 ⊢ (((𝐴[,)𝐵) ∈ dom vol ∧ {𝐴} ∈ dom vol) → ((𝐴[,)𝐵) ∖ {𝐴}) ∈ dom vol)
36	29, 34, 35	syl2anc 596	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,)𝐵) ∖ {𝐴}) ∈ dom vol)
37	21, 36	eqeltrrd 2862	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴(,)𝐵) ∈ dom vol)
38	37	expr 462	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ < 𝐴 → (𝐴(,)𝐵) ∈ dom vol))
39		uncom 4105	. . . . . . . . 9 ⊢ ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ((-∞(,)𝐵) ∪ (𝐵[,)+∞))
40	22	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → -∞ ∈ ℝ^)
41		simplr 781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^)
42		pnfxr 11344	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
43	42	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → +∞ ∈ ℝ^)
44		simpll 779	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^)
45		mnfle 13245	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
46	45	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → -∞ ≤ 𝐴)
47		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
48	40, 44, 41, 46, 47	xrlelttrd 13270	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → -∞ < 𝐵)
49		pnfge 13240	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
50	41, 49	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐵 ≤ +∞)
51		df-ico 13463	. . . . . . . . . . 11 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
52		xrlenlt 11355	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵))
53		xrltletr 13267	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ +∞) → 𝑤 < +∞))
54		xrltletr 13267	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝑤) → -∞ < 𝑤))
55	14, 51, 52, 14, 53, 54	ixxun 13473	. . . . . . . . . 10 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 𝐵 ∧ 𝐵 ≤ +∞)) → ((-∞(,)𝐵) ∪ (𝐵[,)+∞)) = (-∞(,)+∞))
56	40, 41, 43, 48, 50, 55	syl32anc 1405	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((-∞(,)𝐵) ∪ (𝐵[,)+∞)) = (-∞(,)+∞))
57	39, 56	eqtrid 2808	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = (-∞(,)+∞))
58		ioomax 13534	. . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ
59	57, 58	eqtrdi 2812	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ)
60		ssun1 4124	. . . . . . . . 9 ⊢ (𝐵[,)+∞) ⊆ ((𝐵[,)+∞) ∪ (-∞(,)𝐵))
61	60, 59	sseqtrid 3973	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ⊆ ℝ)
62		incom 4155	. . . . . . . . 9 ⊢ ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ((-∞(,)𝐵) ∩ (𝐵[,)+∞))
63	14, 51, 52	ixxdisj 13472	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞(,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
64	40, 41, 43, 63	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((-∞(,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
65	62, 64	eqtrid 2808	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ∅)
66		uneqdifeq 4448	. . . . . . . 8 ⊢ (((𝐵[,)+∞) ⊆ ℝ ∧ ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ∅) → (((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ ↔ (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵)))
67	61, 65, 66	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ ↔ (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵)))
68	59, 67	mpbid 235	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵))
69		rembl 25841	. . . . . . 7 ⊢ ℝ ∈ dom vol
70		xrleloe 13254	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
71	41, 42, 70	sylancl 598	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
72	50, 71	mpbid 235	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ ∨ 𝐵 = +∞))
73		xrre2 13281	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < +∞)) → 𝐵 ∈ ℝ)
74	73	expr 462	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
75	42, 74	mp3anl3 1486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
76	75	orim1d 981	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵 < +∞ ∨ 𝐵 = +∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞)))
77	72, 76	mpd 16	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
78		icombl1 25864	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (𝐵[,)+∞) ∈ dom vol)
79		oveq1 7419	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = (+∞[,)+∞))
80		pnfge 13240	. . . . . . . . . . . . 13 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
81	42, 80	ax-mp 5	. . . . . . . . . . . 12 ⊢ +∞ ≤ +∞
82		ico0 13503	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞))
83	42, 42, 82	mp2an 705	. . . . . . . . . . . 12 ⊢ ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞)
84	81, 83	mpbir 234	. . . . . . . . . . 11 ⊢ (+∞[,)+∞) = ∅
85	79, 84	eqtrdi 2812	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = ∅)
86		0mbl 25840	. . . . . . . . . 10 ⊢ ∅ ∈ dom vol
87	85, 86	eqeltrdi 2869	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) ∈ dom vol)
88	78, 87	jaoi 871	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = +∞) → (𝐵[,)+∞) ∈ dom vol)
89	77, 88	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ∈ dom vol)
90		difmbl 25844	. . . . . . 7 ⊢ ((ℝ ∈ dom vol ∧ (𝐵[,)+∞) ∈ dom vol) → (ℝ ∖ (𝐵[,)+∞)) ∈ dom vol)
91	69, 89, 90	sylancr 599	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (ℝ ∖ (𝐵[,)+∞)) ∈ dom vol)
92	68, 91	eqeltrrd 2862	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞(,)𝐵) ∈ dom vol)
93		oveq1 7419	. . . . . 6 ⊢ (-∞ = 𝐴 → (-∞(,)𝐵) = (𝐴(,)𝐵))
94	93	eleq1d 2846	. . . . 5 ⊢ (-∞ = 𝐴 → ((-∞(,)𝐵) ∈ dom vol ↔ (𝐴(,)𝐵) ∈ dom vol))
95	92, 94	syl5ibcom 248	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ = 𝐴 → (𝐴(,)𝐵) ∈ dom vol))
96		xrleloe 13254	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
97	22, 44, 96	sylancr 599	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
98	46, 97	mpbid 235	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ < 𝐴 ∨ -∞ = 𝐴))
99	38, 95, 98	mpjaod 874	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
100		ioo0 13482	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
101		xrlenlt 11355	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
102	101	ancoms 464	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
103	100, 102	bitrd 282	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ ¬ 𝐴 < 𝐵))
104	103	biimpar 483	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴(,)𝐵) = ∅)
105	104, 86	eqeltrdi 2869	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
106	99, 105	pm2.61dan 825	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ dom vol)
107		ndmioo 13484	. . 3 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
108	107, 86	eqeltrdi 2869	. 2 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ dom vol)
109	106, 108	pm2.61i 184	1 ⊢ (𝐴(,)𝐵) ∈ dom vol

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∖ cdif 3896 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {csn 4584 class class class wbr 5103 dom cdm 5651 ‘cfv 6531 (class class class)co 7412 ℝcr 11180 0cc0 11181 +∞cpnf 11321 -∞cmnf 11322 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 (,)cioo 13457 [,)cico 13459 [,]cicc 13460 vol*covol 25763 volcvol 25764

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-pm 8834 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9418 df-inf 9419 df-oi 9488 df-dju 9963 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-n0 12588 df-z 12675 df-uz 12947 df-q 13057 df-rp 13102 df-xadd 13223 df-ioo 13461 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-seq 14125 df-exp 14185 df-hash 14455 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-clim 15635 df-rlim 15636 df-sum 15834 df-xmet 21651 df-met 21652 df-ovol 25765 df-vol 25766

This theorem is used by: iccmbl 25867 ovolioo 25869 volioo 25870 ioovolcl 25871 uniioovol 25880 uniioombllem4 25887 uniioombllem5 25888 opnmblALT 25904 mbfid 25936 ditgcl 26158 ditgswap 26159 ditgsplitlem 26160 ftc1lem1 26335 ftc1lem2 26336 ftc1a 26337 ftc1lem4 26339 ftc2 26344 ftc2ditglem 26345 itgsubstlem 26348 itgpowd 26350 ftc2re 35210 fdvposlt 35211 fdvposle 35213 itgexpif 35218 circlemeth 35252 itg2gt0cn 38561 ftc1cnnclem 38577 ftc1anclem7 38585 ftc1anclem8 38586 ftc1anc 38587 ftc2nc 38588 areacirc 38599 lcmineqlem10 43056 lcmineqlem12 43058 iocmbl 44173 cnioobibld 44174 lhe4.4ex1a 45272 itgsin0pilem1 46904 iblioosinexp 46907 itgsinexplem1 46908 itgsinexp 46909 itgcoscmulx 46923 volioc 46926 itgsincmulx 46928 iblcncfioo 46932 itgiccshift 46934 itgperiod 46935 itgsbtaddcnst 46936 volico 46937 volioof 46941 wallispilem2 47020 dirkeritg 47056 fourierdlem16 47077 fourierdlem21 47082 fourierdlem22 47083 fourierdlem39 47100 fourierdlem73 47133 fourierdlem83 47143 fourierdlem103 47163 fourierdlem104 47164 fourierdlem111 47171 fourierdlem112 47172 sqwvfoura 47182 sqwvfourb 47183 etransclem18 47206 etransclem23 47211 ovolval4lem1 47603 ovolval5lem1 47606

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