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Theorem ioombl 25619

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 13538	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
2	1	3expa 1118	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
3	2	adantrr 716	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
4		lbico1 13461	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
5	4	3expa 1118	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
6	5	adantrr 716	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ (𝐴[,)𝐵))
7	6	snssd 4834	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ⊆ (𝐴[,)𝐵))
8		iccid 13452	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → (𝐴[,]𝐴) = {𝐴})
9	8	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴[,]𝐴) = {𝐴})
10	9	ineq1d 4240	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ({𝐴} ∩ (𝐴(,)𝐵)))
11		simpll 766	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ ℝ^)
12		simplr 768	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐵 ∈ ℝ^)
13		df-icc 13414	. . . . . . . . . . 11 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
14		df-ioo 13411	. . . . . . . . . . 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 13422	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ∅)
17	11, 11, 12, 16	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ∅)
18	10, 17	eqtr3d 2782	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ({𝐴} ∩ (𝐴(,)𝐵)) = ∅)
19		uneqdifeq 4516	. . . . . . . 8 ⊢ (({𝐴} ⊆ (𝐴[,)𝐵) ∧ ({𝐴} ∩ (𝐴(,)𝐵)) = ∅) → (({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵)))
20	7, 18, 19	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵)))
21	3, 20	mpbid 232	. . . . . 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 772	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → -∞ < 𝐴)
25		simprl 770	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 < 𝐵)
26		xrre2 13232	. . . . . . . . 9 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (-∞ < 𝐴 ∧ 𝐴 < 𝐵)) → 𝐴 ∈ ℝ)
27	23, 11, 12, 24, 25, 26	syl32anc 1378	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ ℝ)
28		icombl 25618	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)
29	27, 12, 28	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴[,)𝐵) ∈ dom vol)
30	27	snssd 4834	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ⊆ ℝ)
31		ovolsn 25549	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)
32	27, 31	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (vol‘{𝐴}) = 0)
33		nulmbl 25589	. . . . . . . 8 ⊢ (({𝐴} ⊆ ℝ ∧ (vol*‘{𝐴}) = 0) → {𝐴} ∈ dom vol)
34	30, 32, 33	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ∈ dom vol)
35		difmbl 25597	. . . . . . 7 ⊢ (((𝐴[,)𝐵) ∈ dom vol ∧ {𝐴} ∈ dom vol) → ((𝐴[,)𝐵) ∖ {𝐴}) ∈ dom vol)
36	29, 34, 35	syl2anc 583	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,)𝐵) ∖ {𝐴}) ∈ dom vol)
37	21, 36	eqeltrrd 2845	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴(,)𝐵) ∈ dom vol)
38	37	expr 456	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ < 𝐴 → (𝐴(,)𝐵) ∈ dom vol))
39		uncom 4181	. . . . . . . . 9 ⊢ ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ((-∞(,)𝐵) ∪ (𝐵[,)+∞))
40	22	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → -∞ ∈ ℝ^)
41		simplr 768	. . . . . . . . . 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 766	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^)
45		mnfle 13197	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
46	45	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → -∞ ≤ 𝐴)
47		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
48	40, 44, 41, 46, 47	xrlelttrd 13222	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → -∞ < 𝐵)
49		pnfge 13193	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
50	41, 49	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐵 ≤ +∞)
51		df-ico 13413	. . . . . . . . . . 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 13219	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ +∞) → 𝑤 < +∞))
54		xrltletr 13219	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝑤) → -∞ < 𝑤))
55	14, 51, 52, 14, 53, 54	ixxun 13423	. . . . . . . . . 10 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 𝐵 ∧ 𝐵 ≤ +∞)) → ((-∞(,)𝐵) ∪ (𝐵[,)+∞)) = (-∞(,)+∞))
56	40, 41, 43, 48, 50, 55	syl32anc 1378	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((-∞(,)𝐵) ∪ (𝐵[,)+∞)) = (-∞(,)+∞))
57	39, 56	eqtrid 2792	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = (-∞(,)+∞))
58		ioomax 13482	. . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ
59	57, 58	eqtrdi 2796	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ)
60		ssun1 4201	. . . . . . . . 9 ⊢ (𝐵[,)+∞) ⊆ ((𝐵[,)+∞) ∪ (-∞(,)𝐵))
61	60, 59	sseqtrid 4061	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ⊆ ℝ)
62		incom 4230	. . . . . . . . 9 ⊢ ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ((-∞(,)𝐵) ∩ (𝐵[,)+∞))
63	14, 51, 52	ixxdisj 13422	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞(,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
64	40, 41, 43, 63	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((-∞(,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
65	62, 64	eqtrid 2792	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ∅)
66		uneqdifeq 4516	. . . . . . . 8 ⊢ (((𝐵[,)+∞) ⊆ ℝ ∧ ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ∅) → (((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ ↔ (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵)))
67	61, 65, 66	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ ↔ (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵)))
68	59, 67	mpbid 232	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵))
69		rembl 25594	. . . . . . 7 ⊢ ℝ ∈ dom vol
70		xrleloe 13206	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
71	41, 42, 70	sylancl 585	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
72	50, 71	mpbid 232	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ ∨ 𝐵 = +∞))
73		xrre2 13232	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < +∞)) → 𝐵 ∈ ℝ)
74	73	expr 456	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
75	42, 74	mp3anl3 1457	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
76	75	orim1d 966	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵 < +∞ ∨ 𝐵 = +∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞)))
77	72, 76	mpd 15	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
78		icombl1 25617	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (𝐵[,)+∞) ∈ dom vol)
79		oveq1 7455	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = (+∞[,)+∞))
80		pnfge 13193	. . . . . . . . . . . . 13 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
81	42, 80	ax-mp 5	. . . . . . . . . . . 12 ⊢ +∞ ≤ +∞
82		ico0 13453	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞))
83	42, 42, 82	mp2an 691	. . . . . . . . . . . 12 ⊢ ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞)
84	81, 83	mpbir 231	. . . . . . . . . . 11 ⊢ (+∞[,)+∞) = ∅
85	79, 84	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = ∅)
86		0mbl 25593	. . . . . . . . . 10 ⊢ ∅ ∈ dom vol
87	85, 86	eqeltrdi 2852	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) ∈ dom vol)
88	78, 87	jaoi 856	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = +∞) → (𝐵[,)+∞) ∈ dom vol)
89	77, 88	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ∈ dom vol)
90		difmbl 25597	. . . . . . 7 ⊢ ((ℝ ∈ dom vol ∧ (𝐵[,)+∞) ∈ dom vol) → (ℝ ∖ (𝐵[,)+∞)) ∈ dom vol)
91	69, 89, 90	sylancr 586	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (ℝ ∖ (𝐵[,)+∞)) ∈ dom vol)
92	68, 91	eqeltrrd 2845	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞(,)𝐵) ∈ dom vol)
93		oveq1 7455	. . . . . 6 ⊢ (-∞ = 𝐴 → (-∞(,)𝐵) = (𝐴(,)𝐵))
94	93	eleq1d 2829	. . . . 5 ⊢ (-∞ = 𝐴 → ((-∞(,)𝐵) ∈ dom vol ↔ (𝐴(,)𝐵) ∈ dom vol))
95	92, 94	syl5ibcom 245	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ = 𝐴 → (𝐴(,)𝐵) ∈ dom vol))
96		xrleloe 13206	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
97	22, 44, 96	sylancr 586	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
98	46, 97	mpbid 232	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ < 𝐴 ∨ -∞ = 𝐴))
99	38, 95, 98	mpjaod 859	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
100		ioo0 13432	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
101		xrlenlt 11355	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
102	101	ancoms 458	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
103	100, 102	bitrd 279	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ ¬ 𝐴 < 𝐵))
104	103	biimpar 477	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴(,)𝐵) = ∅)
105	104, 86	eqeltrdi 2852	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
106	99, 105	pm2.61dan 812	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ dom vol)
107		ndmioo 13434	. . 3 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
108	107, 86	eqeltrdi 2852	. 2 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ dom vol)
109	106, 108	pm2.61i 182	1 ⊢ (𝐴(,)𝐵) ∈ dom vol

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∖ cdif 3973 ∪ cun 3974 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 {csn 4648 class class class wbr 5166 dom cdm 5700 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 +∞cpnf 11321 -∞cmnf 11322 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 (,)cioo 13407 [,)cico 13409 [,]cicc 13410 vol*covol 25516 volcvol 25517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-xadd 13176 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-xmet 21380 df-met 21381 df-ovol 25518 df-vol 25519

This theorem is referenced by: iccmbl 25620 ovolioo 25622 volioo 25623 ioovolcl 25624 uniioovol 25633 uniioombllem4 25640 uniioombllem5 25641 opnmblALT 25657 mbfid 25689 ditgcl 25913 ditgswap 25914 ditgsplitlem 25915 ftc1lem1 26096 ftc1lem2 26097 ftc1a 26098 ftc1lem4 26100 ftc2 26105 ftc2ditglem 26106 itgsubstlem 26109 itgpowd 26111 ftc2re 34575 fdvposlt 34576 fdvposle 34578 itgexpif 34583 circlemeth 34617 itg2gt0cn 37635 ftc1cnnclem 37651 ftc1anclem7 37659 ftc1anclem8 37660 ftc1anc 37661 ftc2nc 37662 areacirc 37673 lcmineqlem10 41995 lcmineqlem12 41997 iocmbl 43174 cnioobibld 43175 lhe4.4ex1a 44298 itgsin0pilem1 45871 iblioosinexp 45874 itgsinexplem1 45875 itgsinexp 45876 itgcoscmulx 45890 volioc 45893 itgsincmulx 45895 iblcncfioo 45899 itgiccshift 45901 itgperiod 45902 itgsbtaddcnst 45903 volico 45904 volioof 45908 wallispilem2 45987 dirkeritg 46023 fourierdlem16 46044 fourierdlem21 46049 fourierdlem22 46050 fourierdlem39 46067 fourierdlem73 46100 fourierdlem83 46110 fourierdlem103 46130 fourierdlem104 46131 fourierdlem111 46138 fourierdlem112 46139 sqwvfoura 46149 sqwvfourb 46150 etransclem18 46173 etransclem23 46178 ovolval4lem1 46570 ovolval5lem1 46573

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