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Theorem ioombl 25557

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 13429	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
2	1	3expa 1124	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
3	2	adantrr 723	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
4		lbico1 13351	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
5	4	3expa 1124	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
6	5	adantrr 723	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ (𝐴[,)𝐵))
7	6	snssd 4725	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ⊆ (𝐴[,)𝐵))
8		iccid 13341	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → (𝐴[,]𝐴) = {𝐴})
9	8	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴[,]𝐴) = {𝐴})
10	9	ineq1d 4155	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ({𝐴} ∩ (𝐴(,)𝐵)))
11		simpll 772	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ ℝ^)
12		simplr 774	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐵 ∈ ℝ^)
13		df-icc 13303	. . . . . . . . . . 11 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
14		df-ioo 13300	. . . . . . . . . . 11 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
15		xrltnle 11210	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴))
16	13, 14, 15	ixxdisj 13311	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ∅)
17	11, 11, 12, 16	syl3anc 1379	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ∅)
18	10, 17	eqtr3d 2777	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ({𝐴} ∩ (𝐴(,)𝐵)) = ∅)
19		uneqdifeq 4427	. . . . . . . 8 ⊢ (({𝐴} ⊆ (𝐴[,)𝐵) ∧ ({𝐴} ∩ (𝐴(,)𝐵)) = ∅) → (({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵)))
20	7, 18, 19	syl2anc 590	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵)))
21	3, 20	mpbid 233	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵))
22		mnfxr 11200	. . . . . . . . . 10 ⊢ -∞ ∈ ℝ^*
23	22	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → -∞ ∈ ℝ^)
24		simprr 778	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → -∞ < 𝐴)
25		simprl 776	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 < 𝐵)
26		xrre2 13120	. . . . . . . . 9 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (-∞ < 𝐴 ∧ 𝐴 < 𝐵)) → 𝐴 ∈ ℝ)
27	23, 11, 12, 24, 25, 26	syl32anc 1386	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ ℝ)
28		icombl 25556	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)
29	27, 12, 28	syl2anc 590	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴[,)𝐵) ∈ dom vol)
30	27	snssd 4725	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ⊆ ℝ)
31		ovolsn 25487	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)
32	27, 31	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (vol‘{𝐴}) = 0)
33		nulmbl 25527	. . . . . . . 8 ⊢ (({𝐴} ⊆ ℝ ∧ (vol*‘{𝐴}) = 0) → {𝐴} ∈ dom vol)
34	30, 32, 33	syl2anc 590	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ∈ dom vol)
35		difmbl 25535	. . . . . . 7 ⊢ (((𝐴[,)𝐵) ∈ dom vol ∧ {𝐴} ∈ dom vol) → ((𝐴[,)𝐵) ∖ {𝐴}) ∈ dom vol)
36	29, 34, 35	syl2anc 590	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,)𝐵) ∖ {𝐴}) ∈ dom vol)
37	21, 36	eqeltrrd 2841	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴(,)𝐵) ∈ dom vol)
38	37	expr 457	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ < 𝐴 → (𝐴(,)𝐵) ∈ dom vol))
39		uncom 4095	. . . . . . . . 9 ⊢ ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ((-∞(,)𝐵) ∪ (𝐵[,)+∞))
40	22	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → -∞ ∈ ℝ^)
41		simplr 774	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^)
42		pnfxr 11197	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
43	42	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → +∞ ∈ ℝ^)
44		simpll 772	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^)
45		mnfle 13084	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
46	45	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → -∞ ≤ 𝐴)
47		simpr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
48	40, 44, 41, 46, 47	xrlelttrd 13109	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → -∞ < 𝐵)
49		pnfge 13079	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
50	41, 49	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐵 ≤ +∞)
51		df-ico 13302	. . . . . . . . . . 11 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
52		xrlenlt 11208	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵))
53		xrltletr 13106	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ +∞) → 𝑤 < +∞))
54		xrltletr 13106	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝑤) → -∞ < 𝑤))
55	14, 51, 52, 14, 53, 54	ixxun 13312	. . . . . . . . . 10 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 𝐵 ∧ 𝐵 ≤ +∞)) → ((-∞(,)𝐵) ∪ (𝐵[,)+∞)) = (-∞(,)+∞))
56	40, 41, 43, 48, 50, 55	syl32anc 1386	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((-∞(,)𝐵) ∪ (𝐵[,)+∞)) = (-∞(,)+∞))
57	39, 56	eqtrid 2787	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = (-∞(,)+∞))
58		ioomax 13373	. . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ
59	57, 58	eqtrdi 2791	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ)
60		ssun1 4114	. . . . . . . . 9 ⊢ (𝐵[,)+∞) ⊆ ((𝐵[,)+∞) ∪ (-∞(,)𝐵))
61	60, 59	sseqtrid 3964	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ⊆ ℝ)
62		incom 4145	. . . . . . . . 9 ⊢ ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ((-∞(,)𝐵) ∩ (𝐵[,)+∞))
63	14, 51, 52	ixxdisj 13311	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞(,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
64	40, 41, 43, 63	syl3anc 1379	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((-∞(,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
65	62, 64	eqtrid 2787	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ∅)
66		uneqdifeq 4427	. . . . . . . 8 ⊢ (((𝐵[,)+∞) ⊆ ℝ ∧ ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ∅) → (((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ ↔ (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵)))
67	61, 65, 66	syl2anc 590	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ ↔ (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵)))
68	59, 67	mpbid 233	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵))
69		rembl 25532	. . . . . . 7 ⊢ ℝ ∈ dom vol
70		xrleloe 13093	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
71	41, 42, 70	sylancl 592	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
72	50, 71	mpbid 233	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ ∨ 𝐵 = +∞))
73		xrre2 13120	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < +∞)) → 𝐵 ∈ ℝ)
74	73	expr 457	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
75	42, 74	mp3anl3 1465	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
76	75	orim1d 973	. . . . . . . . 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 25555	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (𝐵[,)+∞) ∈ dom vol)
79		oveq1 7370	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = (+∞[,)+∞))
80		pnfge 13079	. . . . . . . . . . . . 13 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
81	42, 80	ax-mp 5	. . . . . . . . . . . 12 ⊢ +∞ ≤ +∞
82		ico0 13342	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞))
83	42, 42, 82	mp2an 698	. . . . . . . . . . . 12 ⊢ ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞)
84	81, 83	mpbir 232	. . . . . . . . . . 11 ⊢ (+∞[,)+∞) = ∅
85	79, 84	eqtrdi 2791	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = ∅)
86		0mbl 25531	. . . . . . . . . 10 ⊢ ∅ ∈ dom vol
87	85, 86	eqeltrdi 2848	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) ∈ dom vol)
88	78, 87	jaoi 863	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = +∞) → (𝐵[,)+∞) ∈ dom vol)
89	77, 88	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ∈ dom vol)
90		difmbl 25535	. . . . . . 7 ⊢ ((ℝ ∈ dom vol ∧ (𝐵[,)+∞) ∈ dom vol) → (ℝ ∖ (𝐵[,)+∞)) ∈ dom vol)
91	69, 89, 90	sylancr 593	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (ℝ ∖ (𝐵[,)+∞)) ∈ dom vol)
92	68, 91	eqeltrrd 2841	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞(,)𝐵) ∈ dom vol)
93		oveq1 7370	. . . . . 6 ⊢ (-∞ = 𝐴 → (-∞(,)𝐵) = (𝐴(,)𝐵))
94	93	eleq1d 2825	. . . . 5 ⊢ (-∞ = 𝐴 → ((-∞(,)𝐵) ∈ dom vol ↔ (𝐴(,)𝐵) ∈ dom vol))
95	92, 94	syl5ibcom 246	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ = 𝐴 → (𝐴(,)𝐵) ∈ dom vol))
96		xrleloe 13093	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
97	22, 44, 96	sylancr 593	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
98	46, 97	mpbid 233	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ < 𝐴 ∨ -∞ = 𝐴))
99	38, 95, 98	mpjaod 866	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
100		ioo0 13321	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
101		xrlenlt 11208	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
102	101	ancoms 459	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
103	100, 102	bitrd 280	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ ¬ 𝐴 < 𝐵))
104	103	biimpar 478	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴(,)𝐵) = ∅)
105	104, 86	eqeltrdi 2848	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
106	99, 105	pm2.61dan 818	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ dom vol)
107		ndmioo 13323	. . 3 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
108	107, 86	eqeltrdi 2848	. 2 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ dom vol)
109	106, 108	pm2.61i 183	1 ⊢ (𝐴(,)𝐵) ∈ dom vol

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 853 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∖ cdif 3887 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4268 {csn 4562 class class class wbr 5079 dom cdm 5625 ‘cfv 6492 (class class class)co 7363 ℝcr 11035 0cc0 11036 +∞cpnf 11174 -∞cmnf 11175 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 (,)cioo 13296 [,)cico 13298 [,]cicc 13299 vol*covol 25454 volcvol 25455

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-inf2 9560 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-pm 8773 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9352 df-inf 9353 df-oi 9422 df-dju 9823 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-n0 12436 df-z 12523 df-uz 12787 df-q 12897 df-rp 12941 df-xadd 13062 df-ioo 13300 df-ico 13302 df-icc 13303 df-fz 13460 df-fzo 13607 df-fl 13749 df-seq 13962 df-exp 14022 df-hash 14291 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-clim 15448 df-rlim 15449 df-sum 15647 df-xmet 21347 df-met 21348 df-ovol 25456 df-vol 25457

This theorem is referenced by: iccmbl 25558 ovolioo 25560 volioo 25561 ioovolcl 25562 uniioovol 25571 uniioombllem4 25578 uniioombllem5 25579 opnmblALT 25595 mbfid 25627 ditgcl 25850 ditgswap 25851 ditgsplitlem 25852 ftc1lem1 26027 ftc1lem2 26028 ftc1a 26029 ftc1lem4 26031 ftc2 26036 ftc2ditglem 26037 itgsubstlem 26040 itgpowd 26042 ftc2re 34789 fdvposlt 34790 fdvposle 34792 itgexpif 34797 circlemeth 34831 itg2gt0cn 38049 ftc1cnnclem 38065 ftc1anclem7 38073 ftc1anclem8 38074 ftc1anc 38075 ftc2nc 38076 areacirc 38087 lcmineqlem10 42530 lcmineqlem12 42532 iocmbl 43665 cnioobibld 43666 lhe4.4ex1a 44780 itgsin0pilem1 46400 iblioosinexp 46403 itgsinexplem1 46404 itgsinexp 46405 itgcoscmulx 46419 volioc 46422 itgsincmulx 46424 iblcncfioo 46428 itgiccshift 46430 itgperiod 46431 itgsbtaddcnst 46432 volico 46433 volioof 46437 wallispilem2 46516 dirkeritg 46552 fourierdlem16 46573 fourierdlem21 46578 fourierdlem22 46579 fourierdlem39 46596 fourierdlem73 46629 fourierdlem83 46639 fourierdlem103 46659 fourierdlem104 46660 fourierdlem111 46667 fourierdlem112 46668 sqwvfoura 46678 sqwvfourb 46679 etransclem18 46702 etransclem23 46707 ovolval4lem1 47099 ovolval5lem1 47102

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