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Theorem ioombl 25600

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 13518	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
2	1	3expa 1119	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
3	2	adantrr 717	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
4		lbico1 13441	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
5	4	3expa 1119	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
6	5	adantrr 717	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ (𝐴[,)𝐵))
7	6	snssd 4809	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ⊆ (𝐴[,)𝐵))
8		iccid 13432	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → (𝐴[,]𝐴) = {𝐴})
9	8	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴[,]𝐴) = {𝐴})
10	9	ineq1d 4219	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ({𝐴} ∩ (𝐴(,)𝐵)))
11		simpll 767	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ ℝ^)
12		simplr 769	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐵 ∈ ℝ^)
13		df-icc 13394	. . . . . . . . . . 11 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
14		df-ioo 13391	. . . . . . . . . . 11 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
15		xrltnle 11328	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴))
16	13, 14, 15	ixxdisj 13402	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ∅)
17	11, 11, 12, 16	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ∅)
18	10, 17	eqtr3d 2779	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ({𝐴} ∩ (𝐴(,)𝐵)) = ∅)
19		uneqdifeq 4493	. . . . . . . 8 ⊢ (({𝐴} ⊆ (𝐴[,)𝐵) ∧ ({𝐴} ∩ (𝐴(,)𝐵)) = ∅) → (({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵)))
20	7, 18, 19	syl2anc 584	. . . . . . 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 11318	. . . . . . . . . 10 ⊢ -∞ ∈ ℝ^*
23	22	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → -∞ ∈ ℝ^)
24		simprr 773	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → -∞ < 𝐴)
25		simprl 771	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 < 𝐵)
26		xrre2 13212	. . . . . . . . 9 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (-∞ < 𝐴 ∧ 𝐴 < 𝐵)) → 𝐴 ∈ ℝ)
27	23, 11, 12, 24, 25, 26	syl32anc 1380	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ ℝ)
28		icombl 25599	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)
29	27, 12, 28	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴[,)𝐵) ∈ dom vol)
30	27	snssd 4809	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ⊆ ℝ)
31		ovolsn 25530	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)
32	27, 31	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (vol‘{𝐴}) = 0)
33		nulmbl 25570	. . . . . . . 8 ⊢ (({𝐴} ⊆ ℝ ∧ (vol*‘{𝐴}) = 0) → {𝐴} ∈ dom vol)
34	30, 32, 33	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ∈ dom vol)
35		difmbl 25578	. . . . . . 7 ⊢ (((𝐴[,)𝐵) ∈ dom vol ∧ {𝐴} ∈ dom vol) → ((𝐴[,)𝐵) ∖ {𝐴}) ∈ dom vol)
36	29, 34, 35	syl2anc 584	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,)𝐵) ∖ {𝐴}) ∈ dom vol)
37	21, 36	eqeltrrd 2842	. . . . 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 4158	. . . . . . . . 9 ⊢ ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ((-∞(,)𝐵) ∪ (𝐵[,)+∞))
40	22	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → -∞ ∈ ℝ^)
41		simplr 769	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^)
42		pnfxr 11315	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
43	42	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → +∞ ∈ ℝ^)
44		simpll 767	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^)
45		mnfle 13177	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
46	45	ad2antrr 726	. . . . . . . . . . 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 13202	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → -∞ < 𝐵)
49		pnfge 13172	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
50	41, 49	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐵 ≤ +∞)
51		df-ico 13393	. . . . . . . . . . 11 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
52		xrlenlt 11326	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵))
53		xrltletr 13199	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ +∞) → 𝑤 < +∞))
54		xrltletr 13199	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝑤) → -∞ < 𝑤))
55	14, 51, 52, 14, 53, 54	ixxun 13403	. . . . . . . . . 10 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 𝐵 ∧ 𝐵 ≤ +∞)) → ((-∞(,)𝐵) ∪ (𝐵[,)+∞)) = (-∞(,)+∞))
56	40, 41, 43, 48, 50, 55	syl32anc 1380	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((-∞(,)𝐵) ∪ (𝐵[,)+∞)) = (-∞(,)+∞))
57	39, 56	eqtrid 2789	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = (-∞(,)+∞))
58		ioomax 13462	. . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ
59	57, 58	eqtrdi 2793	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ)
60		ssun1 4178	. . . . . . . . 9 ⊢ (𝐵[,)+∞) ⊆ ((𝐵[,)+∞) ∪ (-∞(,)𝐵))
61	60, 59	sseqtrid 4026	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ⊆ ℝ)
62		incom 4209	. . . . . . . . 9 ⊢ ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ((-∞(,)𝐵) ∩ (𝐵[,)+∞))
63	14, 51, 52	ixxdisj 13402	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞(,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
64	40, 41, 43, 63	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((-∞(,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
65	62, 64	eqtrid 2789	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ∅)
66		uneqdifeq 4493	. . . . . . . 8 ⊢ (((𝐵[,)+∞) ⊆ ℝ ∧ ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ∅) → (((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ ↔ (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵)))
67	61, 65, 66	syl2anc 584	. . . . . . 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 25575	. . . . . . 7 ⊢ ℝ ∈ dom vol
70		xrleloe 13186	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
71	41, 42, 70	sylancl 586	. . . . . . . . . 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 13212	. . . . . . . . . . . 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 1459	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
76	75	orim1d 968	. . . . . . . . 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 25598	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (𝐵[,)+∞) ∈ dom vol)
79		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = (+∞[,)+∞))
80		pnfge 13172	. . . . . . . . . . . . 13 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
81	42, 80	ax-mp 5	. . . . . . . . . . . 12 ⊢ +∞ ≤ +∞
82		ico0 13433	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞))
83	42, 42, 82	mp2an 692	. . . . . . . . . . . 12 ⊢ ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞)
84	81, 83	mpbir 231	. . . . . . . . . . 11 ⊢ (+∞[,)+∞) = ∅
85	79, 84	eqtrdi 2793	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = ∅)
86		0mbl 25574	. . . . . . . . . 10 ⊢ ∅ ∈ dom vol
87	85, 86	eqeltrdi 2849	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) ∈ dom vol)
88	78, 87	jaoi 858	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = +∞) → (𝐵[,)+∞) ∈ dom vol)
89	77, 88	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ∈ dom vol)
90		difmbl 25578	. . . . . . 7 ⊢ ((ℝ ∈ dom vol ∧ (𝐵[,)+∞) ∈ dom vol) → (ℝ ∖ (𝐵[,)+∞)) ∈ dom vol)
91	69, 89, 90	sylancr 587	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (ℝ ∖ (𝐵[,)+∞)) ∈ dom vol)
92	68, 91	eqeltrrd 2842	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞(,)𝐵) ∈ dom vol)
93		oveq1 7438	. . . . . 6 ⊢ (-∞ = 𝐴 → (-∞(,)𝐵) = (𝐴(,)𝐵))
94	93	eleq1d 2826	. . . . 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 13186	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
97	22, 44, 96	sylancr 587	. . . . 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 861	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
100		ioo0 13412	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
101		xrlenlt 11326	. . . . . . 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 2849	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
106	99, 105	pm2.61dan 813	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ dom vol)
107		ndmioo 13414	. . 3 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
108	107, 86	eqeltrdi 2849	. 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 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 +∞cpnf 11292 -∞cmnf 11293 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 (,)cioo 13387 [,)cico 13389 [,]cicc 13390 vol*covol 25497 volcvol 25498

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xadd 13155 df-ioo 13391 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-xmet 21357 df-met 21358 df-ovol 25499 df-vol 25500

This theorem is referenced by: iccmbl 25601 ovolioo 25603 volioo 25604 ioovolcl 25605 uniioovol 25614 uniioombllem4 25621 uniioombllem5 25622 opnmblALT 25638 mbfid 25670 ditgcl 25893 ditgswap 25894 ditgsplitlem 25895 ftc1lem1 26076 ftc1lem2 26077 ftc1a 26078 ftc1lem4 26080 ftc2 26085 ftc2ditglem 26086 itgsubstlem 26089 itgpowd 26091 ftc2re 34613 fdvposlt 34614 fdvposle 34616 itgexpif 34621 circlemeth 34655 itg2gt0cn 37682 ftc1cnnclem 37698 ftc1anclem7 37706 ftc1anclem8 37707 ftc1anc 37708 ftc2nc 37709 areacirc 37720 lcmineqlem10 42039 lcmineqlem12 42041 iocmbl 43225 cnioobibld 43226 lhe4.4ex1a 44348 itgsin0pilem1 45965 iblioosinexp 45968 itgsinexplem1 45969 itgsinexp 45970 itgcoscmulx 45984 volioc 45987 itgsincmulx 45989 iblcncfioo 45993 itgiccshift 45995 itgperiod 45996 itgsbtaddcnst 45997 volico 45998 volioof 46002 wallispilem2 46081 dirkeritg 46117 fourierdlem16 46138 fourierdlem21 46143 fourierdlem22 46144 fourierdlem39 46161 fourierdlem73 46194 fourierdlem83 46204 fourierdlem103 46224 fourierdlem104 46225 fourierdlem111 46232 fourierdlem112 46233 sqwvfoura 46243 sqwvfourb 46244 etransclem18 46267 etransclem23 46272 ovolval4lem1 46664 ovolval5lem1 46667

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