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Theorem ioombl 23140

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 12169	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
2	1	3expa 1257	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
3	2	adantrr 749	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵))
4		lbico1 12099	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
5	4	3expa 1257	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))
6	5	adantrr 749	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ (𝐴[,)𝐵))
7	6	snssd 4281	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ⊆ (𝐴[,)𝐵))
8		iccid 12091	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → (𝐴[,]𝐴) = {𝐴})
9	8	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴[,]𝐴) = {𝐴})
10	9	ineq1d 3775	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ({𝐴} ∩ (𝐴(,)𝐵)))
11		simpll 786	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ ℝ^)
12		simplr 788	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐵 ∈ ℝ^)
13		df-icc 12053	. . . . . . . . . . 11 ⊢ [,] = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)})
14		df-ioo 12050	. . . . . . . . . . 11 ⊢ (,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)})
15		xrltnle 9984	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐴 < 𝑤 ↔ ¬ 𝑤 ≤ 𝐴))
16	13, 14, 15	ixxdisj 12061	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ∅)
17	11, 11, 12, 16	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,]𝐴) ∩ (𝐴(,)𝐵)) = ∅)
18	10, 17	eqtr3d 2646	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ({𝐴} ∩ (𝐴(,)𝐵)) = ∅)
19		uneqdifeq 4009	. . . . . . . 8 ⊢ (({𝐴} ⊆ (𝐴[,)𝐵) ∧ ({𝐴} ∩ (𝐴(,)𝐵)) = ∅) → (({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵)))
20	7, 18, 19	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (({𝐴} ∪ (𝐴(,)𝐵)) = (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵)))
21	3, 20	mpbid 221	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,)𝐵) ∖ {𝐴}) = (𝐴(,)𝐵))
22		mnfxr 9975	. . . . . . . . . 10 ⊢ -∞ ∈ ℝ^*
23	22	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → -∞ ∈ ℝ^)
24		simprr 792	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → -∞ < 𝐴)
25		simprl 790	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 < 𝐵)
26		xrre2 11875	. . . . . . . . 9 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (-∞ < 𝐴 ∧ 𝐴 < 𝐵)) → 𝐴 ∈ ℝ)
27	23, 11, 12, 24, 25, 26	syl32anc 1326	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → 𝐴 ∈ ℝ)
28		icombl 23139	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴[,)𝐵) ∈ dom vol)
29	27, 12, 28	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴[,)𝐵) ∈ dom vol)
30	27	snssd 4281	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ⊆ ℝ)
31		ovolsn 23070	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0)
32	27, 31	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (vol‘{𝐴}) = 0)
33		nulmbl 23110	. . . . . . . 8 ⊢ (({𝐴} ⊆ ℝ ∧ (vol*‘{𝐴}) = 0) → {𝐴} ∈ dom vol)
34	30, 32, 33	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → {𝐴} ∈ dom vol)
35		difmbl 23118	. . . . . . 7 ⊢ (((𝐴[,)𝐵) ∈ dom vol ∧ {𝐴} ∈ dom vol) → ((𝐴[,)𝐵) ∖ {𝐴}) ∈ dom vol)
36	29, 34, 35	syl2anc 691	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → ((𝐴[,)𝐵) ∖ {𝐴}) ∈ dom vol)
37	21, 36	eqeltrrd 2689	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ -∞ < 𝐴)) → (𝐴(,)𝐵) ∈ dom vol)
38	37	expr 641	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ < 𝐴 → (𝐴(,)𝐵) ∈ dom vol))
39		uncom 3719	. . . . . . . . 9 ⊢ ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ((-∞(,)𝐵) ∪ (𝐵[,)+∞))
40	22	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → -∞ ∈ ℝ^)
41		simplr 788	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^)
42		pnfxr 9971	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ^*
43	42	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → +∞ ∈ ℝ^)
44		simpll 786	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^)
45		mnfle 11845	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
46	45	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → -∞ ≤ 𝐴)
47		simpr 476	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
48	40, 44, 41, 46, 47	xrlelttrd 11867	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → -∞ < 𝐵)
49		pnfge 11840	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
50	41, 49	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → 𝐵 ≤ +∞)
51		df-ico 12052	. . . . . . . . . . 11 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
52		xrlenlt 9982	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → (𝐵 ≤ 𝑤 ↔ ¬ 𝑤 < 𝐵))
53		xrltletr 11864	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑤 < 𝐵 ∧ 𝐵 ≤ +∞) → 𝑤 < +∞))
54		xrltletr 11864	. . . . . . . . . . 11 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝑤) → -∞ < 𝑤))
55	14, 51, 52, 14, 53, 54	ixxun 12062	. . . . . . . . . 10 ⊢ (((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < 𝐵 ∧ 𝐵 ≤ +∞)) → ((-∞(,)𝐵) ∪ (𝐵[,)+∞)) = (-∞(,)+∞))
56	40, 41, 43, 48, 50, 55	syl32anc 1326	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((-∞(,)𝐵) ∪ (𝐵[,)+∞)) = (-∞(,)+∞))
57	39, 56	syl5eq 2656	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = (-∞(,)+∞))
58		ioomax 12119	. . . . . . . 8 ⊢ (-∞(,)+∞) = ℝ
59	57, 58	syl6eq 2660	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ)
60		ssun1 3738	. . . . . . . . 9 ⊢ (𝐵[,)+∞) ⊆ ((𝐵[,)+∞) ∪ (-∞(,)𝐵))
61	60, 59	syl5sseq 3616	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ⊆ ℝ)
62		incom 3767	. . . . . . . . 9 ⊢ ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ((-∞(,)𝐵) ∩ (𝐵[,)+∞))
63	14, 51, 52	ixxdisj 12061	. . . . . . . . . 10 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞(,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
64	40, 41, 43, 63	syl3anc 1318	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((-∞(,)𝐵) ∩ (𝐵[,)+∞)) = ∅)
65	62, 64	syl5eq 2656	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ∅)
66		uneqdifeq 4009	. . . . . . . 8 ⊢ (((𝐵[,)+∞) ⊆ ℝ ∧ ((𝐵[,)+∞) ∩ (-∞(,)𝐵)) = ∅) → (((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ ↔ (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵)))
67	61, 65, 66	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (((𝐵[,)+∞) ∪ (-∞(,)𝐵)) = ℝ ↔ (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵)))
68	59, 67	mpbid 221	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (ℝ ∖ (𝐵[,)+∞)) = (-∞(,)𝐵))
69		rembl 23115	. . . . . . 7 ⊢ ℝ ∈ dom vol
70		xrleloe 11853	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
71	41, 42, 70	sylancl 693	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 ≤ +∞ ↔ (𝐵 < +∞ ∨ 𝐵 = +∞)))
72	50, 71	mpbid 221	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ ∨ 𝐵 = +∞))
73		xrre2 11875	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < +∞)) → 𝐵 ∈ ℝ)
74	73	expr 641	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
75	42, 74	mp3anl3 1412	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵 < +∞ → 𝐵 ∈ ℝ))
76	75	orim1d 880	. . . . . . . . 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 23138	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → (𝐵[,)+∞) ∈ dom vol)
79		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = (+∞[,)+∞))
80		pnfge 11840	. . . . . . . . . . . . 13 ⊢ (+∞ ∈ ℝ^* → +∞ ≤ +∞)
81	42, 80	ax-mp 5	. . . . . . . . . . . 12 ⊢ +∞ ≤ +∞
82		ico0 12092	. . . . . . . . . . . . 13 ⊢ ((+∞ ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞))
83	42, 42, 82	mp2an 704	. . . . . . . . . . . 12 ⊢ ((+∞[,)+∞) = ∅ ↔ +∞ ≤ +∞)
84	81, 83	mpbir 220	. . . . . . . . . . 11 ⊢ (+∞[,)+∞) = ∅
85	79, 84	syl6eq 2660	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) = ∅)
86		0mbl 23114	. . . . . . . . . 10 ⊢ ∅ ∈ dom vol
87	85, 86	syl6eqel 2696	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐵[,)+∞) ∈ dom vol)
88	78, 87	jaoi 393	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∨ 𝐵 = +∞) → (𝐵[,)+∞) ∈ dom vol)
89	77, 88	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐵[,)+∞) ∈ dom vol)
90		difmbl 23118	. . . . . . 7 ⊢ ((ℝ ∈ dom vol ∧ (𝐵[,)+∞) ∈ dom vol) → (ℝ ∖ (𝐵[,)+∞)) ∈ dom vol)
91	69, 89, 90	sylancr 694	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (ℝ ∖ (𝐵[,)+∞)) ∈ dom vol)
92	68, 91	eqeltrrd 2689	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞(,)𝐵) ∈ dom vol)
93		oveq1 6556	. . . . . 6 ⊢ (-∞ = 𝐴 → (-∞(,)𝐵) = (𝐴(,)𝐵))
94	93	eleq1d 2672	. . . . 5 ⊢ (-∞ = 𝐴 → ((-∞(,)𝐵) ∈ dom vol ↔ (𝐴(,)𝐵) ∈ dom vol))
95	92, 94	syl5ibcom 234	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ = 𝐴 → (𝐴(,)𝐵) ∈ dom vol))
96		xrleloe 11853	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
97	22, 44, 96	sylancr 694	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ ≤ 𝐴 ↔ (-∞ < 𝐴 ∨ -∞ = 𝐴)))
98	46, 97	mpbid 221	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (-∞ < 𝐴 ∨ -∞ = 𝐴))
99	38, 95, 98	mpjaod 395	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐴 < 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
100		ioo0 12071	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴))
101		xrlenlt 9982	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
102	101	ancoms 468	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐵 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐵))
103	100, 102	bitrd 267	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴(,)𝐵) = ∅ ↔ ¬ 𝐴 < 𝐵))
104	103	biimpar 501	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴(,)𝐵) = ∅)
105	104, 86	syl6eqel 2696	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ ¬ 𝐴 < 𝐵) → (𝐴(,)𝐵) ∈ dom vol)
106	99, 105	pm2.61dan 828	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ dom vol)
107		ndmioo 12073	. . 3 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
108	107, 86	syl6eqel 2696	. 2 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) ∈ dom vol)
109	106, 108	pm2.61i 175	1 ⊢ (𝐴(,)𝐵) ∈ dom vol

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 (,)cioo 12046 [,)cico 12048 [,]cicc 12049 vol*covol 23038 volcvol 23039

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041

This theorem is referenced by: iccmbl 23141 ovolioo 23143 ioovolcl 23144 uniioovol 23153 uniioombllem4 23160 uniioombllem5 23161 opnmblALT 23177 mbfid 23209 ditgcl 23428 ditgswap 23429 ditgsplitlem 23430 ftc1lem1 23602 ftc1lem2 23603 ftc1a 23604 ftc1lem4 23606 ftc2 23611 ftc2ditglem 23612 itgsubstlem 23615 itg2gt0cn 32635 ftc1cnnclem 32653 ftc1anclem7 32661 ftc1anclem8 32662 ftc1anc 32663 ftc2nc 32664 areacirc 32675 iocmbl 36817 cnioobibld 36818 itgpowd 36819 lhe4.4ex1a 37550 volioo 38840 itgsin0pilem1 38841 iblioosinexp 38844 itgsinexplem1 38845 itgsinexp 38846 itgcoscmulx 38861 volioc 38864 itgsincmulx 38866 iblcncfioo 38870 itgiccshift 38872 itgperiod 38873 itgsbtaddcnst 38874 volico 38876 volioof 38880 wallispilem2 38959 dirkeritg 38995 fourierdlem16 39016 fourierdlem21 39021 fourierdlem22 39022 fourierdlem39 39039 fourierdlem73 39072 fourierdlem83 39082 fourierdlem103 39102 fourierdlem104 39103 fourierdlem111 39110 fourierdlem112 39111 sqwvfoura 39121 sqwvfourb 39122 etransclem18 39145 etransclem23 39150 ovolval4lem1 39539 ovolval5lem1 39542

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