iooiinioc - Mathbox for Glauco Siliprandi

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Theorem iooiinioc 45474

Description: A left-open, right-closed interval expressed as the indexed intersection of open intervals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
iooiinioc.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
iooiinioc.2	⊢ (𝜑 → 𝐵 ∈ ℝ)

**Assertion**
Ref	Expression
iooiinioc	⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵))

Distinct variable groups: 𝐴,𝑛 𝐵,𝑛 𝜑,𝑛

Proof of Theorem iooiinioc Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iooiinioc.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐴 ∈ ℝ^*)
3		iooiinioc.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐵 ∈ ℝ)
5	4	rexrd 11340	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐵 ∈ ℝ^*)
6		1nn 12304	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
7		ioossre 13468	. . . . . . . . . 10 ⊢ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ
8		oveq2 7456	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
9	8	oveq2d 7464	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (𝐵 + (1 / 𝑛)) = (𝐵 + (1 / 1)))
10	9	oveq2d 7464	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,)(𝐵 + (1 / 1))))
11	10	sseq1d 4040	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ ↔ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ))
12	11	rspcev 3635	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ) → ∃𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ)
13	6, 7, 12	mp2an 691	. . . . . . . . 9 ⊢ ∃𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ
14		iinss 5079	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ)
15	13, 14	ax-mp 5	. . . . . . . 8 ⊢ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ
16	15	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ)
17		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
18	16, 17	sseldd 4009	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ℝ)
19	18	rexrd 11340	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ℝ^*)
20		1red 11291	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
21		ax-1ne0 11253	. . . . . . . . . . 11 ⊢ 1 ≠ 0
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
23	20, 20, 22	redivcld 12122	. . . . . . . . 9 ⊢ (𝜑 → (1 / 1) ∈ ℝ)
24	3, 23	readdcld 11319	. . . . . . . 8 ⊢ (𝜑 → (𝐵 + (1 / 1)) ∈ ℝ)
25	24	rexrd 11340	. . . . . . 7 ⊢ (𝜑 → (𝐵 + (1 / 1)) ∈ ℝ^*)
26	25	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → (𝐵 + (1 / 1)) ∈ ℝ^*)
27		id 22	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
28	6	a1i 11	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) → 1 ∈ ℕ)
29	10	eleq2d 2830	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) ↔ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1)))))
30	27, 28, 29	eliind 44973	. . . . . . 7 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1))))
31	30	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1))))
32		ioogtlb 45413	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 + (1 / 1)) ∈ ℝ^* ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1)))) → 𝐴 < 𝑥)
33	2, 26, 31, 32	syl3anc 1371	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐴 < 𝑥)
34		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑛𝜑
35		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑥
36		nfii1 5052	. . . . . . . . 9 ⊢ Ⅎ𝑛∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))
37	35, 36	nfel 2923	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))
38	34, 37	nfan 1898	. . . . . . 7 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
39		simpll 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝜑)
40		iinss2 5080	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
41	40	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
42		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
43	41, 42	sseldd 4009	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
44	43	adantll 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
45		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
46		elioore 13437	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) → 𝑥 ∈ ℝ)
47	46	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
48	47	adantll 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
49	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)
50		nnrecre 12335	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
51	50	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
52	49, 51	readdcld 11319	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ)
53	52	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ)
54	1	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ^*)
55	54	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ^*)
56	52	rexrd 11340	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ^*)
57	56	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ^*)
58		simplr 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
59		iooltub 45428	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 + (1 / 𝑛)) ∈ ℝ^* ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 < (𝐵 + (1 / 𝑛)))
60	55, 57, 58, 59	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 < (𝐵 + (1 / 𝑛)))
61	48, 53, 60	ltled 11438	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ≤ (𝐵 + (1 / 𝑛)))
62	39, 44, 45, 61	syl21anc 837	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ≤ (𝐵 + (1 / 𝑛)))
63	62	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → (𝑛 ∈ ℕ → 𝑥 ≤ (𝐵 + (1 / 𝑛))))
64	38, 63	ralrimi 3263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → ∀𝑛 ∈ ℕ 𝑥 ≤ (𝐵 + (1 / 𝑛)))
65	38, 19, 4	xrralrecnnle 45298	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → (𝑥 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ 𝑥 ≤ (𝐵 + (1 / 𝑛))))
66	64, 65	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ≤ 𝐵)
67	2, 5, 19, 33, 66	eliocd 45425	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ (𝐴(,]𝐵))
68	67	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))𝑥 ∈ (𝐴(,]𝐵))
69		dfss3 3997	. . 3 ⊢ (∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,]𝐵) ↔ ∀𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))𝑥 ∈ (𝐴(,]𝐵))
70	68, 69	sylibr 234	. 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,]𝐵))
71	1	xrleidd 13214	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
72	71	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ 𝐴)
73		1rp 13061	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
74	73	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ⁺)
75		nnrp 13068	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
76	74, 75	rpdivcld 13116	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ⁺)
77	76	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
78	49, 77	ltaddrpd 13132	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 < (𝐵 + (1 / 𝑛)))
79		iocssioo 13499	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ (𝐵 + (1 / 𝑛)) ∈ ℝ^*) ∧ (𝐴 ≤ 𝐴 ∧ 𝐵 < (𝐵 + (1 / 𝑛)))) → (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
80	54, 56, 72, 78, 79	syl22anc 838	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
81	80	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
82		ssiin 5078	. . 3 ⊢ ((𝐴(,]𝐵) ⊆ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ↔ ∀𝑛 ∈ ℕ (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
83	81, 82	sylibr 234	. 2 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
84	70, 83	eqssd 4026	1 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 ⊆ wss 3976 ∩ ciin 5016 class class class wbr 5166 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 / cdiv 11947 ℕcn 12293 ℝ⁺crp 13057 (,)cioo 13407 (,]cioc 13408

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-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 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-iun 5017 df-iin 5018 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-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-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 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-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-ioo 13411 df-ioc 13412 df-fl 13843

This theorem is referenced by: iocborel 46277

W3C validator