iooiinioc - Mathbox for Glauco Siliprandi

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Theorem iooiinioc 46008

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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐴 ∈ ℝ^*)
3		iooiinioc.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐵 ∈ ℝ)
5	4	rexrd 11193	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐵 ∈ ℝ^*)
6		1nn 12183	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
7		ioossre 13358	. . . . . . . . . 10 ⊢ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ
8		oveq2 7371	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
9	8	oveq2d 7379	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (𝐵 + (1 / 𝑛)) = (𝐵 + (1 / 1)))
10	9	oveq2d 7379	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,)(𝐵 + (1 / 1))))
11	10	sseq1d 3953	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ ↔ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ))
12	11	rspcev 3567	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ) → ∃𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ)
13	6, 7, 12	mp2an 698	. . . . . . . . 9 ⊢ ∃𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ
14		iinss 4993	. . . . . . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
18	16, 17	sseldd 3923	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ℝ)
19	18	rexrd 11193	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ℝ^*)
20		1red 11143	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
21		ax-1ne0 11105	. . . . . . . . . . 11 ⊢ 1 ≠ 0
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
23	20, 20, 22	redivcld 11981	. . . . . . . . 9 ⊢ (𝜑 → (1 / 1) ∈ ℝ)
24	3, 23	readdcld 11172	. . . . . . . 8 ⊢ (𝜑 → (𝐵 + (1 / 1)) ∈ ℝ)
25	24	rexrd 11193	. . . . . . 7 ⊢ (𝜑 → (𝐵 + (1 / 1)) ∈ ℝ^*)
26	25	adantr 481	. . . . . 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 2826	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) ↔ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1)))))
30	27, 28, 29	eliind 45526	. . . . . . 7 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1))))
31	30	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1))))
32		ioogtlb 45947	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 + (1 / 1)) ∈ ℝ^* ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1)))) → 𝐴 < 𝑥)
33	2, 26, 31, 32	syl3anc 1379	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐴 < 𝑥)
34		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑛𝜑
35		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑥
36		nfii1 4965	. . . . . . . . 9 ⊢ Ⅎ𝑛∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))
37	35, 36	nfel 2916	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))
38	34, 37	nfan 1906	. . . . . . 7 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
39		simpll 772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝜑)
40		iinss2 4994	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
41	40	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
42		simpl 483	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
43	41, 42	sseldd 3923	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
44	43	adantll 720	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
45		simpr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
46		elioore 13326	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) → 𝑥 ∈ ℝ)
47	46	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
48	47	adantll 720	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
49	3	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)
50		nnrecre 12217	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
51	50	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
52	49, 51	readdcld 11172	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ)
53	52	adantlr 721	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ)
54	1	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ^*)
55	54	adantlr 721	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ^*)
56	52	rexrd 11193	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ^*)
57	56	adantlr 721	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ^*)
58		simplr 774	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
59		iooltub 45962	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 + (1 / 𝑛)) ∈ ℝ^* ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 < (𝐵 + (1 / 𝑛)))
60	55, 57, 58, 59	syl3anc 1379	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 < (𝐵 + (1 / 𝑛)))
61	48, 53, 60	ltled 11292	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ≤ (𝐵 + (1 / 𝑛)))
62	39, 44, 45, 61	syl21anc 843	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ≤ (𝐵 + (1 / 𝑛)))
63	62	ex 413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → (𝑛 ∈ ℕ → 𝑥 ≤ (𝐵 + (1 / 𝑛))))
64	38, 63	ralrimi 3238	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → ∀𝑛 ∈ ℕ 𝑥 ≤ (𝐵 + (1 / 𝑛)))
65	38, 19, 4	xrralrecnnle 45834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → (𝑥 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ 𝑥 ≤ (𝐵 + (1 / 𝑛))))
66	64, 65	mpbird 258	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ≤ 𝐵)
67	2, 5, 19, 33, 66	eliocd 45959	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ (𝐴(,]𝐵))
68	67	ralrimiva 3132	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))𝑥 ∈ (𝐴(,]𝐵))
69		dfss3 3911	. . 3 ⊢ (∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,]𝐵) ↔ ∀𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))𝑥 ∈ (𝐴(,]𝐵))
70	68, 69	sylibr 235	. 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,]𝐵))
71	1	xrleidd 13101	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
72	71	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ 𝐴)
73		1rp 12944	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
74	73	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ⁺)
75		nnrp 12952	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
76	74, 75	rpdivcld 13001	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ⁺)
77	76	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
78	49, 77	ltaddrpd 13017	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 < (𝐵 + (1 / 𝑛)))
79		iocssioo 13390	. . . . 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 844	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
81	80	ralrimiva 3132	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
82		ssiin 4992	. . 3 ⊢ ((𝐴(,]𝐵) ⊆ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ↔ ∀𝑛 ∈ ℕ (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
83	81, 82	sylibr 235	. 2 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
84	70, 83	eqssd 3939	1 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∀wral 3054 ∃wrex 3064 ⊆ wss 3890 ∩ ciin 4929 class class class wbr 5079 (class class class)co 7363 ℝcr 11035 0cc0 11036 1c1 11037 + caddc 11039 ℝ^*cxr 11176 < clt 11177 ≤ cle 11178 / cdiv 11805 ℕcn 12172 ℝ⁺crp 12940 (,)cioo 13296 (,]cioc 13297

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-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 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-iun 4930 df-iin 4931 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-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-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-er 8640 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9352 df-inf 9353 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-n0 12436 df-z 12523 df-uz 12787 df-q 12897 df-rp 12941 df-ioo 13300 df-ioc 13301 df-fl 13749

This theorem is referenced by: iocborel 46806

W3C validator