iooiinioc - Mathbox for Glauco Siliprandi

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Theorem iooiinioc 46255

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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐴 ∈ ℝ^*)
3		iooiinioc.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐵 ∈ ℝ)
5	4	rexrd 11260	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐵 ∈ ℝ^*)
6		1nn 12245	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
7		ioossre 13435	. . . . . . . . . 10 ⊢ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ
8		oveq2 7420	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
9	8	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (𝐵 + (1 / 𝑛)) = (𝐵 + (1 / 1)))
10	9	oveq2d 7428	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,)(𝐵 + (1 / 1))))
11	10	sseq1d 3969	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ ↔ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ))
12	11	rspcev 3582	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ) → ∃𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ)
13	6, 7, 12	mp2an 704	. . . . . . . . 9 ⊢ ∃𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ
14		iinss 5022	. . . . . . . . 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 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
18	16, 17	sseldd 3939	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ℝ)
19	18	rexrd 11260	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ℝ^*)
20		1red 11210	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
21		ax-1ne0 11170	. . . . . . . . . . 11 ⊢ 1 ≠ 0
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
23	20, 20, 22	redivcld 12044	. . . . . . . . 9 ⊢ (𝜑 → (1 / 1) ∈ ℝ)
24	3, 23	readdcld 11239	. . . . . . . 8 ⊢ (𝜑 → (𝐵 + (1 / 1)) ∈ ℝ)
25	24	rexrd 11260	. . . . . . 7 ⊢ (𝜑 → (𝐵 + (1 / 1)) ∈ ℝ^*)
26	25	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → (𝐵 + (1 / 1)) ∈ ℝ^*)
27		id 23	. . . . . . . 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 2849	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) ↔ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1)))))
30	27, 28, 29	eliind 45774	. . . . . . 7 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1))))
31	30	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1))))
32		ioogtlb 46194	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 + (1 / 1)) ∈ ℝ^* ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1)))) → 𝐴 < 𝑥)
33	2, 26, 31, 32	syl3anc 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐴 < 𝑥)
34		nfv 1944	. . . . . . . 8 ⊢ Ⅎ𝑛𝜑
35		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑥
36		nfii1 4994	. . . . . . . . 9 ⊢ Ⅎ𝑛∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))
37	35, 36	nfel 2939	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))
38	34, 37	nfan 1929	. . . . . . 7 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
39		simpll 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝜑)
40		iinss2 5023	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
41	40	adantl 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
42		simpl 487	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
43	41, 42	sseldd 3939	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
44	43	adantll 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
45		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
46		elioore 13403	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) → 𝑥 ∈ ℝ)
47	46	adantr 485	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
48	47	adantll 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
49	3	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)
50		nnrecre 12279	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
51	50	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
52	49, 51	readdcld 11239	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ)
53	52	adantlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ)
54	1	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ^*)
55	54	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ^*)
56	52	rexrd 11260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ^*)
57	56	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ^*)
58		simplr 780	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
59		iooltub 46209	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐵 + (1 / 𝑛)) ∈ ℝ^* ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 < (𝐵 + (1 / 𝑛)))
60	55, 57, 58, 59	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 < (𝐵 + (1 / 𝑛)))
61	48, 53, 60	ltled 11359	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ≤ (𝐵 + (1 / 𝑛)))
62	39, 44, 45, 61	syl21anc 850	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ≤ (𝐵 + (1 / 𝑛)))
63	62	ex 417	. . . . . . 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 46081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → (𝑥 ≤ 𝐵 ↔ ∀𝑛 ∈ ℕ 𝑥 ≤ (𝐵 + (1 / 𝑛))))
66	64, 65	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ≤ 𝐵)
67	2, 5, 19, 33, 66	eliocd 46206	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ (𝐴(,]𝐵))
68	67	ralrimiva 3157	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))𝑥 ∈ (𝐴(,]𝐵))
69		dfss3 3927	. . 3 ⊢ (∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,]𝐵) ↔ ∀𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))𝑥 ∈ (𝐴(,]𝐵))
70	68, 69	sylibr 237	. 2 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,]𝐵))
71	1	xrleidd 13178	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
72	71	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ 𝐴)
73		1rp 13021	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
74	73	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ⁺)
75		nnrp 13029	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
76	74, 75	rpdivcld 13078	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ⁺)
77	76	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
78	49, 77	ltaddrpd 13094	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 < (𝐵 + (1 / 𝑛)))
79		iocssioo 13467	. . . . 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 851	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
81	80	ralrimiva 3157	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
82		ssiin 5021	. . 3 ⊢ ((𝐴(,]𝐵) ⊆ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ↔ ∀𝑛 ∈ ℕ (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
83	81, 82	sylibr 237	. 2 ⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
84	70, 83	eqssd 3955	1 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 ⊆ wss 3906 ∩ ciin 4958 class class class wbr 5110 (class class class)co 7412 ℝcr 11100 0cc0 11101 1c1 11102 + caddc 11104 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 / cdiv 11872 ℕcn 12234 ℝ⁺crp 13017 (,)cioo 13373 (,]cioc 13374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-sup 9403 df-inf 9404 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-n0 12506 df-z 12593 df-uz 12864 df-q 12974 df-rp 13018 df-ioo 13377 df-ioc 13378 df-fl 13827

This theorem is referenced by: iocborel 47053

W3C validator