iooiinioc - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > iooiinioc		Structured version Visualization version GIF version

Theorem iooiinioc 46313

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 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐴 ∈ ℝ^*)
3		iooiinioc.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
4	3	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐵 ∈ ℝ)
5	4	rexrd 11277	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝐵 ∈ ℝ^*)
6		1nn 12262	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
7		ioossre 13452	. . . . . . . . . 10 ⊢ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ
8		oveq2 7431	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 1 → (1 / 𝑛) = (1 / 1))
9	8	oveq2d 7439	. . . . . . . . . . . . 13 ⊢ (𝑛 = 1 → (𝐵 + (1 / 𝑛)) = (𝐵 + (1 / 1)))
10	9	oveq2d 7439	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,)(𝐵 + (1 / 1))))
11	10	sseq1d 3971	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → ((𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ ↔ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ))
12	11	rspcev 3584	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ (𝐴(,)(𝐵 + (1 / 1))) ⊆ ℝ) → ∃𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ)
13	6, 7, 12	mp2an 705	. . . . . . . . 9 ⊢ ∃𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ ℝ
14		iinss 5026	. . . . . . . . 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 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
18	16, 17	sseldd 3941	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ℝ)
19	18	rexrd 11277	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ ℝ^*)
20		1red 11227	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
21		ax-1ne0 11187	. . . . . . . . . . 11 ⊢ 1 ≠ 0
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 1 ≠ 0)
23	20, 20, 22	redivcld 12061	. . . . . . . . 9 ⊢ (𝜑 → (1 / 1) ∈ ℝ)
24	3, 23	readdcld 11256	. . . . . . . 8 ⊢ (𝜑 → (𝐵 + (1 / 1)) ∈ ℝ)
25	24	rexrd 11277	. . . . . . 7 ⊢ (𝜑 → (𝐵 + (1 / 1)) ∈ ℝ^*)
26	25	adantr 486	. . . . . 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 2852	. . . . . . . 8 ⊢ (𝑛 = 1 → (𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) ↔ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1)))))
30	27, 28, 29	eliind 45832	. . . . . . 7 ⊢ (𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1))))
31	30	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 1))))
32		ioogtlb 46252	. . . . . 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 1947	. . . . . . . 8 ⊢ Ⅎ𝑛𝜑
35		nfcv 2928	. . . . . . . . 9 ⊢ Ⅎ𝑛𝑥
36		nfii1 4998	. . . . . . . . 9 ⊢ Ⅎ𝑛∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))
37	35, 36	nfel 2942	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))
38	34, 37	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
39		simpll 779	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝜑)
40		iinss2 5027	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
41	40	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
42		simpl 488	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))))
43	41, 42	sseldd 3941	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
44	43	adantll 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
45		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
46		elioore 13420	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) → 𝑥 ∈ ℝ)
47	46	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
48	47	adantll 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ)
49	3	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 ∈ ℝ)
50		nnrecre 12296	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
51	50	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
52	49, 51	readdcld 11256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ)
53	52	adantlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ)
54	1	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ^*)
55	54	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ^*)
56	52	rexrd 11277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ^*)
57	56	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → (𝐵 + (1 / 𝑛)) ∈ ℝ^*)
58		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛))))
59		iooltub 46267	. . . . . . . . . . 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 11376	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ≤ (𝐵 + (1 / 𝑛)))
62	39, 44, 45, 61	syl21anc 851	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) ∧ 𝑛 ∈ ℕ) → 𝑥 ≤ (𝐵 + (1 / 𝑛)))
63	62	ex 418	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → (𝑛 ∈ ℕ → 𝑥 ≤ (𝐵 + (1 / 𝑛))))
64	38, 63	ralrimi 3266	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → ∀𝑛 ∈ ℕ 𝑥 ≤ (𝐵 + (1 / 𝑛)))
65	38, 19, 4	xrralrecnnle 46139	. . . . . 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 46264	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))) → 𝑥 ∈ (𝐴(,]𝐵))
68	67	ralrimiva 3160	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛)))𝑥 ∈ (𝐴(,]𝐵))
69		dfss3 3929	. . 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 13195	. . . . . 6 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
72	71	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ≤ 𝐴)
73		1rp 13038	. . . . . . . . 9 ⊢ 1 ∈ ℝ⁺
74	73	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 1 ∈ ℝ⁺)
75		nnrp 13046	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ⁺)
76	74, 75	rpdivcld 13095	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ⁺)
77	76	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ⁺)
78	49, 77	ltaddrpd 13111	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵 < (𝐵 + (1 / 𝑛)))
79		iocssioo 13484	. . . . 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 852	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
81	80	ralrimiva 3160	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐴(,]𝐵) ⊆ (𝐴(,)(𝐵 + (1 / 𝑛))))
82		ssiin 5025	. . 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 3957	1 ⊢ (𝜑 → ∩ 𝑛 ∈ ℕ (𝐴(,)(𝐵 + (1 / 𝑛))) = (𝐴(,]𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 ∃wrex 3092 ⊆ wss 3908 ∩ ciin 4962 class class class wbr 5114 (class class class)co 7423 ℝcr 11117 0cc0 11118 1c1 11119 + caddc 11121 ℝ^*cxr 11260 < clt 11261 ≤ cle 11262 / cdiv 11889 ℕcn 12251 ℝ⁺crp 13034 (,)cioo 13390 (,]cioc 13391

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-n0 12523 df-z 12610 df-uz 12881 df-q 12991 df-rp 13035 df-ioo 13394 df-ioc 13395 df-fl 13845

This theorem is used by: iocborel 47111

W3C validator