Theorem vitali 25667

Description: If the reals can be well-ordered, then there are non-measurable sets. The proof uses "Vitali sets", named for Giuseppe Vitali (1905). (Contributed by Mario Carneiro, 16-Jun-2014.)

Assertion

Ref	Expression
vitali	⊢ ( < We ℝ → dom vol ⊊ 𝒫 ℝ)

Proof of Theorem vitali

Dummy variables 𝑎 𝑏 𝑐 𝑓 𝑔 𝑚 𝑛 𝑠 𝑡 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 11275	. . . 4 ⊢ ℝ ∈ V
2	1	pwex 5398	. . 3 ⊢ 𝒫 ℝ ∈ V
3		weinxp 5784	. . . . 5 ⊢ ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
4		unipw 5470	. . . . . 6 ⊢ ∪ 𝒫 ℝ = ℝ
5		weeq2 5688	. . . . . 6 ⊢ (∪ 𝒫 ℝ = ℝ → (( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ))
6	4, 5	ax-mp 5	. . . . 5 ⊢ (( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
7	3, 6	bitr4i 278	. . . 4 ⊢ ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ)
8	1, 1	xpex 7788	. . . . . 6 ⊢ (ℝ × ℝ) ∈ V
9	8	inex2 5336	. . . . 5 ⊢ ( < ∩ (ℝ × ℝ)) ∈ V
10		weeq1 5687	. . . . 5 ⊢ (𝑥 = ( < ∩ (ℝ × ℝ)) → (𝑥 We ∪ 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ))
11	9, 10	spcev 3619	. . . 4 ⊢ (( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ → ∃𝑥 𝑥 We ∪ 𝒫 ℝ)
12	7, 11	sylbi 217	. . 3 ⊢ ( < We ℝ → ∃𝑥 𝑥 We ∪ 𝒫 ℝ)
13		dfac8c 10102	. . 3 ⊢ (𝒫 ℝ ∈ V → (∃𝑥 𝑥 We ∪ 𝒫 ℝ → ∃𝑓∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
14	2, 12, 13	mpsyl 68	. 2 ⊢ ( < We ℝ → ∃𝑓∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
15		qex 13026	. . . . . . 7 ⊢ ℚ ∈ V
16	15	inex1 5335	. . . . . 6 ⊢ (ℚ ∩ (-1[,]1)) ∈ V
17		nnrecq 13037	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℚ)
18		nnrecre 12335	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ)
19		neg1rr 12408	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → -1 ∈ ℝ)
21		0re 11292	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ∈ ℝ)
23		neg1lt0 12410	. . . . . . . . . . . 12 ⊢ -1 < 0
24	19, 21, 23	ltleii 11413	. . . . . . . . . . 11 ⊢ -1 ≤ 0
25	24	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → -1 ≤ 0)
26		nnrp 13068	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
27	26	rpreccld 13109	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ+)
28	27	rpge0d 13103	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ≤ (1 / 𝑥))
29	20, 22, 18, 25, 28	letrd 11447	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → -1 ≤ (1 / 𝑥))
30		nnge1 12321	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 1 ≤ 𝑥)
31		nnre 12300	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
32		nngt0 12324	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 0 < 𝑥)
33		1re 11290	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
34		0lt1 11812	. . . . . . . . . . . . 13 ⊢ 0 < 1
35		lerec 12178	. . . . . . . . . . . . 13 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
36	33, 34, 35	mpanl12 701	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
37	31, 32, 36	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
38	30, 37	mpbid 232	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ≤ (1 / 1))
39		1div1e1 11985	. . . . . . . . . 10 ⊢ (1 / 1) = 1
40	38, 39	breqtrdi 5207	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ≤ 1)
41	19, 33	elicc2i 13473	. . . . . . . . 9 ⊢ ((1 / 𝑥) ∈ (-1[,]1) ↔ ((1 / 𝑥) ∈ ℝ ∧ -1 ≤ (1 / 𝑥) ∧ (1 / 𝑥) ≤ 1))
42	18, 29, 40, 41	syl3anbrc 1343	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (-1[,]1))
43	17, 42	elind 4223	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (ℚ ∩ (-1[,]1)))
44		oveq2 7456	. . . . . . . . 9 ⊢ ((1 / 𝑥) = (1 / 𝑦) → (1 / (1 / 𝑥)) = (1 / (1 / 𝑦)))
45		nncn 12301	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
46		nnne0 12327	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
47	45, 46	recrecd 12067	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (1 / (1 / 𝑥)) = 𝑥)
48		nncn 12301	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
49		nnne0 12327	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
50	48, 49	recrecd 12067	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (1 / (1 / 𝑦)) = 𝑦)
51	47, 50	eqeqan12d 2754	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / (1 / 𝑥)) = (1 / (1 / 𝑦)) ↔ 𝑥 = 𝑦))
52	44, 51	imbitrid 244	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) → 𝑥 = 𝑦))
53		oveq2 7456	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
54	52, 53	impbid1 225	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) ↔ 𝑥 = 𝑦))
55	43, 54	dom2 9055	. . . . . 6 ⊢ ((ℚ ∩ (-1[,]1)) ∈ V → ℕ ≼ (ℚ ∩ (-1[,]1)))
56	16, 55	ax-mp 5	. . . . 5 ⊢ ℕ ≼ (ℚ ∩ (-1[,]1))
57		inss1 4258	. . . . . . 7 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
58		ssdomg 9060	. . . . . . 7 ⊢ (ℚ ∈ V → ((ℚ ∩ (-1[,]1)) ⊆ ℚ → (ℚ ∩ (-1[,]1)) ≼ ℚ))
59	15, 57, 58	mp2 9	. . . . . 6 ⊢ (ℚ ∩ (-1[,]1)) ≼ ℚ
60		qnnen 16261	. . . . . 6 ⊢ ℚ ≈ ℕ
61		domentr 9073	. . . . . 6 ⊢ (((ℚ ∩ (-1[,]1)) ≼ ℚ ∧ ℚ ≈ ℕ) → (ℚ ∩ (-1[,]1)) ≼ ℕ)
62	59, 60, 61	mp2an 691	. . . . 5 ⊢ (ℚ ∩ (-1[,]1)) ≼ ℕ
63		sbth 9159	. . . . 5 ⊢ ((ℕ ≼ (ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ≼ ℕ) → ℕ ≈ (ℚ ∩ (-1[,]1)))
64	56, 62, 63	mp2an 691	. . . 4 ⊢ ℕ ≈ (ℚ ∩ (-1[,]1))
65		bren 9013	. . . 4 ⊢ (ℕ ≈ (ℚ ∩ (-1[,]1)) ↔ ∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
66	64, 65	mpbi 230	. . 3 ⊢ ∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))
67		eleq1w 2827	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ (0[,]1) ↔ 𝑥 ∈ (0[,]1)))
68		eleq1w 2827	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑦 → (𝑏 ∈ (0[,]1) ↔ 𝑦 ∈ (0[,]1)))
69	67, 68	bi2anan9 637	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))))
70		oveq12 7457	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 − 𝑏) = (𝑥 − 𝑦))
71	70	eleq1d 2829	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 − 𝑏) ∈ ℚ ↔ (𝑥 − 𝑦) ∈ ℚ))
72	69, 71	anbi12d 631	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ) ↔ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)))
73	72	cbvopabv 5239	. . . . . . . . . 10 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥 − 𝑦) ∈ ℚ)}
74		eqid 2740	. . . . . . . . . 10 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) = ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)})
75		fvex 6933	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑐) ∈ V
76		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) = (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))
77	75, 76	fnmpti 6723	. . . . . . . . . . 11 ⊢ (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) Fn ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)})
78	77	a1i 11	. . . . . . . . . 10 ⊢ ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol))) → (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) Fn ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}))
79		neeq1 3009	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
80		fveq2 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑓‘𝑧) = (𝑓‘𝑤))
81		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
82	80, 81	eleq12d 2838	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑤) ∈ 𝑤))
83	79, 82	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)))
84	83	cbvralvw 3243	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
85	73	vitalilem1 25662	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)} Er (0[,]1)
86	85	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)} Er (0[,]1))
87	86	qsss 8836	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1))
88	87	mptru 1544	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1)
89		unitssre 13559	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) ⊆ ℝ
90	89	sspwi 4634	. . . . . . . . . . . . . . 15 ⊢ 𝒫 (0[,]1) ⊆ 𝒫 ℝ
91	88, 90	sstri 4018	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ
92		ssralv 4077	. . . . . . . . . . . . . 14 ⊢ (((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ → (∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)))
93	91, 92	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
94	84, 93	sylbi 217	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
95		fveq2 6920	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑤 → (𝑓‘𝑐) = (𝑓‘𝑤))
96		fvex 6933	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑤) ∈ V
97	95, 76, 96	fvmpt 7029	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) → ((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))‘𝑤) = (𝑓‘𝑤))
98	97	eleq1d 2829	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) → (((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))‘𝑤) ∈ 𝑤 ↔ (𝑓‘𝑤) ∈ 𝑤))
99	98	imbi2d 340	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) → ((𝑤 ≠ ∅ → ((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))‘𝑤) ∈ 𝑤) ↔ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)))
100	99	ralbiia 3097	. . . . . . . . . . . 12 ⊢ (∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → ((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))‘𝑤) ∈ 𝑤) ↔ ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
101	94, 100	sylibr 234	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → ((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))‘𝑤) ∈ 𝑤))
102	101	ad2antlr 726	. . . . . . . . . 10 ⊢ ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol))) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → ((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))‘𝑤) ∈ 𝑤))
103		simprl 770	. . . . . . . . . 10 ⊢ ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol))) → 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
104		oveq1 7455	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑠 → (𝑡 − (𝑔‘𝑚)) = (𝑠 − (𝑔‘𝑚)))
105	104	eleq1d 2829	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → ((𝑡 − (𝑔‘𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ↔ (𝑠 − (𝑔‘𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))))
106	105	cbvrabv 3454	. . . . . . . . . . . 12 ⊢ {𝑡 ∈ ℝ ∣ (𝑡 − (𝑔‘𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝑔‘𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))}
107		fveq2 6920	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝑔‘𝑚) = (𝑔‘𝑛))
108	107	oveq2d 7464	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑠 − (𝑔‘𝑚)) = (𝑠 − (𝑔‘𝑛)))
109	108	eleq1d 2829	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → ((𝑠 − (𝑔‘𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ↔ (𝑠 − (𝑔‘𝑛)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))))
110	109	rabbidv 3451	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝑔‘𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝑔‘𝑛)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))})
111	106, 110	eqtrid 2792	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → {𝑡 ∈ ℝ ∣ (𝑡 − (𝑔‘𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝑔‘𝑛)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))})
112	111	cbvmptv 5279	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ ↦ {𝑡 ∈ ℝ ∣ (𝑡 − (𝑔‘𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))}) = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝑔‘𝑛)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐))})
113		simprr 772	. . . . . . . . . 10 ⊢ ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol))) → ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol))
114	73, 74, 78, 102, 103, 112, 113	vitalilem5 25666	. . . . . . . . 9 ⊢ ¬ (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol)))
115	114	pm2.21i 119	. . . . . . . 8 ⊢ ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol))) → ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol))
116	115	expr 456	. . . . . . 7 ⊢ ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ∧ 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) → (¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol) → ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol)))
117	116	pm2.18d 127	. . . . . 6 ⊢ ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ∧ 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) → ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol))
118		eldif 3986	. . . . . . 7 ⊢ (ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol) ↔ (ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ 𝒫 ℝ ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ dom vol))
119		mblss 25585	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
120		velpw 4627	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
121	119, 120	sylibr 234	. . . . . . . . 9 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
122	121	ssriv 4012	. . . . . . . 8 ⊢ dom vol ⊆ 𝒫 ℝ
123		ssnelpss 4137	. . . . . . . 8 ⊢ (dom vol ⊆ 𝒫 ℝ → ((ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ 𝒫 ℝ ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ dom vol) → dom vol ⊊ 𝒫 ℝ))
124	122, 123	ax-mp 5	. . . . . . 7 ⊢ ((ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ 𝒫 ℝ ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ dom vol) → dom vol ⊊ 𝒫 ℝ)
125	118, 124	sylbi 217	. . . . . 6 ⊢ (ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ↦ (𝑓‘𝑐)) ∈ (𝒫 ℝ ∖ dom vol) → dom vol ⊊ 𝒫 ℝ)
126	117, 125	syl 17	. . . . 5 ⊢ ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ∧ 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) → dom vol ⊊ 𝒫 ℝ)
127	126	ex 412	. . . 4 ⊢ (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → dom vol ⊊ 𝒫 ℝ))
128	127	exlimdv 1932	. . 3 ⊢ (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → (∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → dom vol ⊊ 𝒫 ℝ))
129	66, 128	mpi 20	. 2 ⊢ (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → dom vol ⊊ 𝒫 ℝ)
130	14, 129	exlimddv 1934	1 ⊢ ( < We ℝ → dom vol ⊊ 𝒫 ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ⊤wtru 1538  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  {crab 3443  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976   ⊊ wpss 3977  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166  {copab 5228   ↦ cmpt 5249   We wwe 5651   × cxp 5698  dom cdm 5700  ran crn 5701   Fn wfn 6568  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   Er wer 8760   / cqs 8762   ≈ cen 9000   ≼ cdom 9001  ℝcr 11183  0cc0 11184  1c1 11185   < clt 11324   ≤ cle 11325   − cmin 11520  -cneg 11521   / cdiv 11947  ℕcn 12293  ℚcq 13013  [,]cicc 13410  volcvol 25517

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-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cc 10504  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-int 4971  df-iun 5017  df-disj 5134  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-se 5653  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-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-oadd 8526  df-omul 8527  df-er 8763  df-ec 8765  df-qs 8769  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fi 9480  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-acn 10011  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-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-rlim 15535  df-sum 15735  df-rest 17482  df-topgen 17503  df-psmet 21379  df-xmet 21380  df-met 21381  df-bl 21382  df-mopn 21383  df-top 22921  df-topon 22938  df-bases 22974  df-cmp 23416  df-ovol 25518  df-vol 25519

This theorem is referenced by:  vitali2  46615