Theorem vitali 25740

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 11190	. . . 4 ⊢ ℝ ∈ V
2	1	pwex 5352	. . 3 ⊢ 𝒫 ℝ ∈ V
3		weinxp 5747	. . . . 5 ⊢ ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
4		unipw 5432	. . . . . 6 ⊢ ∪ 𝒫 ℝ = ℝ
5		weeq2 5650	. . . . . 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 281	. . . 4 ⊢ ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ)
8	1, 1	xpex 7751	. . . . . 6 ⊢ (ℝ × ℝ) ∈ V
9	8	inex2 5289	. . . . 5 ⊢ ( < ∩ (ℝ × ℝ)) ∈ V
10		weeq1 5649	. . . . 5 ⊢ (𝑥 = ( < ∩ (ℝ × ℝ)) → (𝑥 We ∪ 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ))
11	9, 10	spcev 3574	. . . 4 ⊢ (( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ → ∃𝑥 𝑥 We ∪ 𝒫 ℝ)
12	7, 11	sylbi 220	. . 3 ⊢ ( < We ℝ → ∃𝑥 𝑥 We ∪ 𝒫 ℝ)
13		dfac8c 10016	. . 3 ⊢ (𝒫 ℝ ∈ V → (∃𝑥 𝑥 We ∪ 𝒫 ℝ → ∃𝑓∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
14	2, 12, 13	mpsyl 69	. 2 ⊢ ( < We ℝ → ∃𝑓∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
15		qex 12984	. . . . . . 7 ⊢ ℚ ∈ V
16	15	inex1 5288	. . . . . 6 ⊢ (ℚ ∩ (-1[,]1)) ∈ V
17		nnrecq 12995	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℚ)
18		nnrecre 12277	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ)
19		neg1rr 12203	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → -1 ∈ ℝ)
21		0re 11209	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ∈ ℝ)
23		neg1lt0 12205	. . . . . . . . . . . 12 ⊢ -1 < 0
24	19, 21, 23	ltleii 11332	. . . . . . . . . . 11 ⊢ -1 ≤ 0
25	24	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → -1 ≤ 0)
26		nnrp 13027	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
27	26	rpreccld 13069	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ+)
28	27	rpge0d 13063	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ≤ (1 / 𝑥))
29	20, 22, 18, 25, 28	letrd 11366	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → -1 ≤ (1 / 𝑥))
30		nnge1 12263	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 1 ≤ 𝑥)
31		nnre 12239	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
32		nngt0 12266	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 0 < 𝑥)
33		1re 11207	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
34		0lt1 11735	. . . . . . . . . . . . 13 ⊢ 0 < 1
35		lerec 12097	. . . . . . . . . . . . 13 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
36	33, 34, 35	mpanl12 714	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
37	31, 32, 36	syl2anc 595	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
38	30, 37	mpbid 235	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ≤ (1 / 1))
39		1div1e1 11904	. . . . . . . . . 10 ⊢ (1 / 1) = 1
40	38, 39	breqtrdi 5156	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ≤ 1)
41	19, 33	elicc2i 13438	. . . . . . . . 9 ⊢ ((1 / 𝑥) ∈ (-1[,]1) ↔ ((1 / 𝑥) ∈ ℝ ∧ -1 ≤ (1 / 𝑥) ∧ (1 / 𝑥) ≤ 1))
42	18, 29, 40, 41	syl3anbrc 1360	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (-1[,]1))
43	17, 42	elind 4161	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (ℚ ∩ (-1[,]1)))
44		oveq2 7419	. . . . . . . . 9 ⊢ ((1 / 𝑥) = (1 / 𝑦) → (1 / (1 / 𝑥)) = (1 / (1 / 𝑦)))
45		nncn 12240	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
46		nnne0 12269	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
47	45, 46	recrecd 11987	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (1 / (1 / 𝑥)) = 𝑥)
48		nncn 12240	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
49		nnne0 12269	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
50	48, 49	recrecd 11987	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (1 / (1 / 𝑦)) = 𝑦)
51	47, 50	eqeqan12d 2783	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / (1 / 𝑥)) = (1 / (1 / 𝑦)) ↔ 𝑥 = 𝑦))
52	44, 51	imbitrid 247	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) → 𝑥 = 𝑦))
53		oveq2 7419	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
54	52, 53	impbid1 228	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) ↔ 𝑥 = 𝑦))
55	43, 54	dom2 8991	. . . . . 6 ⊢ ((ℚ ∩ (-1[,]1)) ∈ V → ℕ ≼ (ℚ ∩ (-1[,]1)))
56	16, 55	ax-mp 5	. . . . 5 ⊢ ℕ ≼ (ℚ ∩ (-1[,]1))
57		inss1 4197	. . . . . . 7 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
58		ssdomg 8996	. . . . . . 7 ⊢ (ℚ ∈ V → ((ℚ ∩ (-1[,]1)) ⊆ ℚ → (ℚ ∩ (-1[,]1)) ≼ ℚ))
59	15, 57, 58	mp2 9	. . . . . 6 ⊢ (ℚ ∩ (-1[,]1)) ≼ ℚ
60		qnnen 16268	. . . . . 6 ⊢ ℚ ≈ ℕ
61		domentr 9009	. . . . . 6 ⊢ (((ℚ ∩ (-1[,]1)) ≼ ℚ ∧ ℚ ≈ ℕ) → (ℚ ∩ (-1[,]1)) ≼ ℕ)
62	59, 60, 61	mp2an 704	. . . . 5 ⊢ (ℚ ∩ (-1[,]1)) ≼ ℕ
63		sbth 9084	. . . . 5 ⊢ ((ℕ ≼ (ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ≼ ℕ) → ℕ ≈ (ℚ ∩ (-1[,]1)))
64	56, 62, 63	mp2an 704	. . . 4 ⊢ ℕ ≈ (ℚ ∩ (-1[,]1))
65		bren 8952	. . . 4 ⊢ (ℕ ≈ (ℚ ∩ (-1[,]1)) ↔ ∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
66	64, 65	mpbi 233	. . 3 ⊢ ∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))
67		eleq1w 2852	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ (0[,]1) ↔ 𝑥 ∈ (0[,]1)))
68		eleq1w 2852	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑦 → (𝑏 ∈ (0[,]1) ↔ 𝑦 ∈ (0[,]1)))
69	67, 68	bi2anan9 649	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))))
70		oveq12 7420	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 − 𝑏) = (𝑥 − 𝑦))
71	70	eleq1d 2854	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 − 𝑏) ∈ ℚ ↔ (𝑥 − 𝑦) ∈ ℚ))
72	69, 71	anbi12d 643	. . . . . . . . . . 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 5188	. . . . . . . . . 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 2769	. . . . . . . . . 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 6895	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑐) ∈ V
76		eqid 2769	. . . . . . . . . . . 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 6679	. . . . . . . . . . 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 3026	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
80		fveq2 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑓‘𝑧) = (𝑓‘𝑤))
81		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
82	80, 81	eleq12d 2863	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑤) ∈ 𝑤))
83	79, 82	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)))
84	83	cbvralvw 3249	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
85	73	vitalilem1 25735	. . . . . . . . . . . . . . . . . 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 8772	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1))
88	87	mptru 1574	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1)
89		unitssre 13525	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) ⊆ ℝ
90	89	sspwi 4579	. . . . . . . . . . . . . . 15 ⊢ 𝒫 (0[,]1) ⊆ 𝒫 ℝ
91	88, 90	sstri 3954	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ
92		ssralv 4014	. . . . . . . . . . . . . 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 220	. . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑤 → (𝑓‘𝑐) = (𝑓‘𝑤))
96		fvex 6895	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑤) ∈ V
97	95, 76, 96	fvmpt 6990	. . . . . . . . . . . . . . 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 2854	. . . . . . . . . . . . . 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 343	. . . . . . . . . . . . 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 3115	. . . . . . . . . . . 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 237	. . . . . . . . . . 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 739	. . . . . . . . . 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 782	. . . . . . . . . 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 7418	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑠 → (𝑡 − (𝑔‘𝑚)) = (𝑠 − (𝑔‘𝑚)))
105	104	eleq1d 2854	. . . . . . . . . . . . 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 3433	. . . . . . . . . . . 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 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝑔‘𝑚) = (𝑔‘𝑛))
108	107	oveq2d 7427	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑠 − (𝑔‘𝑚)) = (𝑠 − (𝑔‘𝑛)))
109	108	eleq1d 2854	. . . . . . . . . . . . 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 3430	. . . . . . . . . . . 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 2816	. . . . . . . . . . 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 5219	. . . . . . . . . 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 784	. . . . . . . . . 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 25739	. . . . . . . . 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 120	. . . . . . . 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 461	. . . . . . 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 128	. . . . . 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 3923	. . . . . . 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 25658	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
120		velpw 4572	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
121	119, 120	sylibr 237	. . . . . . . . 9 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
122	121	ssriv 3949	. . . . . . . 8 ⊢ dom vol ⊆ 𝒫 ℝ
123		ssnelpss 4077	. . . . . . . 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 220	. . . . . 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 18	. . . . 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 417	. . . 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 1960	. . 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 21	. 2 ⊢ (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) → dom vol ⊊ 𝒫 ℝ)
130	14, 129	exlimddv 1962	1 ⊢ ( < We ℝ → dom vol ⊊ 𝒫 ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ⊤wtru 1568  ∃wex 1806   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  {crab 3423  Vcvv 3463   ∖ cdif 3910   ∩ cin 3912   ⊆ wss 3913   ⊊ wpss 3914  ∅c0 4294  𝒫 cpw 4567  ∪ cuni 4876   class class class wbr 5113  {copab 5177   ↦ cmpt 5196   We wwe 5614   × cxp 5660  dom cdm 5662  ran crn 5663   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411   Er wer 8690   / cqs 8692   ≈ cen 8939   ≼ cdom 8940  ℝcr 11098  0cc0 11099  1c1 11100   < clt 11242   ≤ cle 11243   − cmin 11440  -cneg 11441   / cdiv 11870  ℕcn 12232  ℚcq 12971  [,]cicc 13374  volcvol 25590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9609  ax-cc 10418  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-addrcl 11160  ax-mulcl 11161  ax-mulrcl 11162  ax-mulcom 11163  ax-addass 11164  ax-mulass 11165  ax-distr 11166  ax-i2m1 11167  ax-1ne0 11168  ax-1rid 11169  ax-rnegex 11170  ax-rrecex 11171  ax-cnre 11172  ax-pre-lttri 11173  ax-pre-lttrn 11174  ax-pre-ltadd 11175  ax-pre-mulgt0 11176  ax-pre-sup 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-disj 5081  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7862  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8452  df-2o 8453  df-oadd 8456  df-omul 8457  df-er 8693  df-ec 8695  df-qs 8699  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fi 9370  df-sup 9401  df-inf 9402  df-oi 9471  df-dju 9886  df-card 9924  df-acn 9927  df-pnf 11244  df-mnf 11245  df-xr 11246  df-ltxr 11247  df-le 11248  df-sub 11442  df-neg 11443  df-div 11871  df-nn 12233  df-2 12302  df-3 12303  df-n0 12504  df-z 12591  df-uz 12862  df-q 12972  df-rp 13016  df-xneg 13136  df-xadd 13137  df-xmul 13138  df-ioo 13375  df-ico 13377  df-icc 13378  df-fz 13535  df-fzo 13682  df-fl 13824  df-seq 14037  df-exp 14097  df-hash 14366  df-cj 15149  df-re 15150  df-im 15151  df-sqrt 15285  df-abs 15286  df-clim 15538  df-rlim 15539  df-sum 15737  df-rest 17474  df-topgen 17495  df-psmet 21482  df-xmet 21483  df-met 21484  df-bl 21485  df-mopn 21486  df-top 23019  df-topon 23036  df-bases 23071  df-cmp 23512  df-ovol 25591  df-vol 25592

This theorem is referenced by:  vitali2  47299