Theorem vitali 25661

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 11243	. . . 4 ⊢ ℝ ∈ V
2	1	pwex 5385	. . 3 ⊢ 𝒫 ℝ ∈ V
3		weinxp 5772	. . . . 5 ⊢ ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
4		unipw 5460	. . . . . 6 ⊢ ∪ 𝒫 ℝ = ℝ
5		weeq2 5676	. . . . . 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 7771	. . . . . 6 ⊢ (ℝ × ℝ) ∈ V
9	8	inex2 5323	. . . . 5 ⊢ ( < ∩ (ℝ × ℝ)) ∈ V
10		weeq1 5675	. . . . 5 ⊢ (𝑥 = ( < ∩ (ℝ × ℝ)) → (𝑥 We ∪ 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ))
11	9, 10	spcev 3605	. . . 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 10070	. . 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 13000	. . . . . . 7 ⊢ ℚ ∈ V
16	15	inex1 5322	. . . . . 6 ⊢ (ℚ ∩ (-1[,]1)) ∈ V
17		nnrecq 13011	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℚ)
18		nnrecre 12305	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ)
19		neg1rr 12378	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → -1 ∈ ℝ)
21		0re 11260	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ∈ ℝ)
23		neg1lt0 12380	. . . . . . . . . . . 12 ⊢ -1 < 0
24	19, 21, 23	ltleii 11381	. . . . . . . . . . 11 ⊢ -1 ≤ 0
25	24	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → -1 ≤ 0)
26		nnrp 13043	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
27	26	rpreccld 13084	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ+)
28	27	rpge0d 13078	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ≤ (1 / 𝑥))
29	20, 22, 18, 25, 28	letrd 11415	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → -1 ≤ (1 / 𝑥))
30		nnge1 12291	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 1 ≤ 𝑥)
31		nnre 12270	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
32		nngt0 12294	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 0 < 𝑥)
33		1re 11258	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
34		0lt1 11782	. . . . . . . . . . . . 13 ⊢ 0 < 1
35		lerec 12148	. . . . . . . . . . . . 13 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
36	33, 34, 35	mpanl12 702	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
37	31, 32, 36	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
38	30, 37	mpbid 232	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ≤ (1 / 1))
39		1div1e1 11955	. . . . . . . . . 10 ⊢ (1 / 1) = 1
40	38, 39	breqtrdi 5188	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ≤ 1)
41	19, 33	elicc2i 13449	. . . . . . . . 9 ⊢ ((1 / 𝑥) ∈ (-1[,]1) ↔ ((1 / 𝑥) ∈ ℝ ∧ -1 ≤ (1 / 𝑥) ∧ (1 / 𝑥) ≤ 1))
42	18, 29, 40, 41	syl3anbrc 1342	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (-1[,]1))
43	17, 42	elind 4209	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (ℚ ∩ (-1[,]1)))
44		oveq2 7438	. . . . . . . . 9 ⊢ ((1 / 𝑥) = (1 / 𝑦) → (1 / (1 / 𝑥)) = (1 / (1 / 𝑦)))
45		nncn 12271	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
46		nnne0 12297	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
47	45, 46	recrecd 12037	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (1 / (1 / 𝑥)) = 𝑥)
48		nncn 12271	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
49		nnne0 12297	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
50	48, 49	recrecd 12037	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (1 / (1 / 𝑦)) = 𝑦)
51	47, 50	eqeqan12d 2748	. . . . . . . . 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 7438	. . . . . . . 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 9033	. . . . . 6 ⊢ ((ℚ ∩ (-1[,]1)) ∈ V → ℕ ≼ (ℚ ∩ (-1[,]1)))
56	16, 55	ax-mp 5	. . . . 5 ⊢ ℕ ≼ (ℚ ∩ (-1[,]1))
57		inss1 4244	. . . . . . 7 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
58		ssdomg 9038	. . . . . . 7 ⊢ (ℚ ∈ V → ((ℚ ∩ (-1[,]1)) ⊆ ℚ → (ℚ ∩ (-1[,]1)) ≼ ℚ))
59	15, 57, 58	mp2 9	. . . . . 6 ⊢ (ℚ ∩ (-1[,]1)) ≼ ℚ
60		qnnen 16245	. . . . . 6 ⊢ ℚ ≈ ℕ
61		domentr 9051	. . . . . 6 ⊢ (((ℚ ∩ (-1[,]1)) ≼ ℚ ∧ ℚ ≈ ℕ) → (ℚ ∩ (-1[,]1)) ≼ ℕ)
62	59, 60, 61	mp2an 692	. . . . 5 ⊢ (ℚ ∩ (-1[,]1)) ≼ ℕ
63		sbth 9131	. . . . 5 ⊢ ((ℕ ≼ (ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ≼ ℕ) → ℕ ≈ (ℚ ∩ (-1[,]1)))
64	56, 62, 63	mp2an 692	. . . 4 ⊢ ℕ ≈ (ℚ ∩ (-1[,]1))
65		bren 8993	. . . 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 2821	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ (0[,]1) ↔ 𝑥 ∈ (0[,]1)))
68		eleq1w 2821	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑦 → (𝑏 ∈ (0[,]1) ↔ 𝑦 ∈ (0[,]1)))
69	67, 68	bi2anan9 638	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))))
70		oveq12 7439	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 − 𝑏) = (𝑥 − 𝑦))
71	70	eleq1d 2823	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 − 𝑏) ∈ ℚ ↔ (𝑥 − 𝑦) ∈ ℚ))
72	69, 71	anbi12d 632	. . . . . . . . . . 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 5220	. . . . . . . . . 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 2734	. . . . . . . . . 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 6919	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑐) ∈ V
76		eqid 2734	. . . . . . . . . . . 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 6711	. . . . . . . . . . 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 3000	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
80		fveq2 6906	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑓‘𝑧) = (𝑓‘𝑤))
81		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
82	80, 81	eleq12d 2832	. . . . . . . . . . . . . . 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 3234	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
85	73	vitalilem1 25656	. . . . . . . . . . . . . . . . . 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 8816	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1))
88	87	mptru 1543	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1)
89		unitssre 13535	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) ⊆ ℝ
90	89	sspwi 4616	. . . . . . . . . . . . . . 15 ⊢ 𝒫 (0[,]1) ⊆ 𝒫 ℝ
91	88, 90	sstri 4004	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ
92		ssralv 4063	. . . . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑤 → (𝑓‘𝑐) = (𝑓‘𝑤))
96		fvex 6919	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑤) ∈ V
97	95, 76, 96	fvmpt 7015	. . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . 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 3088	. . . . . . . . . . . 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 727	. . . . . . . . . 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 771	. . . . . . . . . 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 7437	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑠 → (𝑡 − (𝑔‘𝑚)) = (𝑠 − (𝑔‘𝑚)))
105	104	eleq1d 2823	. . . . . . . . . . . . 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 3443	. . . . . . . . . . . 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 6906	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝑔‘𝑚) = (𝑔‘𝑛))
108	107	oveq2d 7446	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑠 − (𝑔‘𝑚)) = (𝑠 − (𝑔‘𝑛)))
109	108	eleq1d 2823	. . . . . . . . . . . . 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 3440	. . . . . . . . . . . 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 2786	. . . . . . . . . . 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 5260	. . . . . . . . . 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 773	. . . . . . . . . 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 25660	. . . . . . . . 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 3972	. . . . . . 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 25579	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
120		velpw 4609	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
121	119, 120	sylibr 234	. . . . . . . . 9 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
122	121	ssriv 3998	. . . . . . . 8 ⊢ dom vol ⊆ 𝒫 ℝ
123		ssnelpss 4123	. . . . . . . 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 1930	. . 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 1932	1 ⊢ ( < We ℝ → dom vol ⊊ 𝒫 ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536  ⊤wtru 1537  ∃wex 1775   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  {crab 3432  Vcvv 3477   ∖ cdif 3959   ∩ cin 3961   ⊆ wss 3962   ⊊ wpss 3963  ∅c0 4338  𝒫 cpw 4604  ∪ cuni 4911   class class class wbr 5147  {copab 5209   ↦ cmpt 5230   We wwe 5639   × cxp 5686  dom cdm 5688  ran crn 5689   Fn wfn 6557  –1-1-onto→wf1o 6561  ‘cfv 6562  (class class class)co 7430   Er wer 8740   / cqs 8742   ≈ cen 8980   ≼ cdom 8981  ℝcr 11151  0cc0 11152  1c1 11153   < clt 11292   ≤ cle 11293   − cmin 11489  -cneg 11490   / cdiv 11917  ℕcn 12263  ℚcq 12987  [,]cicc 13386  volcvol 25511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cc 10472  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-er 8743  df-ec 8745  df-qs 8749  df-map 8866  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-acn 9979  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-rlim 15521  df-sum 15719  df-rest 17468  df-topgen 17489  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-top 22915  df-topon 22932  df-bases 22968  df-cmp 23410  df-ovol 25512  df-vol 25513

This theorem is referenced by:  vitali2  46649