Theorem vitali 25512

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 11100	. . . 4 ⊢ ℝ ∈ V
2	1	pwex 5319	. . 3 ⊢ 𝒫 ℝ ∈ V
3		weinxp 5704	. . . . 5 ⊢ ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
4		unipw 5393	. . . . . 6 ⊢ ∪ 𝒫 ℝ = ℝ
5		weeq2 5607	. . . . . 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 7689	. . . . . 6 ⊢ (ℝ × ℝ) ∈ V
9	8	inex2 5257	. . . . 5 ⊢ ( < ∩ (ℝ × ℝ)) ∈ V
10		weeq1 5606	. . . . 5 ⊢ (𝑥 = ( < ∩ (ℝ × ℝ)) → (𝑥 We ∪ 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ))
11	9, 10	spcev 3561	. . . 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 9927	. . 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 12862	. . . . . . 7 ⊢ ℚ ∈ V
16	15	inex1 5256	. . . . . 6 ⊢ (ℚ ∩ (-1[,]1)) ∈ V
17		nnrecq 12873	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℚ)
18		nnrecre 12170	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ)
19		neg1rr 12114	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → -1 ∈ ℝ)
21		0re 11117	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ∈ ℝ)
23		neg1lt0 12116	. . . . . . . . . . . 12 ⊢ -1 < 0
24	19, 21, 23	ltleii 11239	. . . . . . . . . . 11 ⊢ -1 ≤ 0
25	24	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → -1 ≤ 0)
26		nnrp 12905	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
27	26	rpreccld 12947	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ+)
28	27	rpge0d 12941	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ≤ (1 / 𝑥))
29	20, 22, 18, 25, 28	letrd 11273	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → -1 ≤ (1 / 𝑥))
30		nnge1 12156	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 1 ≤ 𝑥)
31		nnre 12135	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
32		nngt0 12159	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 0 < 𝑥)
33		1re 11115	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
34		0lt1 11642	. . . . . . . . . . . . 13 ⊢ 0 < 1
35		lerec 12008	. . . . . . . . . . . . 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 11815	. . . . . . . . . 10 ⊢ (1 / 1) = 1
40	38, 39	breqtrdi 5133	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ≤ 1)
41	19, 33	elicc2i 13315	. . . . . . . . 9 ⊢ ((1 / 𝑥) ∈ (-1[,]1) ↔ ((1 / 𝑥) ∈ ℝ ∧ -1 ≤ (1 / 𝑥) ∧ (1 / 𝑥) ≤ 1))
42	18, 29, 40, 41	syl3anbrc 1344	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (-1[,]1))
43	17, 42	elind 4151	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (ℚ ∩ (-1[,]1)))
44		oveq2 7357	. . . . . . . . 9 ⊢ ((1 / 𝑥) = (1 / 𝑦) → (1 / (1 / 𝑥)) = (1 / (1 / 𝑦)))
45		nncn 12136	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
46		nnne0 12162	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
47	45, 46	recrecd 11897	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (1 / (1 / 𝑥)) = 𝑥)
48		nncn 12136	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
49		nnne0 12162	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
50	48, 49	recrecd 11897	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (1 / (1 / 𝑦)) = 𝑦)
51	47, 50	eqeqan12d 2743	. . . . . . . . 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 7357	. . . . . . . 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 8920	. . . . . 6 ⊢ ((ℚ ∩ (-1[,]1)) ∈ V → ℕ ≼ (ℚ ∩ (-1[,]1)))
56	16, 55	ax-mp 5	. . . . 5 ⊢ ℕ ≼ (ℚ ∩ (-1[,]1))
57		inss1 4188	. . . . . . 7 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
58		ssdomg 8925	. . . . . . 7 ⊢ (ℚ ∈ V → ((ℚ ∩ (-1[,]1)) ⊆ ℚ → (ℚ ∩ (-1[,]1)) ≼ ℚ))
59	15, 57, 58	mp2 9	. . . . . 6 ⊢ (ℚ ∩ (-1[,]1)) ≼ ℚ
60		qnnen 16122	. . . . . 6 ⊢ ℚ ≈ ℕ
61		domentr 8938	. . . . . 6 ⊢ (((ℚ ∩ (-1[,]1)) ≼ ℚ ∧ ℚ ≈ ℕ) → (ℚ ∩ (-1[,]1)) ≼ ℕ)
62	59, 60, 61	mp2an 692	. . . . 5 ⊢ (ℚ ∩ (-1[,]1)) ≼ ℕ
63		sbth 9014	. . . . 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 8882	. . . 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 2811	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ (0[,]1) ↔ 𝑥 ∈ (0[,]1)))
68		eleq1w 2811	. . . . . . . . . . . . 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 7358	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 − 𝑏) = (𝑥 − 𝑦))
71	70	eleq1d 2813	. . . . . . . . . . . 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 5165	. . . . . . . . . 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 2729	. . . . . . . . . 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 6835	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑐) ∈ V
76		eqid 2729	. . . . . . . . . . . 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 6625	. . . . . . . . . . 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 2987	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
80		fveq2 6822	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑓‘𝑧) = (𝑓‘𝑤))
81		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
82	80, 81	eleq12d 2822	. . . . . . . . . . . . . . 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 3207	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
85	73	vitalilem1 25507	. . . . . . . . . . . . . . . . . 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 8703	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1))
88	87	mptru 1547	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1)
89		unitssre 13402	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) ⊆ ℝ
90	89	sspwi 4563	. . . . . . . . . . . . . . 15 ⊢ 𝒫 (0[,]1) ⊆ 𝒫 ℝ
91	88, 90	sstri 3945	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ
92		ssralv 4004	. . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑤 → (𝑓‘𝑐) = (𝑓‘𝑤))
96		fvex 6835	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑤) ∈ V
97	95, 76, 96	fvmpt 6930	. . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . 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 3073	. . . . . . . . . . . 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 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 7356	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑠 → (𝑡 − (𝑔‘𝑚)) = (𝑠 − (𝑔‘𝑚)))
105	104	eleq1d 2813	. . . . . . . . . . . . 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 3405	. . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝑔‘𝑚) = (𝑔‘𝑛))
108	107	oveq2d 7365	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑠 − (𝑔‘𝑚)) = (𝑠 − (𝑔‘𝑛)))
109	108	eleq1d 2813	. . . . . . . . . . . . 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 3402	. . . . . . . . . . . 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 2776	. . . . . . . . . . 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 5196	. . . . . . . . . 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 25511	. . . . . . . . 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 3913	. . . . . . 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 25430	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
120		velpw 4556	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
121	119, 120	sylibr 234	. . . . . . . . 9 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
122	121	ssriv 3939	. . . . . . . 8 ⊢ dom vol ⊆ 𝒫 ℝ
123		ssnelpss 4065	. . . . . . . 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 1933	. . 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 1935	1 ⊢ ( < We ℝ → dom vol ⊊ 𝒫 ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3394  Vcvv 3436   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903   ⊊ wpss 3904  ∅c0 4284  𝒫 cpw 4551  ∪ cuni 4858   class class class wbr 5092  {copab 5154   ↦ cmpt 5173   We wwe 5571   × cxp 5617  dom cdm 5619  ran crn 5620   Fn wfn 6477  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349   Er wer 8622   / cqs 8624   ≈ cen 8869   ≼ cdom 8870  ℝcr 11008  0cc0 11009  1c1 11010   < clt 11149   ≤ cle 11150   − cmin 11347  -cneg 11348   / cdiv 11777  ℕcn 12128  ℚcq 12849  [,]cicc 13251  volcvol 25362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537  ax-cc 10329  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-disj 5060  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-isom 6491  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-of 7613  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-omul 8393  df-er 8625  df-ec 8627  df-qs 8631  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-fi 9301  df-sup 9332  df-inf 9333  df-oi 9402  df-dju 9797  df-card 9835  df-acn 9838  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-q 12850  df-rp 12894  df-xneg 13014  df-xadd 13015  df-xmul 13016  df-ioo 13252  df-ico 13254  df-icc 13255  df-fz 13411  df-fzo 13558  df-fl 13696  df-seq 13909  df-exp 13969  df-hash 14238  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-sum 15594  df-rest 17326  df-topgen 17347  df-psmet 21253  df-xmet 21254  df-met 21255  df-bl 21256  df-mopn 21257  df-top 22779  df-topon 22796  df-bases 22831  df-cmp 23272  df-ovol 25363  df-vol 25364

This theorem is referenced by:  vitali2  46675