Theorem vitali 24326

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 10679	. . . 4 ⊢ ℝ ∈ V
2	1	pwex 5253	. . 3 ⊢ 𝒫 ℝ ∈ V
3		weinxp 5610	. . . . 5 ⊢ ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
4		unipw 5315	. . . . . 6 ⊢ ∪ 𝒫 ℝ = ℝ
5		weeq2 5517	. . . . . 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 7480	. . . . . 6 ⊢ (ℝ × ℝ) ∈ V
9	8	inex2 5192	. . . . 5 ⊢ ( < ∩ (ℝ × ℝ)) ∈ V
10		weeq1 5516	. . . . 5 ⊢ (𝑥 = ( < ∩ (ℝ × ℝ)) → (𝑥 We ∪ 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We ∪ 𝒫 ℝ))
11	9, 10	spcev 3527	. . . 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 9506	. . 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 12414	. . . . . . 7 ⊢ ℚ ∈ V
16	15	inex1 5191	. . . . . 6 ⊢ (ℚ ∩ (-1[,]1)) ∈ V
17		nnrecq 12425	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℚ)
18		nnrecre 11729	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ)
19		neg1rr 11802	. . . . . . . . . . 11 ⊢ -1 ∈ ℝ
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → -1 ∈ ℝ)
21		0re 10694	. . . . . . . . . . 11 ⊢ 0 ∈ ℝ
22	21	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ∈ ℝ)
23		neg1lt0 11804	. . . . . . . . . . . 12 ⊢ -1 < 0
24	19, 21, 23	ltleii 10814	. . . . . . . . . . 11 ⊢ -1 ≤ 0
25	24	a1i 11	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → -1 ≤ 0)
26		nnrp 12454	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
27	26	rpreccld 12495	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ+)
28	27	rpge0d 12489	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → 0 ≤ (1 / 𝑥))
29	20, 22, 18, 25, 28	letrd 10848	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → -1 ≤ (1 / 𝑥))
30		nnge1 11715	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 1 ≤ 𝑥)
31		nnre 11694	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
32		nngt0 11718	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 0 < 𝑥)
33		1re 10692	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ
34		0lt1 11213	. . . . . . . . . . . . 13 ⊢ 0 < 1
35		lerec 11574	. . . . . . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
38	30, 37	mpbid 235	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ≤ (1 / 1))
39		1div1e1 11381	. . . . . . . . . 10 ⊢ (1 / 1) = 1
40	38, 39	breqtrdi 5077	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ≤ 1)
41	19, 33	elicc2i 12858	. . . . . . . . 9 ⊢ ((1 / 𝑥) ∈ (-1[,]1) ↔ ((1 / 𝑥) ∈ ℝ ∧ -1 ≤ (1 / 𝑥) ∧ (1 / 𝑥) ≤ 1))
42	18, 29, 40, 41	syl3anbrc 1340	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (-1[,]1))
43	17, 42	elind 4101	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (ℚ ∩ (-1[,]1)))
44		oveq2 7164	. . . . . . . . 9 ⊢ ((1 / 𝑥) = (1 / 𝑦) → (1 / (1 / 𝑥)) = (1 / (1 / 𝑦)))
45		nncn 11695	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
46		nnne0 11721	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → 𝑥 ≠ 0)
47	45, 46	recrecd 11464	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ → (1 / (1 / 𝑥)) = 𝑥)
48		nncn 11695	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
49		nnne0 11721	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0)
50	48, 49	recrecd 11464	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (1 / (1 / 𝑦)) = 𝑦)
51	47, 50	eqeqan12d 2775	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / (1 / 𝑥)) = (1 / (1 / 𝑦)) ↔ 𝑥 = 𝑦))
52	44, 51	syl5ib 247	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) → 𝑥 = 𝑦))
53		oveq2 7164	. . . . . . . 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 8583	. . . . . 6 ⊢ ((ℚ ∩ (-1[,]1)) ∈ V → ℕ ≼ (ℚ ∩ (-1[,]1)))
56	16, 55	ax-mp 5	. . . . 5 ⊢ ℕ ≼ (ℚ ∩ (-1[,]1))
57		inss1 4135	. . . . . . 7 ⊢ (ℚ ∩ (-1[,]1)) ⊆ ℚ
58		ssdomg 8586	. . . . . . 7 ⊢ (ℚ ∈ V → ((ℚ ∩ (-1[,]1)) ⊆ ℚ → (ℚ ∩ (-1[,]1)) ≼ ℚ))
59	15, 57, 58	mp2 9	. . . . . 6 ⊢ (ℚ ∩ (-1[,]1)) ≼ ℚ
60		qnnen 15627	. . . . . 6 ⊢ ℚ ≈ ℕ
61		domentr 8599	. . . . . 6 ⊢ (((ℚ ∩ (-1[,]1)) ≼ ℚ ∧ ℚ ≈ ℕ) → (ℚ ∩ (-1[,]1)) ≼ ℕ)
62	59, 60, 61	mp2an 691	. . . . 5 ⊢ (ℚ ∩ (-1[,]1)) ≼ ℕ
63		sbth 8672	. . . . 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 8549	. . . 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 2834	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ (0[,]1) ↔ 𝑥 ∈ (0[,]1)))
68		eleq1w 2834	. . . . . . . . . . . . 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 7165	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑎 − 𝑏) = (𝑥 − 𝑦))
71	70	eleq1d 2836	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑎 − 𝑏) ∈ ℚ ↔ (𝑥 − 𝑦) ∈ ℚ))
72	69, 71	anbi12d 633	. . . . . . . . . . 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 5108	. . . . . . . . . 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 2758	. . . . . . . . . 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 6676	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑐) ∈ V
76		eqid 2758	. . . . . . . . . . . 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 6479	. . . . . . . . . . 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 3013	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
80		fveq2 6663	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → (𝑓‘𝑧) = (𝑓‘𝑤))
81		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
82	80, 81	eleq12d 2846	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → ((𝑓‘𝑧) ∈ 𝑧 ↔ (𝑓‘𝑤) ∈ 𝑤))
83	79, 82	imbi12d 348	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤)))
84	83	cbvralvw 3361	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓‘𝑤) ∈ 𝑤))
85	73	vitalilem1 24321	. . . . . . . . . . . . . . . . . 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 8374	. . . . . . . . . . . . . . . 16 ⊢ (⊤ → ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1))
88	87	mptru 1545	. . . . . . . . . . . . . . 15 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1)
89		unitssre 12944	. . . . . . . . . . . . . . . 16 ⊢ (0[,]1) ⊆ ℝ
90	89	sspwi 4511	. . . . . . . . . . . . . . 15 ⊢ 𝒫 (0[,]1) ⊆ 𝒫 ℝ
91	88, 90	sstri 3903	. . . . . . . . . . . . . 14 ⊢ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎 − 𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ
92		ssralv 3960	. . . . . . . . . . . . . 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 6663	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑤 → (𝑓‘𝑐) = (𝑓‘𝑤))
96		fvex 6676	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑤) ∈ V
97	95, 76, 96	fvmpt 6764	. . . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . . 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 344	. . . . . . . . . . . . 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 3096	. . . . . . . . . . . 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 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 7163	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑠 → (𝑡 − (𝑔‘𝑚)) = (𝑠 − (𝑔‘𝑚)))
105	104	eleq1d 2836	. . . . . . . . . . . . 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 3404	. . . . . . . . . . . 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 6663	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → (𝑔‘𝑚) = (𝑔‘𝑛))
108	107	oveq2d 7172	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑠 − (𝑔‘𝑚)) = (𝑠 − (𝑔‘𝑛)))
109	108	eleq1d 2836	. . . . . . . . . . . . 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 3392	. . . . . . . . . . . 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	syl5eq 2805	. . . . . . . . . . 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 5139	. . . . . . . . . 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 24325	. . . . . . . . 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 460	. . . . . . 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 3870	. . . . . . 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 24244	. . . . . . . . . 10 ⊢ (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
120		velpw 4502	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
121	119, 120	sylibr 237	. . . . . . . . 9 ⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
122	121	ssriv 3898	. . . . . . . 8 ⊢ dom vol ⊆ 𝒫 ℝ
123		ssnelpss 4019	. . . . . . . 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 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 416	. . . 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 1934	. . 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 1936	1 ⊢ ( < We ℝ → dom vol ⊊ 𝒫 ℝ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ⊤wtru 1539  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  {crab 3074  Vcvv 3409   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860   ⊊ wpss 3861  ∅c0 4227  𝒫 cpw 4497  ∪ cuni 4801   class class class wbr 5036  {copab 5098   ↦ cmpt 5116   We wwe 5486   × cxp 5526  dom cdm 5528  ran crn 5529   Fn wfn 6335  –1-1-onto→wf1o 6339  ‘cfv 6340  (class class class)co 7156   Er wer 8302   / cqs 8304   ≈ cen 8537   ≼ cdom 8538  ℝcr 10587  0cc0 10588  1c1 10589   < clt 10726   ≤ cle 10727   − cmin 10921  -cneg 10922   / cdiv 11348  ℕcn 11687  ℚcq 12401  [,]cicc 12795  volcvol 24176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-inf2 9150  ax-cc 9908  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665  ax-pre-sup 10666

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-disj 5002  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-2o 8119  df-oadd 8122  df-omul 8123  df-er 8305  df-ec 8307  df-qs 8311  df-map 8424  df-pm 8425  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-fi 8921  df-sup 8952  df-inf 8953  df-oi 9020  df-dju 9376  df-card 9414  df-acn 9417  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-div 11349  df-nn 11688  df-2 11750  df-3 11751  df-n0 11948  df-z 12034  df-uz 12296  df-q 12402  df-rp 12444  df-xneg 12561  df-xadd 12562  df-xmul 12563  df-ioo 12796  df-ico 12798  df-icc 12799  df-fz 12953  df-fzo 13096  df-fl 13224  df-seq 13432  df-exp 13493  df-hash 13754  df-cj 14519  df-re 14520  df-im 14521  df-sqrt 14655  df-abs 14656  df-clim 14906  df-rlim 14907  df-sum 15104  df-rest 16767  df-topgen 16788  df-psmet 20171  df-xmet 20172  df-met 20173  df-bl 20174  df-mopn 20175  df-top 21607  df-topon 21624  df-bases 21659  df-cmp 22100  df-ovol 24177  df-vol 24178

This theorem is referenced by:  vitali2  43734