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Theorem vitali 24148
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.
StepHypRef Expression
1 reex 10622 . . . 4 ℝ ∈ V
21pwex 5278 . . 3 𝒫 ℝ ∈ V
3 weinxp 5635 . . . . 5 ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
4 unipw 5339 . . . . . 6 𝒫 ℝ = ℝ
5 weeq2 5543 . . . . . 6 ( 𝒫 ℝ = ℝ → (( < ∩ (ℝ × ℝ)) We 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ))
64, 5ax-mp 5 . . . . 5 (( < ∩ (ℝ × ℝ)) We 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
73, 6bitr4i 279 . . . 4 ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We 𝒫 ℝ)
81, 1xpex 7469 . . . . . 6 (ℝ × ℝ) ∈ V
98inex2 5219 . . . . 5 ( < ∩ (ℝ × ℝ)) ∈ V
10 weeq1 5542 . . . . 5 (𝑥 = ( < ∩ (ℝ × ℝ)) → (𝑥 We 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We 𝒫 ℝ))
119, 10spcev 3611 . . . 4 (( < ∩ (ℝ × ℝ)) We 𝒫 ℝ → ∃𝑥 𝑥 We 𝒫 ℝ)
127, 11sylbi 218 . . 3 ( < We ℝ → ∃𝑥 𝑥 We 𝒫 ℝ)
13 dfac8c 9453 . . 3 (𝒫 ℝ ∈ V → (∃𝑥 𝑥 We 𝒫 ℝ → ∃𝑓𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
142, 12, 13mpsyl 68 . 2 ( < We ℝ → ∃𝑓𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
15 qex 12355 . . . . . . 7 ℚ ∈ V
1615inex1 5218 . . . . . 6 (ℚ ∩ (-1[,]1)) ∈ V
17 nnrecq 12366 . . . . . . . 8 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℚ)
18 nnrecre 11673 . . . . . . . . 9 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ)
19 neg1rr 11746 . . . . . . . . . . 11 -1 ∈ ℝ
2019a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → -1 ∈ ℝ)
21 0re 10637 . . . . . . . . . . 11 0 ∈ ℝ
2221a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → 0 ∈ ℝ)
23 neg1lt0 11748 . . . . . . . . . . . 12 -1 < 0
2419, 21, 23ltleii 10757 . . . . . . . . . . 11 -1 ≤ 0
2524a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → -1 ≤ 0)
26 nnrp 12395 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
2726rpreccld 12436 . . . . . . . . . . 11 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ+)
2827rpge0d 12430 . . . . . . . . . 10 (𝑥 ∈ ℕ → 0 ≤ (1 / 𝑥))
2920, 22, 18, 25, 28letrd 10791 . . . . . . . . 9 (𝑥 ∈ ℕ → -1 ≤ (1 / 𝑥))
30 nnge1 11659 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 1 ≤ 𝑥)
31 nnre 11639 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
32 nngt0 11662 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 0 < 𝑥)
33 1re 10635 . . . . . . . . . . . . 13 1 ∈ ℝ
34 0lt1 11156 . . . . . . . . . . . . 13 0 < 1
35 lerec 11517 . . . . . . . . . . . . 13 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3633, 34, 35mpanl12 698 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3731, 32, 36syl2anc 584 . . . . . . . . . . 11 (𝑥 ∈ ℕ → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3830, 37mpbid 233 . . . . . . . . . 10 (𝑥 ∈ ℕ → (1 / 𝑥) ≤ (1 / 1))
39 1div1e1 11324 . . . . . . . . . 10 (1 / 1) = 1
4038, 39breqtrdi 5104 . . . . . . . . 9 (𝑥 ∈ ℕ → (1 / 𝑥) ≤ 1)
4119, 33elicc2i 12797 . . . . . . . . 9 ((1 / 𝑥) ∈ (-1[,]1) ↔ ((1 / 𝑥) ∈ ℝ ∧ -1 ≤ (1 / 𝑥) ∧ (1 / 𝑥) ≤ 1))
4218, 29, 40, 41syl3anbrc 1337 . . . . . . . 8 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (-1[,]1))
4317, 42elind 4175 . . . . . . 7 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (ℚ ∩ (-1[,]1)))
44 oveq2 7158 . . . . . . . . 9 ((1 / 𝑥) = (1 / 𝑦) → (1 / (1 / 𝑥)) = (1 / (1 / 𝑦)))
45 nncn 11640 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
46 nnne0 11665 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 𝑥 ≠ 0)
4745, 46recrecd 11407 . . . . . . . . . 10 (𝑥 ∈ ℕ → (1 / (1 / 𝑥)) = 𝑥)
48 nncn 11640 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
49 nnne0 11665 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
5048, 49recrecd 11407 . . . . . . . . . 10 (𝑦 ∈ ℕ → (1 / (1 / 𝑦)) = 𝑦)
5147, 50eqeqan12d 2843 . . . . . . . . 9 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / (1 / 𝑥)) = (1 / (1 / 𝑦)) ↔ 𝑥 = 𝑦))
5244, 51syl5ib 245 . . . . . . . 8 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) → 𝑥 = 𝑦))
53 oveq2 7158 . . . . . . . 8 (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
5452, 53impbid1 226 . . . . . . 7 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) ↔ 𝑥 = 𝑦))
5543, 54dom2 8546 . . . . . 6 ((ℚ ∩ (-1[,]1)) ∈ V → ℕ ≼ (ℚ ∩ (-1[,]1)))
5616, 55ax-mp 5 . . . . 5 ℕ ≼ (ℚ ∩ (-1[,]1))
57 inss1 4209 . . . . . . 7 (ℚ ∩ (-1[,]1)) ⊆ ℚ
58 ssdomg 8549 . . . . . . 7 (ℚ ∈ V → ((ℚ ∩ (-1[,]1)) ⊆ ℚ → (ℚ ∩ (-1[,]1)) ≼ ℚ))
5915, 57, 58mp2 9 . . . . . 6 (ℚ ∩ (-1[,]1)) ≼ ℚ
60 qnnen 15561 . . . . . 6 ℚ ≈ ℕ
61 domentr 8562 . . . . . 6 (((ℚ ∩ (-1[,]1)) ≼ ℚ ∧ ℚ ≈ ℕ) → (ℚ ∩ (-1[,]1)) ≼ ℕ)
6259, 60, 61mp2an 688 . . . . 5 (ℚ ∩ (-1[,]1)) ≼ ℕ
63 sbth 8631 . . . . 5 ((ℕ ≼ (ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ≼ ℕ) → ℕ ≈ (ℚ ∩ (-1[,]1)))
6456, 62, 63mp2an 688 . . . 4 ℕ ≈ (ℚ ∩ (-1[,]1))
65 bren 8512 . . . 4 (ℕ ≈ (ℚ ∩ (-1[,]1)) ↔ ∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
6664, 65mpbi 231 . . 3 𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))
67 eleq1w 2900 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑎 ∈ (0[,]1) ↔ 𝑥 ∈ (0[,]1)))
68 eleq1w 2900 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝑏 ∈ (0[,]1) ↔ 𝑦 ∈ (0[,]1)))
6967, 68bi2anan9 635 . . . . . . . . . . . 12 ((𝑎 = 𝑥𝑏 = 𝑦) → ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))))
70 oveq12 7159 . . . . . . . . . . . . 13 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝑎𝑏) = (𝑥𝑦))
7170eleq1d 2902 . . . . . . . . . . . 12 ((𝑎 = 𝑥𝑏 = 𝑦) → ((𝑎𝑏) ∈ ℚ ↔ (𝑥𝑦) ∈ ℚ))
7269, 71anbi12d 630 . . . . . . . . . . 11 ((𝑎 = 𝑥𝑏 = 𝑦) → (((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ) ↔ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)))
7372cbvopabv 5135 . . . . . . . . . 10 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
74 eqid 2826 . . . . . . . . . 10 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) = ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)})
75 fvex 6682 . . . . . . . . . . . 12 (𝑓𝑐) ∈ V
76 eqid 2826 . . . . . . . . . . . 12 (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) = (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))
7775, 76fnmpti 6490 . . . . . . . . . . 11 (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) Fn ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)})
7877a1i 11 . . . . . . . . . 10 ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol))) → (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) Fn ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}))
79 neeq1 3083 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
80 fveq2 6669 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑓𝑧) = (𝑓𝑤))
81 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤𝑧 = 𝑤)
8280, 81eleq12d 2912 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑤) ∈ 𝑤))
8379, 82imbi12d 346 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)))
8483cbvralvw 3455 . . . . . . . . . . . . 13 (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
8573vitalilem1 24143 . . . . . . . . . . . . . . . . . 18 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)} Er (0[,]1)
8685a1i 11 . . . . . . . . . . . . . . . . 17 (⊤ → {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)} Er (0[,]1))
8786qsss 8353 . . . . . . . . . . . . . . . 16 (⊤ → ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1))
8887mptru 1537 . . . . . . . . . . . . . . 15 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1)
89 unitssre 12880 . . . . . . . . . . . . . . . 16 (0[,]1) ⊆ ℝ
90 sspwb 5338 . . . . . . . . . . . . . . . 16 ((0[,]1) ⊆ ℝ ↔ 𝒫 (0[,]1) ⊆ 𝒫 ℝ)
9189, 90mpbi 231 . . . . . . . . . . . . . . 15 𝒫 (0[,]1) ⊆ 𝒫 ℝ
9288, 91sstri 3980 . . . . . . . . . . . . . 14 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ
93 ssralv 4037 . . . . . . . . . . . . . 14 (((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ → (∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)))
9492, 93ax-mp 5 . . . . . . . . . . . . 13 (∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
9584, 94sylbi 218 . . . . . . . . . . . 12 (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
96 fveq2 6669 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑤 → (𝑓𝑐) = (𝑓𝑤))
97 fvex 6682 . . . . . . . . . . . . . . . 16 (𝑓𝑤) ∈ V
9896, 76, 97fvmpt 6767 . . . . . . . . . . . . . . 15 (𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) → ((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))‘𝑤) = (𝑓𝑤))
9998eleq1d 2902 . . . . . . . . . . . . . 14 (𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) → (((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))‘𝑤) ∈ 𝑤 ↔ (𝑓𝑤) ∈ 𝑤))
10099imbi2d 342 . . . . . . . . . . . . 13 (𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) → ((𝑤 ≠ ∅ → ((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))‘𝑤) ∈ 𝑤) ↔ (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)))
101100ralbiia 3169 . . . . . . . . . . . 12 (∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → ((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))‘𝑤) ∈ 𝑤) ↔ ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
10295, 101sylibr 235 . . . . . . . . . . 11 (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → ((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))‘𝑤) ∈ 𝑤))
103102ad2antlr 723 . . . . . . . . . 10 ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol))) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → ((𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))‘𝑤) ∈ 𝑤))
104 simprl 767 . . . . . . . . . 10 ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol))) → 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
105 oveq1 7157 . . . . . . . . . . . . . 14 (𝑡 = 𝑠 → (𝑡 − (𝑔𝑚)) = (𝑠 − (𝑔𝑚)))
106105eleq1d 2902 . . . . . . . . . . . . 13 (𝑡 = 𝑠 → ((𝑡 − (𝑔𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ↔ (𝑠 − (𝑔𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))))
107106cbvrabv 3497 . . . . . . . . . . . 12 {𝑡 ∈ ℝ ∣ (𝑡 − (𝑔𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝑔𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))}
108 fveq2 6669 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝑔𝑚) = (𝑔𝑛))
109108oveq2d 7166 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝑠 − (𝑔𝑚)) = (𝑠 − (𝑔𝑛)))
110109eleq1d 2902 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((𝑠 − (𝑔𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ↔ (𝑠 − (𝑔𝑛)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))))
111110rabbidv 3486 . . . . . . . . . . . 12 (𝑚 = 𝑛 → {𝑠 ∈ ℝ ∣ (𝑠 − (𝑔𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝑔𝑛)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))})
112107, 111syl5eq 2873 . . . . . . . . . . 11 (𝑚 = 𝑛 → {𝑡 ∈ ℝ ∣ (𝑡 − (𝑔𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))} = {𝑠 ∈ ℝ ∣ (𝑠 − (𝑔𝑛)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))})
113112cbvmptv 5166 . . . . . . . . . 10 (𝑚 ∈ ℕ ↦ {𝑡 ∈ ℝ ∣ (𝑡 − (𝑔𝑚)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))}) = (𝑛 ∈ ℕ ↦ {𝑠 ∈ ℝ ∣ (𝑠 − (𝑔𝑛)) ∈ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐))})
114 simprr 769 . . . . . . . . . 10 ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol))) → ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol))
11573, 74, 78, 103, 104, 113, 114vitalilem5 24147 . . . . . . . . 9 ¬ (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol)))
116115pm2.21i 119 . . . . . . . 8 ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ∧ (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol))) → ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol))
117116expr 457 . . . . . . 7 ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ∧ 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) → (¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol) → ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol)))
118117pm2.18d 127 . . . . . 6 ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ∧ 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) → ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol))
119 eldif 3950 . . . . . . 7 (ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol) ↔ (ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ 𝒫 ℝ ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ dom vol))
120 mblss 24066 . . . . . . . . . 10 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
121 velpw 4550 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
122120, 121sylibr 235 . . . . . . . . 9 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
123122ssriv 3975 . . . . . . . 8 dom vol ⊆ 𝒫 ℝ
124 ssnelpss 4092 . . . . . . . 8 (dom vol ⊆ 𝒫 ℝ → ((ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ 𝒫 ℝ ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ dom vol) → dom vol ⊊ 𝒫 ℝ))
125123, 124ax-mp 5 . . . . . . 7 ((ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ 𝒫 ℝ ∧ ¬ ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ dom vol) → dom vol ⊊ 𝒫 ℝ)
126119, 125sylbi 218 . . . . . 6 (ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol) → dom vol ⊊ 𝒫 ℝ)
127118, 126syl 17 . . . . 5 ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ∧ 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) → dom vol ⊊ 𝒫 ℝ)
128127ex 413 . . . 4 (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → dom vol ⊊ 𝒫 ℝ))
129128exlimdv 1927 . . 3 (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → dom vol ⊊ 𝒫 ℝ))
13066, 129mpi 20 . 2 (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → dom vol ⊊ 𝒫 ℝ)
13114, 130exlimddv 1929 1 ( < We ℝ → dom vol ⊊ 𝒫 ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wtru 1531  wex 1773  wcel 2107  wne 3021  wral 3143  {crab 3147  Vcvv 3500  cdif 3937  cin 3939  wss 3940  wpss 3941  c0 4295  𝒫 cpw 4542   cuni 4837   class class class wbr 5063  {copab 5125  cmpt 5143   We wwe 5512   × cxp 5552  dom cdm 5554  ran crn 5555   Fn wfn 6349  1-1-ontowf1o 6353  cfv 6354  (class class class)co 7150   Er wer 8281   / cqs 8283  cen 8500  cdom 8501  cr 10530  0cc0 10531  1c1 10532   < clt 10669  cle 10670  cmin 10864  -cneg 10865   / cdiv 11291  cn 11632  cq 12342  [,]cicc 12736  volcvol 23998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-inf2 9098  ax-cc 9851  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-disj 5029  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-omul 8103  df-er 8284  df-ec 8286  df-qs 8290  df-map 8403  df-pm 8404  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-sup 8900  df-inf 8901  df-oi 8968  df-dju 9324  df-card 9362  df-acn 9365  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-q 12343  df-rp 12385  df-xneg 12502  df-xadd 12503  df-xmul 12504  df-ioo 12737  df-ico 12739  df-icc 12740  df-fz 12888  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13425  df-hash 13686  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-clim 14840  df-rlim 14841  df-sum 15038  df-rest 16691  df-topgen 16712  df-psmet 20472  df-xmet 20473  df-met 20474  df-bl 20475  df-mopn 20476  df-top 21437  df-topon 21454  df-bases 21489  df-cmp 21930  df-ovol 23999  df-vol 24000
This theorem is referenced by:  vitali2  42861
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