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Theorem vitali 23600
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 10228 . . . 4 ℝ ∈ V
21pwex 4976 . . 3 𝒫 ℝ ∈ V
3 weinxp 5326 . . . . 5 ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
4 unipw 5046 . . . . . 6 𝒫 ℝ = ℝ
5 weeq2 5238 . . . . . 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 267 . . . 4 ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We 𝒫 ℝ)
81, 1xpex 7108 . . . . . 6 (ℝ × ℝ) ∈ V
98inex2 4931 . . . . 5 ( < ∩ (ℝ × ℝ)) ∈ V
10 weeq1 5237 . . . . 5 (𝑥 = ( < ∩ (ℝ × ℝ)) → (𝑥 We 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We 𝒫 ℝ))
119, 10spcev 3449 . . . 4 (( < ∩ (ℝ × ℝ)) We 𝒫 ℝ → ∃𝑥 𝑥 We 𝒫 ℝ)
127, 11sylbi 207 . . 3 ( < We ℝ → ∃𝑥 𝑥 We 𝒫 ℝ)
13 dfac8c 9055 . . 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 12002 . . . . . . 7 ℚ ∈ V
1615inex1 4930 . . . . . 6 (ℚ ∩ (-1[,]1)) ∈ V
17 nnrecq 12013 . . . . . . . 8 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℚ)
18 nnrecre 11258 . . . . . . . . 9 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ)
19 neg1rr 11326 . . . . . . . . . . 11 -1 ∈ ℝ
2019a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → -1 ∈ ℝ)
21 0re 10241 . . . . . . . . . . 11 0 ∈ ℝ
2221a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → 0 ∈ ℝ)
23 neg1lt0 11328 . . . . . . . . . . . 12 -1 < 0
2419, 21, 23ltleii 10361 . . . . . . . . . . 11 -1 ≤ 0
2524a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → -1 ≤ 0)
26 nnrp 12044 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
2726rpreccld 12084 . . . . . . . . . . 11 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ+)
2827rpge0d 12078 . . . . . . . . . 10 (𝑥 ∈ ℕ → 0 ≤ (1 / 𝑥))
2920, 22, 18, 25, 28letrd 10395 . . . . . . . . 9 (𝑥 ∈ ℕ → -1 ≤ (1 / 𝑥))
30 nnge1 11247 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 1 ≤ 𝑥)
31 nnre 11228 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
32 nngt0 11250 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 0 < 𝑥)
33 1re 10240 . . . . . . . . . . . . 13 1 ∈ ℝ
34 0lt1 10751 . . . . . . . . . . . . 13 0 < 1
35 lerec 11107 . . . . . . . . . . . . 13 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3633, 34, 35mpanl12 674 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3731, 32, 36syl2anc 565 . . . . . . . . . . 11 (𝑥 ∈ ℕ → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3830, 37mpbid 222 . . . . . . . . . 10 (𝑥 ∈ ℕ → (1 / 𝑥) ≤ (1 / 1))
39 1div1e1 10918 . . . . . . . . . 10 (1 / 1) = 1
4038, 39syl6breq 4825 . . . . . . . . 9 (𝑥 ∈ ℕ → (1 / 𝑥) ≤ 1)
4119, 33elicc2i 12443 . . . . . . . . 9 ((1 / 𝑥) ∈ (-1[,]1) ↔ ((1 / 𝑥) ∈ ℝ ∧ -1 ≤ (1 / 𝑥) ∧ (1 / 𝑥) ≤ 1))
4218, 29, 40, 41syl3anbrc 1427 . . . . . . . 8 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (-1[,]1))
4317, 42elind 3947 . . . . . . 7 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (ℚ ∩ (-1[,]1)))
44 oveq2 6800 . . . . . . . . 9 ((1 / 𝑥) = (1 / 𝑦) → (1 / (1 / 𝑥)) = (1 / (1 / 𝑦)))
45 nncn 11229 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
46 nnne0 11254 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 𝑥 ≠ 0)
4745, 46recrecd 10999 . . . . . . . . . 10 (𝑥 ∈ ℕ → (1 / (1 / 𝑥)) = 𝑥)
48 nncn 11229 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
49 nnne0 11254 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
5048, 49recrecd 10999 . . . . . . . . . 10 (𝑦 ∈ ℕ → (1 / (1 / 𝑦)) = 𝑦)
5147, 50eqeqan12d 2786 . . . . . . . . 9 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / (1 / 𝑥)) = (1 / (1 / 𝑦)) ↔ 𝑥 = 𝑦))
5244, 51syl5ib 234 . . . . . . . 8 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) → 𝑥 = 𝑦))
53 oveq2 6800 . . . . . . . 8 (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
5452, 53impbid1 215 . . . . . . 7 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) ↔ 𝑥 = 𝑦))
5543, 54dom2 8151 . . . . . 6 ((ℚ ∩ (-1[,]1)) ∈ V → ℕ ≼ (ℚ ∩ (-1[,]1)))
5616, 55ax-mp 5 . . . . 5 ℕ ≼ (ℚ ∩ (-1[,]1))
57 inss1 3979 . . . . . . 7 (ℚ ∩ (-1[,]1)) ⊆ ℚ
58 ssdomg 8154 . . . . . . 7 (ℚ ∈ V → ((ℚ ∩ (-1[,]1)) ⊆ ℚ → (ℚ ∩ (-1[,]1)) ≼ ℚ))
5915, 57, 58mp2 9 . . . . . 6 (ℚ ∩ (-1[,]1)) ≼ ℚ
60 qnnen 15147 . . . . . 6 ℚ ≈ ℕ
61 domentr 8167 . . . . . 6 (((ℚ ∩ (-1[,]1)) ≼ ℚ ∧ ℚ ≈ ℕ) → (ℚ ∩ (-1[,]1)) ≼ ℕ)
6259, 60, 61mp2an 664 . . . . 5 (ℚ ∩ (-1[,]1)) ≼ ℕ
63 sbth 8235 . . . . 5 ((ℕ ≼ (ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ≼ ℕ) → ℕ ≈ (ℚ ∩ (-1[,]1)))
6456, 62, 63mp2an 664 . . . 4 ℕ ≈ (ℚ ∩ (-1[,]1))
65 bren 8117 . . . 4 (ℕ ≈ (ℚ ∩ (-1[,]1)) ↔ ∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
6664, 65mpbi 220 . . 3 𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))
67 eleq1w 2832 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑎 ∈ (0[,]1) ↔ 𝑥 ∈ (0[,]1)))
68 eleq1w 2832 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝑏 ∈ (0[,]1) ↔ 𝑦 ∈ (0[,]1)))
6967, 68bi2anan9 612 . . . . . . . . . . . 12 ((𝑎 = 𝑥𝑏 = 𝑦) → ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))))
70 oveq12 6801 . . . . . . . . . . . . 13 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝑎𝑏) = (𝑥𝑦))
7170eleq1d 2834 . . . . . . . . . . . 12 ((𝑎 = 𝑥𝑏 = 𝑦) → ((𝑎𝑏) ∈ ℚ ↔ (𝑥𝑦) ∈ ℚ))
7269, 71anbi12d 608 . . . . . . . . . . 11 ((𝑎 = 𝑥𝑏 = 𝑦) → (((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ) ↔ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)))
7372cbvopabv 4854 . . . . . . . . . 10 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
74 eqid 2770 . . . . . . . . . 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 6342 . . . . . . . . . . . 12 (𝑓𝑐) ∈ V
76 eqid 2770 . . . . . . . . . . . 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 6162 . . . . . . . . . . 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 3004 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
80 fveq2 6332 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑓𝑧) = (𝑓𝑤))
81 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤𝑧 = 𝑤)
8280, 81eleq12d 2843 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑤) ∈ 𝑤))
8379, 82imbi12d 333 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)))
8483cbvralv 3319 . . . . . . . . . . . . 13 (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
8573vitalilem1 23595 . . . . . . . . . . . . . . . . . 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 7959 . . . . . . . . . . . . . . . 16 (⊤ → ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1))
8887trud 1640 . . . . . . . . . . . . . . 15 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1)
89 unitssre 12525 . . . . . . . . . . . . . . . 16 (0[,]1) ⊆ ℝ
90 sspwb 5045 . . . . . . . . . . . . . . . 16 ((0[,]1) ⊆ ℝ ↔ 𝒫 (0[,]1) ⊆ 𝒫 ℝ)
9189, 90mpbi 220 . . . . . . . . . . . . . . 15 𝒫 (0[,]1) ⊆ 𝒫 ℝ
9288, 91sstri 3759 . . . . . . . . . . . . . 14 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ
93 ssralv 3813 . . . . . . . . . . . . . 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 207 . . . . . . . . . . . 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 6332 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑤 → (𝑓𝑐) = (𝑓𝑤))
97 fvex 6342 . . . . . . . . . . . . . . . 16 (𝑓𝑤) ∈ V
9896, 76, 97fvmpt 6424 . . . . . . . . . . . . . . 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 2834 . . . . . . . . . . . . . 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 329 . . . . . . . . . . . . 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 3127 . . . . . . . . . . . 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 224 . . . . . . . . . . 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 698 . . . . . . . . . 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 746 . . . . . . . . . 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 6799 . . . . . . . . . . . . . 14 (𝑡 = 𝑠 → (𝑡 − (𝑔𝑚)) = (𝑠 − (𝑔𝑚)))
106105eleq1d 2834 . . . . . . . . . . . . 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 3348 . . . . . . . . . . . 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 6332 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝑔𝑚) = (𝑔𝑛))
109108oveq2d 6808 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝑠 − (𝑔𝑚)) = (𝑠 − (𝑔𝑛)))
110109eleq1d 2834 . . . . . . . . . . . . 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 3338 . . . . . . . . . . . 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 2816 . . . . . . . . . . 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 4882 . . . . . . . . . 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 748 . . . . . . . . . 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 23599 . . . . . . . . 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 117 . . . . . . . 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 444 . . . . . . 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 125 . . . . . 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 3731 . . . . . . 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 23518 . . . . . . . . . 10 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
121 selpw 4302 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
122120, 121sylibr 224 . . . . . . . . 9 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
123122ssriv 3754 . . . . . . . 8 dom vol ⊆ 𝒫 ℝ
124 ssnelpss 3866 . . . . . . . 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 207 . . . . . 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 397 . . . 4 (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → dom vol ⊊ 𝒫 ℝ))
129128exlimdv 2012 . . 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 2014 1 ( < We ℝ → dom vol ⊊ 𝒫 ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1630  wtru 1631  wex 1851  wcel 2144  wne 2942  wral 3060  {crab 3064  Vcvv 3349  cdif 3718  cin 3720  wss 3721  wpss 3722  c0 4061  𝒫 cpw 4295   cuni 4572   class class class wbr 4784  {copab 4844  cmpt 4861   We wwe 5207   × cxp 5247  dom cdm 5249  ran crn 5250   Fn wfn 6026  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792   Er wer 7892   / cqs 7894  cen 8105  cdom 8106  cr 10136  0cc0 10137  1c1 10138   < clt 10275  cle 10276  cmin 10467  -cneg 10468   / cdiv 10885  cn 11221  cq 11990  [,]cicc 12382  volcvol 23450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cc 9458  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-disj 4753  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-omul 7717  df-er 7895  df-ec 7897  df-qs 7901  df-map 8010  df-pm 8011  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fi 8472  df-sup 8503  df-inf 8504  df-oi 8570  df-card 8964  df-acn 8967  df-cda 9191  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-z 11579  df-uz 11888  df-q 11991  df-rp 12035  df-xneg 12150  df-xadd 12151  df-xmul 12152  df-ioo 12383  df-ico 12385  df-icc 12386  df-fz 12533  df-fzo 12673  df-fl 12800  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-rlim 14427  df-sum 14624  df-rest 16290  df-topgen 16311  df-psmet 19952  df-xmet 19953  df-met 19954  df-bl 19955  df-mopn 19956  df-top 20918  df-topon 20935  df-bases 20970  df-cmp 21410  df-ovol 23451  df-vol 23452
This theorem is referenced by:  vitali2  41422
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