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Theorem vitali 25130
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 11201 . . . 4 ℝ ∈ V
21pwex 5379 . . 3 𝒫 ℝ ∈ V
3 weinxp 5761 . . . . 5 ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
4 unipw 5451 . . . . . 6 𝒫 ℝ = ℝ
5 weeq2 5666 . . . . . 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 278 . . . 4 ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We 𝒫 ℝ)
81, 1xpex 7740 . . . . . 6 (ℝ × ℝ) ∈ V
98inex2 5319 . . . . 5 ( < ∩ (ℝ × ℝ)) ∈ V
10 weeq1 5665 . . . . 5 (𝑥 = ( < ∩ (ℝ × ℝ)) → (𝑥 We 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We 𝒫 ℝ))
119, 10spcev 3597 . . . 4 (( < ∩ (ℝ × ℝ)) We 𝒫 ℝ → ∃𝑥 𝑥 We 𝒫 ℝ)
127, 11sylbi 216 . . 3 ( < We ℝ → ∃𝑥 𝑥 We 𝒫 ℝ)
13 dfac8c 10028 . . 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 12945 . . . . . . 7 ℚ ∈ V
1615inex1 5318 . . . . . 6 (ℚ ∩ (-1[,]1)) ∈ V
17 nnrecq 12956 . . . . . . . 8 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℚ)
18 nnrecre 12254 . . . . . . . . 9 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ)
19 neg1rr 12327 . . . . . . . . . . 11 -1 ∈ ℝ
2019a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → -1 ∈ ℝ)
21 0re 11216 . . . . . . . . . . 11 0 ∈ ℝ
2221a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → 0 ∈ ℝ)
23 neg1lt0 12329 . . . . . . . . . . . 12 -1 < 0
2419, 21, 23ltleii 11337 . . . . . . . . . . 11 -1 ≤ 0
2524a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → -1 ≤ 0)
26 nnrp 12985 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
2726rpreccld 13026 . . . . . . . . . . 11 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ+)
2827rpge0d 13020 . . . . . . . . . 10 (𝑥 ∈ ℕ → 0 ≤ (1 / 𝑥))
2920, 22, 18, 25, 28letrd 11371 . . . . . . . . 9 (𝑥 ∈ ℕ → -1 ≤ (1 / 𝑥))
30 nnge1 12240 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 1 ≤ 𝑥)
31 nnre 12219 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
32 nngt0 12243 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 0 < 𝑥)
33 1re 11214 . . . . . . . . . . . . 13 1 ∈ ℝ
34 0lt1 11736 . . . . . . . . . . . . 13 0 < 1
35 lerec 12097 . . . . . . . . . . . . 13 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3633, 34, 35mpanl12 701 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3731, 32, 36syl2anc 585 . . . . . . . . . . 11 (𝑥 ∈ ℕ → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3830, 37mpbid 231 . . . . . . . . . 10 (𝑥 ∈ ℕ → (1 / 𝑥) ≤ (1 / 1))
39 1div1e1 11904 . . . . . . . . . 10 (1 / 1) = 1
4038, 39breqtrdi 5190 . . . . . . . . 9 (𝑥 ∈ ℕ → (1 / 𝑥) ≤ 1)
4119, 33elicc2i 13390 . . . . . . . . 9 ((1 / 𝑥) ∈ (-1[,]1) ↔ ((1 / 𝑥) ∈ ℝ ∧ -1 ≤ (1 / 𝑥) ∧ (1 / 𝑥) ≤ 1))
4218, 29, 40, 41syl3anbrc 1344 . . . . . . . 8 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (-1[,]1))
4317, 42elind 4195 . . . . . . 7 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (ℚ ∩ (-1[,]1)))
44 oveq2 7417 . . . . . . . . 9 ((1 / 𝑥) = (1 / 𝑦) → (1 / (1 / 𝑥)) = (1 / (1 / 𝑦)))
45 nncn 12220 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
46 nnne0 12246 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 𝑥 ≠ 0)
4745, 46recrecd 11987 . . . . . . . . . 10 (𝑥 ∈ ℕ → (1 / (1 / 𝑥)) = 𝑥)
48 nncn 12220 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
49 nnne0 12246 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
5048, 49recrecd 11987 . . . . . . . . . 10 (𝑦 ∈ ℕ → (1 / (1 / 𝑦)) = 𝑦)
5147, 50eqeqan12d 2747 . . . . . . . . 9 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / (1 / 𝑥)) = (1 / (1 / 𝑦)) ↔ 𝑥 = 𝑦))
5244, 51imbitrid 243 . . . . . . . 8 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) → 𝑥 = 𝑦))
53 oveq2 7417 . . . . . . . 8 (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
5452, 53impbid1 224 . . . . . . 7 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) ↔ 𝑥 = 𝑦))
5543, 54dom2 8991 . . . . . 6 ((ℚ ∩ (-1[,]1)) ∈ V → ℕ ≼ (ℚ ∩ (-1[,]1)))
5616, 55ax-mp 5 . . . . 5 ℕ ≼ (ℚ ∩ (-1[,]1))
57 inss1 4229 . . . . . . 7 (ℚ ∩ (-1[,]1)) ⊆ ℚ
58 ssdomg 8996 . . . . . . 7 (ℚ ∈ V → ((ℚ ∩ (-1[,]1)) ⊆ ℚ → (ℚ ∩ (-1[,]1)) ≼ ℚ))
5915, 57, 58mp2 9 . . . . . 6 (ℚ ∩ (-1[,]1)) ≼ ℚ
60 qnnen 16156 . . . . . 6 ℚ ≈ ℕ
61 domentr 9009 . . . . . 6 (((ℚ ∩ (-1[,]1)) ≼ ℚ ∧ ℚ ≈ ℕ) → (ℚ ∩ (-1[,]1)) ≼ ℕ)
6259, 60, 61mp2an 691 . . . . 5 (ℚ ∩ (-1[,]1)) ≼ ℕ
63 sbth 9093 . . . . 5 ((ℕ ≼ (ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ≼ ℕ) → ℕ ≈ (ℚ ∩ (-1[,]1)))
6456, 62, 63mp2an 691 . . . 4 ℕ ≈ (ℚ ∩ (-1[,]1))
65 bren 8949 . . . 4 (ℕ ≈ (ℚ ∩ (-1[,]1)) ↔ ∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
6664, 65mpbi 229 . . 3 𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))
67 eleq1w 2817 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑎 ∈ (0[,]1) ↔ 𝑥 ∈ (0[,]1)))
68 eleq1w 2817 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝑏 ∈ (0[,]1) ↔ 𝑦 ∈ (0[,]1)))
6967, 68bi2anan9 638 . . . . . . . . . . . 12 ((𝑎 = 𝑥𝑏 = 𝑦) → ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))))
70 oveq12 7418 . . . . . . . . . . . . 13 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝑎𝑏) = (𝑥𝑦))
7170eleq1d 2819 . . . . . . . . . . . 12 ((𝑎 = 𝑥𝑏 = 𝑦) → ((𝑎𝑏) ∈ ℚ ↔ (𝑥𝑦) ∈ ℚ))
7269, 71anbi12d 632 . . . . . . . . . . 11 ((𝑎 = 𝑥𝑏 = 𝑦) → (((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ) ↔ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)))
7372cbvopabv 5222 . . . . . . . . . 10 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
74 eqid 2733 . . . . . . . . . 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 6905 . . . . . . . . . . . 12 (𝑓𝑐) ∈ V
76 eqid 2733 . . . . . . . . . . . 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 6694 . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑓𝑧) = (𝑓𝑤))
81 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤𝑧 = 𝑤)
8280, 81eleq12d 2828 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑤) ∈ 𝑤))
8379, 82imbi12d 345 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)))
8483cbvralvw 3235 . . . . . . . . . . . . 13 (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
8573vitalilem1 25125 . . . . . . . . . . . . . . . . . 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 8772 . . . . . . . . . . . . . . . 16 (⊤ → ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1))
8887mptru 1549 . . . . . . . . . . . . . . 15 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1)
89 unitssre 13476 . . . . . . . . . . . . . . . 16 (0[,]1) ⊆ ℝ
9089sspwi 4615 . . . . . . . . . . . . . . 15 𝒫 (0[,]1) ⊆ 𝒫 ℝ
9188, 90sstri 3992 . . . . . . . . . . . . . 14 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ
92 ssralv 4051 . . . . . . . . . . . . . 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)))
9391, 92ax-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))
9484, 93sylbi 216 . . . . . . . . . . . 12 (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∀𝑤 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)})(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
95 fveq2 6892 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑤 → (𝑓𝑐) = (𝑓𝑤))
96 fvex 6905 . . . . . . . . . . . . . . . 16 (𝑓𝑤) ∈ V
9795, 76, 96fvmpt 6999 . . . . . . . . . . . . . . 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))
9897eleq1d 2819 . . . . . . . . . . . . . 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))
9998imbi2d 341 . . . . . . . . . . . . 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)))
10099ralbiia 3092 . . . . . . . . . . . 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))
10194, 100sylibr 233 . . . . . . . . . . 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))
102101ad2antlr 726 . . . . . . . . . 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))
103 simprl 770 . . . . . . . . . 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)))
104 oveq1 7416 . . . . . . . . . . . . . 14 (𝑡 = 𝑠 → (𝑡 − (𝑔𝑚)) = (𝑠 − (𝑔𝑚)))
105104eleq1d 2819 . . . . . . . . . . . . 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))))
106105cbvrabv 3443 . . . . . . . . . . . 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))}
107 fveq2 6892 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝑔𝑚) = (𝑔𝑛))
108107oveq2d 7425 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝑠 − (𝑔𝑚)) = (𝑠 − (𝑔𝑛)))
109108eleq1d 2819 . . . . . . . . . . . . 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))))
110109rabbidv 3441 . . . . . . . . . . . 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))})
111106, 110eqtrid 2785 . . . . . . . . . . 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))})
112111cbvmptv 5262 . . . . . . . . . 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))})
113 simprr 772 . . . . . . . . . 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))
11473, 74, 78, 102, 103, 112, 113vitalilem5 25129 . . . . . . . . 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)))
115114pm2.21i 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))
116115expr 458 . . . . . . 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)))
117116pm2.18d 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))
118 eldif 3959 . . . . . . 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))
119 mblss 25048 . . . . . . . . . 10 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
120 velpw 4608 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
121119, 120sylibr 233 . . . . . . . . 9 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
122121ssriv 3987 . . . . . . . 8 dom vol ⊆ 𝒫 ℝ
123 ssnelpss 4112 . . . . . . . 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))
124122, 123ax-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)
125118, 124sylbi 216 . . . . . 6 (ran (𝑐 ∈ ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ↦ (𝑓𝑐)) ∈ (𝒫 ℝ ∖ dom vol) → dom vol ⊊ 𝒫 ℝ)
126117, 125syl 17 . . . . 5 ((( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ∧ 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))) → dom vol ⊊ 𝒫 ℝ)
127126ex 414 . . . 4 (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → dom vol ⊊ 𝒫 ℝ))
128127exlimdv 1937 . . 3 (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → dom vol ⊊ 𝒫 ℝ))
12966, 128mpi 20 . 2 (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → dom vol ⊊ 𝒫 ℝ)
13014, 129exlimddv 1939 1 ( < We ℝ → dom vol ⊊ 𝒫 ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wtru 1543  wex 1782  wcel 2107  wne 2941  wral 3062  {crab 3433  Vcvv 3475  cdif 3946  cin 3948  wss 3949  wpss 3950  c0 4323  𝒫 cpw 4603   cuni 4909   class class class wbr 5149  {copab 5211  cmpt 5232   We wwe 5631   × cxp 5675  dom cdm 5677  ran crn 5678   Fn wfn 6539  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409   Er wer 8700   / cqs 8702  cen 8936  cdom 8937  cr 11109  0cc0 11110  1c1 11111   < clt 11248  cle 11249  cmin 11444  -cneg 11445   / cdiv 11871  cn 12212  cq 12932  [,]cicc 13327  volcvol 24980
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cc 10430  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-disj 5115  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-oadd 8470  df-omul 8471  df-er 8703  df-ec 8705  df-qs 8709  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-dju 9896  df-card 9934  df-acn 9937  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ioo 13328  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-fl 13757  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-clim 15432  df-rlim 15433  df-sum 15633  df-rest 17368  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-bases 22449  df-cmp 22891  df-ovol 24981  df-vol 24982
This theorem is referenced by:  vitali2  45458
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