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Theorem vitali 25648
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 11246 . . . 4 ℝ ∈ V
21pwex 5380 . . 3 𝒫 ℝ ∈ V
3 weinxp 5770 . . . . 5 ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
4 unipw 5455 . . . . . 6 𝒫 ℝ = ℝ
5 weeq2 5673 . . . . . 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 7773 . . . . . 6 (ℝ × ℝ) ∈ V
98inex2 5318 . . . . 5 ( < ∩ (ℝ × ℝ)) ∈ V
10 weeq1 5672 . . . . 5 (𝑥 = ( < ∩ (ℝ × ℝ)) → (𝑥 We 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We 𝒫 ℝ))
119, 10spcev 3606 . . . 4 (( < ∩ (ℝ × ℝ)) We 𝒫 ℝ → ∃𝑥 𝑥 We 𝒫 ℝ)
127, 11sylbi 217 . . 3 ( < We ℝ → ∃𝑥 𝑥 We 𝒫 ℝ)
13 dfac8c 10073 . . 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 13003 . . . . . . 7 ℚ ∈ V
1615inex1 5317 . . . . . 6 (ℚ ∩ (-1[,]1)) ∈ V
17 nnrecq 13014 . . . . . . . 8 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℚ)
18 nnrecre 12308 . . . . . . . . 9 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ)
19 neg1rr 12381 . . . . . . . . . . 11 -1 ∈ ℝ
2019a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → -1 ∈ ℝ)
21 0re 11263 . . . . . . . . . . 11 0 ∈ ℝ
2221a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → 0 ∈ ℝ)
23 neg1lt0 12383 . . . . . . . . . . . 12 -1 < 0
2419, 21, 23ltleii 11384 . . . . . . . . . . 11 -1 ≤ 0
2524a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → -1 ≤ 0)
26 nnrp 13046 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
2726rpreccld 13087 . . . . . . . . . . 11 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ+)
2827rpge0d 13081 . . . . . . . . . 10 (𝑥 ∈ ℕ → 0 ≤ (1 / 𝑥))
2920, 22, 18, 25, 28letrd 11418 . . . . . . . . 9 (𝑥 ∈ ℕ → -1 ≤ (1 / 𝑥))
30 nnge1 12294 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 1 ≤ 𝑥)
31 nnre 12273 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ)
32 nngt0 12297 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 0 < 𝑥)
33 1re 11261 . . . . . . . . . . . . 13 1 ∈ ℝ
34 0lt1 11785 . . . . . . . . . . . . 13 0 < 1
35 lerec 12151 . . . . . . . . . . . . 13 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3633, 34, 35mpanl12 702 . . . . . . . . . . . 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 232 . . . . . . . . . 10 (𝑥 ∈ ℕ → (1 / 𝑥) ≤ (1 / 1))
39 1div1e1 11958 . . . . . . . . . 10 (1 / 1) = 1
4038, 39breqtrdi 5184 . . . . . . . . 9 (𝑥 ∈ ℕ → (1 / 𝑥) ≤ 1)
4119, 33elicc2i 13453 . . . . . . . . 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 4200 . . . . . . 7 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (ℚ ∩ (-1[,]1)))
44 oveq2 7439 . . . . . . . . 9 ((1 / 𝑥) = (1 / 𝑦) → (1 / (1 / 𝑥)) = (1 / (1 / 𝑦)))
45 nncn 12274 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 𝑥 ∈ ℂ)
46 nnne0 12300 . . . . . . . . . . 11 (𝑥 ∈ ℕ → 𝑥 ≠ 0)
4745, 46recrecd 12040 . . . . . . . . . 10 (𝑥 ∈ ℕ → (1 / (1 / 𝑥)) = 𝑥)
48 nncn 12274 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
49 nnne0 12300 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
5048, 49recrecd 12040 . . . . . . . . . 10 (𝑦 ∈ ℕ → (1 / (1 / 𝑦)) = 𝑦)
5147, 50eqeqan12d 2751 . . . . . . . . 9 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / (1 / 𝑥)) = (1 / (1 / 𝑦)) ↔ 𝑥 = 𝑦))
5244, 51imbitrid 244 . . . . . . . 8 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) → 𝑥 = 𝑦))
53 oveq2 7439 . . . . . . . 8 (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
5452, 53impbid1 225 . . . . . . 7 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) ↔ 𝑥 = 𝑦))
5543, 54dom2 9035 . . . . . 6 ((ℚ ∩ (-1[,]1)) ∈ V → ℕ ≼ (ℚ ∩ (-1[,]1)))
5616, 55ax-mp 5 . . . . 5 ℕ ≼ (ℚ ∩ (-1[,]1))
57 inss1 4237 . . . . . . 7 (ℚ ∩ (-1[,]1)) ⊆ ℚ
58 ssdomg 9040 . . . . . . 7 (ℚ ∈ V → ((ℚ ∩ (-1[,]1)) ⊆ ℚ → (ℚ ∩ (-1[,]1)) ≼ ℚ))
5915, 57, 58mp2 9 . . . . . 6 (ℚ ∩ (-1[,]1)) ≼ ℚ
60 qnnen 16249 . . . . . 6 ℚ ≈ ℕ
61 domentr 9053 . . . . . 6 (((ℚ ∩ (-1[,]1)) ≼ ℚ ∧ ℚ ≈ ℕ) → (ℚ ∩ (-1[,]1)) ≼ ℕ)
6259, 60, 61mp2an 692 . . . . 5 (ℚ ∩ (-1[,]1)) ≼ ℕ
63 sbth 9133 . . . . 5 ((ℕ ≼ (ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ≼ ℕ) → ℕ ≈ (ℚ ∩ (-1[,]1)))
6456, 62, 63mp2an 692 . . . 4 ℕ ≈ (ℚ ∩ (-1[,]1))
65 bren 8995 . . . 4 (ℕ ≈ (ℚ ∩ (-1[,]1)) ↔ ∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
6664, 65mpbi 230 . . 3 𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1))
67 eleq1w 2824 . . . . . . . . . . . . 13 (𝑎 = 𝑥 → (𝑎 ∈ (0[,]1) ↔ 𝑥 ∈ (0[,]1)))
68 eleq1w 2824 . . . . . . . . . . . . 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 7440 . . . . . . . . . . . . 13 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝑎𝑏) = (𝑥𝑦))
7170eleq1d 2826 . . . . . . . . . . . 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 5216 . . . . . . . . . 10 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
74 eqid 2737 . . . . . . . . . 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 6919 . . . . . . . . . . . 12 (𝑓𝑐) ∈ V
76 eqid 2737 . . . . . . . . . . . 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 6711 . . . . . . . . . . 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 3003 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
80 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑓𝑧) = (𝑓𝑤))
81 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤𝑧 = 𝑤)
8280, 81eleq12d 2835 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑤) ∈ 𝑤))
8379, 82imbi12d 344 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)))
8483cbvralvw 3237 . . . . . . . . . . . . 13 (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
8573vitalilem1 25643 . . . . . . . . . . . . . . . . . 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 8818 . . . . . . . . . . . . . . . 16 (⊤ → ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1))
8887mptru 1547 . . . . . . . . . . . . . . 15 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1)
89 unitssre 13539 . . . . . . . . . . . . . . . 16 (0[,]1) ⊆ ℝ
9089sspwi 4612 . . . . . . . . . . . . . . 15 𝒫 (0[,]1) ⊆ 𝒫 ℝ
9188, 90sstri 3993 . . . . . . . . . . . . . 14 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ
92 ssralv 4052 . . . . . . . . . . . . . 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 217 . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑤 → (𝑓𝑐) = (𝑓𝑤))
96 fvex 6919 . . . . . . . . . . . . . . . 16 (𝑓𝑤) ∈ V
9795, 76, 96fvmpt 7016 . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . 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 340 . . . . . . . . . . . . 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 3091 . . . . . . . . . . . 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 234 . . . . . . . . . . 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 727 . . . . . . . . . 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 771 . . . . . . . . . 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 7438 . . . . . . . . . . . . . 14 (𝑡 = 𝑠 → (𝑡 − (𝑔𝑚)) = (𝑠 − (𝑔𝑚)))
105104eleq1d 2826 . . . . . . . . . . . . 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 3447 . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝑔𝑚) = (𝑔𝑛))
108107oveq2d 7447 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝑠 − (𝑔𝑚)) = (𝑠 − (𝑔𝑛)))
109108eleq1d 2826 . . . . . . . . . . . . 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 3444 . . . . . . . . . . . 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 2789 . . . . . . . . . . 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 5255 . . . . . . . . . 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 773 . . . . . . . . . 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 25647 . . . . . . . . 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 456 . . . . . . 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 3961 . . . . . . 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 25566 . . . . . . . . . 10 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
120 velpw 4605 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
121119, 120sylibr 234 . . . . . . . . 9 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
122121ssriv 3987 . . . . . . . 8 dom vol ⊆ 𝒫 ℝ
123 ssnelpss 4114 . . . . . . . 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 217 . . . . . 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 412 . . . 4 (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → dom vol ⊊ 𝒫 ℝ))
128127exlimdv 1933 . . 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 1935 1 ( < We ℝ → dom vol ⊊ 𝒫 ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wtru 1541  wex 1779  wcel 2108  wne 2940  wral 3061  {crab 3436  Vcvv 3480  cdif 3948  cin 3950  wss 3951  wpss 3952  c0 4333  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  {copab 5205  cmpt 5225   We wwe 5636   × cxp 5683  dom cdm 5685  ran crn 5686   Fn wfn 6556  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431   Er wer 8742   / cqs 8744  cen 8982  cdom 8983  cr 11154  0cc0 11155  1c1 11156   < clt 11295  cle 11296  cmin 11492  -cneg 11493   / cdiv 11920  cn 12266  cq 12990  [,]cicc 13390  volcvol 25498
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500
This theorem is referenced by:  vitali2  46709
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