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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.
StepHypRef Expression
1 reex 10679 . . . 4 ℝ ∈ V
21pwex 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))
64, 5ax-mp 5 . . . . 5 (( < ∩ (ℝ × ℝ)) We 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We ℝ)
73, 6bitr4i 281 . . . 4 ( < We ℝ ↔ ( < ∩ (ℝ × ℝ)) We 𝒫 ℝ)
81, 1xpex 7480 . . . . . 6 (ℝ × ℝ) ∈ V
98inex2 5192 . . . . 5 ( < ∩ (ℝ × ℝ)) ∈ V
10 weeq1 5516 . . . . 5 (𝑥 = ( < ∩ (ℝ × ℝ)) → (𝑥 We 𝒫 ℝ ↔ ( < ∩ (ℝ × ℝ)) We 𝒫 ℝ))
119, 10spcev 3527 . . . 4 (( < ∩ (ℝ × ℝ)) We 𝒫 ℝ → ∃𝑥 𝑥 We 𝒫 ℝ)
127, 11sylbi 220 . . 3 ( < We ℝ → ∃𝑥 𝑥 We 𝒫 ℝ)
13 dfac8c 9506 . . 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 12414 . . . . . . 7 ℚ ∈ V
1615inex1 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
2019a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → -1 ∈ ℝ)
21 0re 10694 . . . . . . . . . . 11 0 ∈ ℝ
2221a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → 0 ∈ ℝ)
23 neg1lt0 11804 . . . . . . . . . . . 12 -1 < 0
2419, 21, 23ltleii 10814 . . . . . . . . . . 11 -1 ≤ 0
2524a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ → -1 ≤ 0)
26 nnrp 12454 . . . . . . . . . . . 12 (𝑥 ∈ ℕ → 𝑥 ∈ ℝ+)
2726rpreccld 12495 . . . . . . . . . . 11 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ ℝ+)
2827rpge0d 12489 . . . . . . . . . 10 (𝑥 ∈ ℕ → 0 ≤ (1 / 𝑥))
2920, 22, 18, 25, 28letrd 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)))
3633, 34, 35mpanl12 701 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ ∧ 0 < 𝑥) → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3731, 32, 36syl2anc 587 . . . . . . . . . . 11 (𝑥 ∈ ℕ → (1 ≤ 𝑥 ↔ (1 / 𝑥) ≤ (1 / 1)))
3830, 37mpbid 235 . . . . . . . . . 10 (𝑥 ∈ ℕ → (1 / 𝑥) ≤ (1 / 1))
39 1div1e1 11381 . . . . . . . . . 10 (1 / 1) = 1
4038, 39breqtrdi 5077 . . . . . . . . 9 (𝑥 ∈ ℕ → (1 / 𝑥) ≤ 1)
4119, 33elicc2i 12858 . . . . . . . . 9 ((1 / 𝑥) ∈ (-1[,]1) ↔ ((1 / 𝑥) ∈ ℝ ∧ -1 ≤ (1 / 𝑥) ∧ (1 / 𝑥) ≤ 1))
4218, 29, 40, 41syl3anbrc 1340 . . . . . . . 8 (𝑥 ∈ ℕ → (1 / 𝑥) ∈ (-1[,]1))
4317, 42elind 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)
4745, 46recrecd 11464 . . . . . . . . . 10 (𝑥 ∈ ℕ → (1 / (1 / 𝑥)) = 𝑥)
48 nncn 11695 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
49 nnne0 11721 . . . . . . . . . . 11 (𝑦 ∈ ℕ → 𝑦 ≠ 0)
5048, 49recrecd 11464 . . . . . . . . . 10 (𝑦 ∈ ℕ → (1 / (1 / 𝑦)) = 𝑦)
5147, 50eqeqan12d 2775 . . . . . . . . 9 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / (1 / 𝑥)) = (1 / (1 / 𝑦)) ↔ 𝑥 = 𝑦))
5244, 51syl5ib 247 . . . . . . . 8 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) → 𝑥 = 𝑦))
53 oveq2 7164 . . . . . . . 8 (𝑥 = 𝑦 → (1 / 𝑥) = (1 / 𝑦))
5452, 53impbid1 228 . . . . . . 7 ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → ((1 / 𝑥) = (1 / 𝑦) ↔ 𝑥 = 𝑦))
5543, 54dom2 8583 . . . . . 6 ((ℚ ∩ (-1[,]1)) ∈ V → ℕ ≼ (ℚ ∩ (-1[,]1)))
5616, 55ax-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))
5915, 57, 58mp2 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)
6259, 60, 61mp2an 691 . . . . 5 (ℚ ∩ (-1[,]1)) ≼ ℕ
63 sbth 8672 . . . . 5 ((ℕ ≼ (ℚ ∩ (-1[,]1)) ∧ (ℚ ∩ (-1[,]1)) ≼ ℕ) → ℕ ≈ (ℚ ∩ (-1[,]1)))
6456, 62, 63mp2an 691 . . . 4 ℕ ≈ (ℚ ∩ (-1[,]1))
65 bren 8549 . . . 4 (ℕ ≈ (ℚ ∩ (-1[,]1)) ↔ ∃𝑔 𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)))
6664, 65mpbi 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)))
6967, 68bi2anan9 638 . . . . . . . . . . . 12 ((𝑎 = 𝑥𝑏 = 𝑦) → ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))))
70 oveq12 7165 . . . . . . . . . . . . 13 ((𝑎 = 𝑥𝑏 = 𝑦) → (𝑎𝑏) = (𝑥𝑦))
7170eleq1d 2836 . . . . . . . . . . . 12 ((𝑎 = 𝑥𝑏 = 𝑦) → ((𝑎𝑏) ∈ ℚ ↔ (𝑥𝑦) ∈ ℚ))
7269, 71anbi12d 633 . . . . . . . . . . 11 ((𝑎 = 𝑥𝑏 = 𝑦) → (((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ) ↔ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)))
7372cbvopabv 5108 . . . . . . . . . 10 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) ∧ (𝑥𝑦) ∈ ℚ)}
74 eqid 2758 . . . . . . . . . 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 6676 . . . . . . . . . . . 12 (𝑓𝑐) ∈ V
76 eqid 2758 . . . . . . . . . . . 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 6479 . . . . . . . . . . 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 3013 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → (𝑧 ≠ ∅ ↔ 𝑤 ≠ ∅))
80 fveq2 6663 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤 → (𝑓𝑧) = (𝑓𝑤))
81 id 22 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑤𝑧 = 𝑤)
8280, 81eleq12d 2846 . . . . . . . . . . . . . . 15 (𝑧 = 𝑤 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑓𝑤) ∈ 𝑤))
8379, 82imbi12d 348 . . . . . . . . . . . . . 14 (𝑧 = 𝑤 → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤)))
8483cbvralvw 3361 . . . . . . . . . . . . 13 (∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑤 ∈ 𝒫 ℝ(𝑤 ≠ ∅ → (𝑓𝑤) ∈ 𝑤))
8573vitalilem1 24321 . . . . . . . . . . . . . . . . . 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 8374 . . . . . . . . . . . . . . . 16 (⊤ → ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1))
8887mptru 1545 . . . . . . . . . . . . . . 15 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 (0[,]1)
89 unitssre 12944 . . . . . . . . . . . . . . . 16 (0[,]1) ⊆ ℝ
9089sspwi 4511 . . . . . . . . . . . . . . 15 𝒫 (0[,]1) ⊆ 𝒫 ℝ
9188, 90sstri 3903 . . . . . . . . . . . . . 14 ((0[,]1) / {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (0[,]1) ∧ 𝑏 ∈ (0[,]1)) ∧ (𝑎𝑏) ∈ ℚ)}) ⊆ 𝒫 ℝ
92 ssralv 3960 . . . . . . . . . . . . . 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 220 . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑤 → (𝑓𝑐) = (𝑓𝑤))
96 fvex 6676 . . . . . . . . . . . . . . . 16 (𝑓𝑤) ∈ V
9795, 76, 96fvmpt 6764 . . . . . . . . . . . . . . 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 2836 . . . . . . . . . . . . . 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 344 . . . . . . . . . . . . 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 3096 . . . . . . . . . . . 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 237 . . . . . . . . . . 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 7163 . . . . . . . . . . . . . 14 (𝑡 = 𝑠 → (𝑡 − (𝑔𝑚)) = (𝑠 − (𝑔𝑚)))
105104eleq1d 2836 . . . . . . . . . . . . 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 3404 . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝑔𝑚) = (𝑔𝑛))
108107oveq2d 7172 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (𝑠 − (𝑔𝑚)) = (𝑠 − (𝑔𝑛)))
109108eleq1d 2836 . . . . . . . . . . . . 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 3392 . . . . . . . . . . . 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, 110syl5eq 2805 . . . . . . . . . . 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 5139 . . . . . . . . . 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 24325 . . . . . . . . 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 460 . . . . . . 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 3870 . . . . . . 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 24244 . . . . . . . . . 10 (𝑥 ∈ dom vol → 𝑥 ⊆ ℝ)
120 velpw 4502 . . . . . . . . . 10 (𝑥 ∈ 𝒫 ℝ ↔ 𝑥 ⊆ ℝ)
121119, 120sylibr 237 . . . . . . . . 9 (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫 ℝ)
122121ssriv 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)) ∧ (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 220 . . . . . 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 416 . . . 4 (( < We ℝ ∧ ∀𝑧 ∈ 𝒫 ℝ(𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → (𝑔:ℕ–1-1-onto→(ℚ ∩ (-1[,]1)) → dom vol ⊊ 𝒫 ℝ))
128127exlimdv 1934 . . 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 1936 1 ( < We ℝ → dom vol ⊊ 𝒫 ℝ)
 Colors of variables: wff setvar class 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
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