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Theorem ovoliunnul 24871
Description: A countable union of nullsets is null. (Contributed by Mario Carneiro, 8-Apr-2015.)
Assertion
Ref Expression
ovoliunnul ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘ 𝑛𝐴 𝐵) = 0)
Distinct variable group:   𝐴,𝑛
Allowed substitution hint:   𝐵(𝑛)

Proof of Theorem ovoliunnul
Dummy variables 𝑓 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iuneq1 4970 . . . . . 6 (𝐴 = ∅ → 𝑛𝐴 𝐵 = 𝑛 ∈ ∅ 𝐵)
2 0iun 5023 . . . . . 6 𝑛 ∈ ∅ 𝐵 = ∅
31, 2eqtrdi 2792 . . . . 5 (𝐴 = ∅ → 𝑛𝐴 𝐵 = ∅)
43fveq2d 6846 . . . 4 (𝐴 = ∅ → (vol*‘ 𝑛𝐴 𝐵) = (vol*‘∅))
5 ovol0 24857 . . . 4 (vol*‘∅) = 0
64, 5eqtrdi 2792 . . 3 (𝐴 = ∅ → (vol*‘ 𝑛𝐴 𝐵) = 0)
76a1i 11 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝐴 = ∅ → (vol*‘ 𝑛𝐴 𝐵) = 0))
8 reldom 8889 . . . . . 6 Rel ≼
98brrelex1i 5688 . . . . 5 (𝐴 ≼ ℕ → 𝐴 ∈ V)
109adantr 481 . . . 4 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → 𝐴 ∈ V)
11 0sdomg 9048 . . . 4 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1210, 11syl 17 . . 3 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴𝐴 ≠ ∅))
13 fodomr 9072 . . . . . 6 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
1413expcom 414 . . . . 5 (𝐴 ≼ ℕ → (∅ ≺ 𝐴 → ∃𝑓 𝑓:ℕ–onto𝐴))
1514adantr 481 . . . 4 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 → ∃𝑓 𝑓:ℕ–onto𝐴))
16 eliun 4958 . . . . . . . . . 10 (𝑥 𝑛𝐴 𝐵 ↔ ∃𝑛𝐴 𝑥𝐵)
17 nfv 1917 . . . . . . . . . . 11 𝑛 𝑓:ℕ–onto𝐴
18 nfcv 2907 . . . . . . . . . . . . 13 𝑛
19 nfcsb1v 3880 . . . . . . . . . . . . 13 𝑛(𝑓𝑘) / 𝑛𝐵
2018, 19nfiun 4984 . . . . . . . . . . . 12 𝑛 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵
2120nfcri 2894 . . . . . . . . . . 11 𝑛 𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵
22 foelrn 7056 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴𝑛𝐴) → ∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘))
2322ex 413 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (𝑛𝐴 → ∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘)))
24 csbeq1a 3869 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = (𝑓𝑘) → 𝐵 = (𝑓𝑘) / 𝑛𝐵)
2524adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ–onto𝐴𝑛 = (𝑓𝑘)) → 𝐵 = (𝑓𝑘) / 𝑛𝐵)
2625eleq2d 2823 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ–onto𝐴𝑛 = (𝑓𝑘)) → (𝑥𝐵𝑥(𝑓𝑘) / 𝑛𝐵))
2726biimpd 228 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ–onto𝐴𝑛 = (𝑓𝑘)) → (𝑥𝐵𝑥(𝑓𝑘) / 𝑛𝐵))
2827impancom 452 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–onto𝐴𝑥𝐵) → (𝑛 = (𝑓𝑘) → 𝑥(𝑓𝑘) / 𝑛𝐵))
2928reximdv 3167 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–onto𝐴𝑥𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘) → ∃𝑘 ∈ ℕ 𝑥(𝑓𝑘) / 𝑛𝐵))
30 eliun 4958 . . . . . . . . . . . . . . 15 (𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ↔ ∃𝑘 ∈ ℕ 𝑥(𝑓𝑘) / 𝑛𝐵)
3129, 30syl6ibr 251 . . . . . . . . . . . . . 14 ((𝑓:ℕ–onto𝐴𝑥𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘) → 𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
3231ex 413 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴 → (𝑥𝐵 → (∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘) → 𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵)))
3332com23 86 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘) → (𝑥𝐵𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵)))
3423, 33syld 47 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (𝑛𝐴 → (𝑥𝐵𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵)))
3517, 21, 34rexlimd 3249 . . . . . . . . . 10 (𝑓:ℕ–onto𝐴 → (∃𝑛𝐴 𝑥𝐵𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
3616, 35biimtrid 241 . . . . . . . . 9 (𝑓:ℕ–onto𝐴 → (𝑥 𝑛𝐴 𝐵𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
3736ssrdv 3950 . . . . . . . 8 (𝑓:ℕ–onto𝐴 𝑛𝐴 𝐵 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵)
3837adantl 482 . . . . . . 7 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → 𝑛𝐴 𝐵 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵)
39 fof 6756 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴𝑓:ℕ⟶𝐴)
4039adantl 482 . . . . . . . . . . . 12 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → 𝑓:ℕ⟶𝐴)
4140ffvelcdmda 7035 . . . . . . . . . . 11 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝐴)
42 simpllr 774 . . . . . . . . . . 11 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0))
43 nfcv 2907 . . . . . . . . . . . . . 14 𝑛
4419, 43nfss 3936 . . . . . . . . . . . . 13 𝑛(𝑓𝑘) / 𝑛𝐵 ⊆ ℝ
45 nfcv 2907 . . . . . . . . . . . . . . 15 𝑛vol*
4645, 19nffv 6852 . . . . . . . . . . . . . 14 𝑛(vol*‘(𝑓𝑘) / 𝑛𝐵)
4746nfeq1 2922 . . . . . . . . . . . . 13 𝑛(vol*‘(𝑓𝑘) / 𝑛𝐵) = 0
4844, 47nfan 1902 . . . . . . . . . . . 12 𝑛((𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)
4924sseq1d 3975 . . . . . . . . . . . . 13 (𝑛 = (𝑓𝑘) → (𝐵 ⊆ ℝ ↔ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ))
5024fveqeq2d 6850 . . . . . . . . . . . . 13 (𝑛 = (𝑓𝑘) → ((vol*‘𝐵) = 0 ↔ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0))
5149, 50anbi12d 631 . . . . . . . . . . . 12 (𝑛 = (𝑓𝑘) → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ↔ ((𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)))
5248, 51rspc 3569 . . . . . . . . . . 11 ((𝑓𝑘) ∈ 𝐴 → (∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)))
5341, 42, 52sylc 65 . . . . . . . . . 10 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → ((𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0))
5453simpld 495 . . . . . . . . 9 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
5554ralrimiva 3143 . . . . . . . 8 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → ∀𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
56 iunss 5005 . . . . . . . 8 ( 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ↔ ∀𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
5755, 56sylibr 233 . . . . . . 7 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
58 eqid 2736 . . . . . . . . . 10 seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)))
59 eqid 2736 . . . . . . . . . 10 (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)) = (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))
6053simprd 496 . . . . . . . . . . 11 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)
61 0re 11157 . . . . . . . . . . 11 0 ∈ ℝ
6260, 61eqeltrdi 2846 . . . . . . . . . 10 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑓𝑘) / 𝑛𝐵) ∈ ℝ)
6360mpteq2dva 5205 . . . . . . . . . . . . 13 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)) = (𝑘 ∈ ℕ ↦ 0))
64 fconstmpt 5694 . . . . . . . . . . . . . 14 (ℕ × {0}) = (𝑘 ∈ ℕ ↦ 0)
65 nnuz 12806 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6665xpeq1i 5659 . . . . . . . . . . . . . 14 (ℕ × {0}) = ((ℤ‘1) × {0})
6764, 66eqtr3i 2766 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ ↦ 0) = ((ℤ‘1) × {0})
6863, 67eqtrdi 2792 . . . . . . . . . . . 12 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)) = ((ℤ‘1) × {0}))
6968seqeq3d 13914 . . . . . . . . . . 11 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))) = seq1( + , ((ℤ‘1) × {0})))
70 1z 12533 . . . . . . . . . . . 12 1 ∈ ℤ
71 serclim0 15459 . . . . . . . . . . . 12 (1 ∈ ℤ → seq1( + , ((ℤ‘1) × {0})) ⇝ 0)
72 seqex 13908 . . . . . . . . . . . . 13 seq1( + , ((ℤ‘1) × {0})) ∈ V
73 c0ex 11149 . . . . . . . . . . . . 13 0 ∈ V
7472, 73breldm 5864 . . . . . . . . . . . 12 (seq1( + , ((ℤ‘1) × {0})) ⇝ 0 → seq1( + , ((ℤ‘1) × {0})) ∈ dom ⇝ )
7570, 71, 74mp2b 10 . . . . . . . . . . 11 seq1( + , ((ℤ‘1) × {0})) ∈ dom ⇝
7669, 75eqeltrdi 2846 . . . . . . . . . 10 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))) ∈ dom ⇝ )
7758, 59, 54, 62, 76ovoliun2 24870 . . . . . . . . 9 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ≤ Σ𝑘 ∈ ℕ (vol*‘(𝑓𝑘) / 𝑛𝐵))
7860sumeq2dv 15588 . . . . . . . . . 10 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → Σ𝑘 ∈ ℕ (vol*‘(𝑓𝑘) / 𝑛𝐵) = Σ𝑘 ∈ ℕ 0)
7965eqimssi 4002 . . . . . . . . . . . 12 ℕ ⊆ (ℤ‘1)
8079orci 863 . . . . . . . . . . 11 (ℕ ⊆ (ℤ‘1) ∨ ℕ ∈ Fin)
81 sumz 15607 . . . . . . . . . . 11 ((ℕ ⊆ (ℤ‘1) ∨ ℕ ∈ Fin) → Σ𝑘 ∈ ℕ 0 = 0)
8280, 81ax-mp 5 . . . . . . . . . 10 Σ𝑘 ∈ ℕ 0 = 0
8378, 82eqtrdi 2792 . . . . . . . . 9 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → Σ𝑘 ∈ ℕ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)
8477, 83breqtrd 5131 . . . . . . . 8 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ≤ 0)
85 ovolge0 24845 . . . . . . . . 9 ( 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ → 0 ≤ (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
8657, 85syl 17 . . . . . . . 8 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → 0 ≤ (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
87 ovolcl 24842 . . . . . . . . . 10 ( 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ∈ ℝ*)
8857, 87syl 17 . . . . . . . . 9 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ∈ ℝ*)
89 0xr 11202 . . . . . . . . 9 0 ∈ ℝ*
90 xrletri3 13073 . . . . . . . . 9 (((vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) = 0 ↔ ((vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ≤ 0 ∧ 0 ≤ (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))))
9188, 89, 90sylancl 586 . . . . . . . 8 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → ((vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) = 0 ↔ ((vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ≤ 0 ∧ 0 ≤ (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))))
9284, 86, 91mpbir2and 711 . . . . . . 7 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) = 0)
93 ovolssnul 24851 . . . . . . 7 (( 𝑛𝐴 𝐵 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) = 0) → (vol*‘ 𝑛𝐴 𝐵) = 0)
9438, 57, 92, 93syl3anc 1371 . . . . . 6 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑛𝐴 𝐵) = 0)
9594ex 413 . . . . 5 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝑓:ℕ–onto𝐴 → (vol*‘ 𝑛𝐴 𝐵) = 0))
9695exlimdv 1936 . . . 4 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∃𝑓 𝑓:ℕ–onto𝐴 → (vol*‘ 𝑛𝐴 𝐵) = 0))
9715, 96syld 47 . . 3 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 → (vol*‘ 𝑛𝐴 𝐵) = 0))
9812, 97sylbird 259 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝐴 ≠ ∅ → (vol*‘ 𝑛𝐴 𝐵) = 0))
997, 98pm2.61dne 3031 1 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘ 𝑛𝐴 𝐵) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  csb 3855  wss 3910  c0 4282  {csn 4586   ciun 4954   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  wf 6492  ontowfo 6494  cfv 6496  cdom 8881  csdm 8882  Fincfn 8883  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  *cxr 11188  cle 11190  cn 12153  cz 12499  cuz 12763  seqcseq 13906  cli 15366  Σcsu 15570  vol*covol 24826
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xadd 13034  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-xmet 20789  df-met 20790  df-ovol 24828
This theorem is referenced by: (None)
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