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Theorem ovoliunnul 25542
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 5008 . . . . . 6 (𝐴 = ∅ → 𝑛𝐴 𝐵 = 𝑛 ∈ ∅ 𝐵)
2 0iun 5063 . . . . . 6 𝑛 ∈ ∅ 𝐵 = ∅
31, 2eqtrdi 2793 . . . . 5 (𝐴 = ∅ → 𝑛𝐴 𝐵 = ∅)
43fveq2d 6910 . . . 4 (𝐴 = ∅ → (vol*‘ 𝑛𝐴 𝐵) = (vol*‘∅))
5 ovol0 25528 . . . 4 (vol*‘∅) = 0
64, 5eqtrdi 2793 . . 3 (𝐴 = ∅ → (vol*‘ 𝑛𝐴 𝐵) = 0)
76a1i 11 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝐴 = ∅ → (vol*‘ 𝑛𝐴 𝐵) = 0))
8 reldom 8991 . . . . . 6 Rel ≼
98brrelex1i 5741 . . . . 5 (𝐴 ≼ ℕ → 𝐴 ∈ V)
109adantr 480 . . . 4 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → 𝐴 ∈ V)
11 0sdomg 9144 . . . 4 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1210, 11syl 17 . . 3 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴𝐴 ≠ ∅))
13 fodomr 9168 . . . . . 6 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
1413expcom 413 . . . . 5 (𝐴 ≼ ℕ → (∅ ≺ 𝐴 → ∃𝑓 𝑓:ℕ–onto𝐴))
1514adantr 480 . . . 4 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 → ∃𝑓 𝑓:ℕ–onto𝐴))
16 eliun 4995 . . . . . . . . . 10 (𝑥 𝑛𝐴 𝐵 ↔ ∃𝑛𝐴 𝑥𝐵)
17 nfv 1914 . . . . . . . . . . 11 𝑛 𝑓:ℕ–onto𝐴
18 nfcv 2905 . . . . . . . . . . . . 13 𝑛
19 nfcsb1v 3923 . . . . . . . . . . . . 13 𝑛(𝑓𝑘) / 𝑛𝐵
2018, 19nfiun 5023 . . . . . . . . . . . 12 𝑛 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵
2120nfcri 2897 . . . . . . . . . . 11 𝑛 𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵
22 foelrn 7127 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴𝑛𝐴) → ∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘))
2322ex 412 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (𝑛𝐴 → ∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘)))
24 csbeq1a 3913 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = (𝑓𝑘) → 𝐵 = (𝑓𝑘) / 𝑛𝐵)
2524adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ–onto𝐴𝑛 = (𝑓𝑘)) → 𝐵 = (𝑓𝑘) / 𝑛𝐵)
2625eleq2d 2827 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ–onto𝐴𝑛 = (𝑓𝑘)) → (𝑥𝐵𝑥(𝑓𝑘) / 𝑛𝐵))
2726biimpd 229 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ–onto𝐴𝑛 = (𝑓𝑘)) → (𝑥𝐵𝑥(𝑓𝑘) / 𝑛𝐵))
2827impancom 451 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–onto𝐴𝑥𝐵) → (𝑛 = (𝑓𝑘) → 𝑥(𝑓𝑘) / 𝑛𝐵))
2928reximdv 3170 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–onto𝐴𝑥𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘) → ∃𝑘 ∈ ℕ 𝑥(𝑓𝑘) / 𝑛𝐵))
30 eliun 4995 . . . . . . . . . . . . . . 15 (𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ↔ ∃𝑘 ∈ ℕ 𝑥(𝑓𝑘) / 𝑛𝐵)
3129, 30imbitrrdi 252 . . . . . . . . . . . . . 14 ((𝑓:ℕ–onto𝐴𝑥𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘) → 𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
3231ex 412 . . . . . . . . . . . . 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 3266 . . . . . . . . . 10 (𝑓:ℕ–onto𝐴 → (∃𝑛𝐴 𝑥𝐵𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
3616, 35biimtrid 242 . . . . . . . . 9 (𝑓:ℕ–onto𝐴 → (𝑥 𝑛𝐴 𝐵𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
3736ssrdv 3989 . . . . . . . 8 (𝑓:ℕ–onto𝐴 𝑛𝐴 𝐵 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵)
3837adantl 481 . . . . . . 7 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → 𝑛𝐴 𝐵 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵)
39 fof 6820 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴𝑓:ℕ⟶𝐴)
4039adantl 481 . . . . . . . . . . . 12 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → 𝑓:ℕ⟶𝐴)
4140ffvelcdmda 7104 . . . . . . . . . . 11 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝐴)
42 simpllr 776 . . . . . . . . . . 11 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0))
43 nfcv 2905 . . . . . . . . . . . . . 14 𝑛
4419, 43nfss 3976 . . . . . . . . . . . . 13 𝑛(𝑓𝑘) / 𝑛𝐵 ⊆ ℝ
45 nfcv 2905 . . . . . . . . . . . . . . 15 𝑛vol*
4645, 19nffv 6916 . . . . . . . . . . . . . 14 𝑛(vol*‘(𝑓𝑘) / 𝑛𝐵)
4746nfeq1 2921 . . . . . . . . . . . . 13 𝑛(vol*‘(𝑓𝑘) / 𝑛𝐵) = 0
4844, 47nfan 1899 . . . . . . . . . . . 12 𝑛((𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)
4924sseq1d 4015 . . . . . . . . . . . . 13 (𝑛 = (𝑓𝑘) → (𝐵 ⊆ ℝ ↔ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ))
5024fveqeq2d 6914 . . . . . . . . . . . . 13 (𝑛 = (𝑓𝑘) → ((vol*‘𝐵) = 0 ↔ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0))
5149, 50anbi12d 632 . . . . . . . . . . . 12 (𝑛 = (𝑓𝑘) → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ↔ ((𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)))
5248, 51rspc 3610 . . . . . . . . . . 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 494 . . . . . . . . 9 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
5554ralrimiva 3146 . . . . . . . 8 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → ∀𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
56 iunss 5045 . . . . . . . 8 ( 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ↔ ∀𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
5755, 56sylibr 234 . . . . . . 7 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
58 eqid 2737 . . . . . . . . . 10 seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)))
59 eqid 2737 . . . . . . . . . 10 (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)) = (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))
6053simprd 495 . . . . . . . . . . 11 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)
61 0re 11263 . . . . . . . . . . 11 0 ∈ ℝ
6260, 61eqeltrdi 2849 . . . . . . . . . 10 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑓𝑘) / 𝑛𝐵) ∈ ℝ)
6360mpteq2dva 5242 . . . . . . . . . . . . 13 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)) = (𝑘 ∈ ℕ ↦ 0))
64 fconstmpt 5747 . . . . . . . . . . . . . 14 (ℕ × {0}) = (𝑘 ∈ ℕ ↦ 0)
65 nnuz 12921 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6665xpeq1i 5711 . . . . . . . . . . . . . 14 (ℕ × {0}) = ((ℤ‘1) × {0})
6764, 66eqtr3i 2767 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ ↦ 0) = ((ℤ‘1) × {0})
6863, 67eqtrdi 2793 . . . . . . . . . . . 12 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)) = ((ℤ‘1) × {0}))
6968seqeq3d 14050 . . . . . . . . . . 11 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))) = seq1( + , ((ℤ‘1) × {0})))
70 1z 12647 . . . . . . . . . . . 12 1 ∈ ℤ
71 serclim0 15613 . . . . . . . . . . . 12 (1 ∈ ℤ → seq1( + , ((ℤ‘1) × {0})) ⇝ 0)
72 seqex 14044 . . . . . . . . . . . . 13 seq1( + , ((ℤ‘1) × {0})) ∈ V
73 c0ex 11255 . . . . . . . . . . . . 13 0 ∈ V
7472, 73breldm 5919 . . . . . . . . . . . 12 (seq1( + , ((ℤ‘1) × {0})) ⇝ 0 → seq1( + , ((ℤ‘1) × {0})) ∈ dom ⇝ )
7570, 71, 74mp2b 10 . . . . . . . . . . 11 seq1( + , ((ℤ‘1) × {0})) ∈ dom ⇝
7669, 75eqeltrdi 2849 . . . . . . . . . 10 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))) ∈ dom ⇝ )
7758, 59, 54, 62, 76ovoliun2 25541 . . . . . . . . 9 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ≤ Σ𝑘 ∈ ℕ (vol*‘(𝑓𝑘) / 𝑛𝐵))
7860sumeq2dv 15738 . . . . . . . . . 10 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → Σ𝑘 ∈ ℕ (vol*‘(𝑓𝑘) / 𝑛𝐵) = Σ𝑘 ∈ ℕ 0)
7965eqimssi 4044 . . . . . . . . . . . 12 ℕ ⊆ (ℤ‘1)
8079orci 866 . . . . . . . . . . 11 (ℕ ⊆ (ℤ‘1) ∨ ℕ ∈ Fin)
81 sumz 15758 . . . . . . . . . . 11 ((ℕ ⊆ (ℤ‘1) ∨ ℕ ∈ Fin) → Σ𝑘 ∈ ℕ 0 = 0)
8280, 81ax-mp 5 . . . . . . . . . 10 Σ𝑘 ∈ ℕ 0 = 0
8378, 82eqtrdi 2793 . . . . . . . . 9 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → Σ𝑘 ∈ ℕ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)
8477, 83breqtrd 5169 . . . . . . . 8 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ≤ 0)
85 ovolge0 25516 . . . . . . . . 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 25513 . . . . . . . . . 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 11308 . . . . . . . . 9 0 ∈ ℝ*
90 xrletri3 13196 . . . . . . . . 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 713 . . . . . . 7 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) = 0)
93 ovolssnul 25522 . . . . . . 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 1373 . . . . . 6 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑛𝐴 𝐵) = 0)
9594ex 412 . . . . 5 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝑓:ℕ–onto𝐴 → (vol*‘ 𝑛𝐴 𝐵) = 0))
9695exlimdv 1933 . . . 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 260 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝐴 ≠ ∅ → (vol*‘ 𝑛𝐴 𝐵) = 0))
997, 98pm2.61dne 3028 1 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘ 𝑛𝐴 𝐵) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  csb 3899  wss 3951  c0 4333  {csn 4626   ciun 4991   class class class wbr 5143  cmpt 5225   × cxp 5683  dom cdm 5685  wf 6557  ontowfo 6559  cfv 6561  cdom 8983  csdm 8984  Fincfn 8985  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  *cxr 11294  cle 11296  cn 12266  cz 12613  cuz 12878  seqcseq 14042  cli 15520  Σcsu 15722  vol*covol 25497
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-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-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  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-xadd 13155  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-xmet 21357  df-met 21358  df-ovol 25499
This theorem is referenced by: (None)
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