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Theorem ovoliunnul 23321
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 4566 . . . . . 6 (𝐴 = ∅ → 𝑛𝐴 𝐵 = 𝑛 ∈ ∅ 𝐵)
2 0iun 4609 . . . . . 6 𝑛 ∈ ∅ 𝐵 = ∅
31, 2syl6eq 2701 . . . . 5 (𝐴 = ∅ → 𝑛𝐴 𝐵 = ∅)
43fveq2d 6233 . . . 4 (𝐴 = ∅ → (vol*‘ 𝑛𝐴 𝐵) = (vol*‘∅))
5 ovol0 23307 . . . 4 (vol*‘∅) = 0
64, 5syl6eq 2701 . . 3 (𝐴 = ∅ → (vol*‘ 𝑛𝐴 𝐵) = 0)
76a1i 11 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝐴 = ∅ → (vol*‘ 𝑛𝐴 𝐵) = 0))
8 reldom 8003 . . . . . 6 Rel ≼
98brrelexi 5192 . . . . 5 (𝐴 ≼ ℕ → 𝐴 ∈ V)
109adantr 480 . . . 4 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → 𝐴 ∈ V)
11 0sdomg 8130 . . . 4 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1210, 11syl 17 . . 3 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴𝐴 ≠ ∅))
13 fodomr 8152 . . . . . 6 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
1413expcom 450 . . . . 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 4556 . . . . . . . . . 10 (𝑥 𝑛𝐴 𝐵 ↔ ∃𝑛𝐴 𝑥𝐵)
17 nfv 1883 . . . . . . . . . . 11 𝑛 𝑓:ℕ–onto𝐴
18 nfcv 2793 . . . . . . . . . . . . 13 𝑛
19 nfcsb1v 3582 . . . . . . . . . . . . 13 𝑛(𝑓𝑘) / 𝑛𝐵
2018, 19nfiun 4580 . . . . . . . . . . . 12 𝑛 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵
2120nfcri 2787 . . . . . . . . . . 11 𝑛 𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵
22 foelrn 6418 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴𝑛𝐴) → ∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘))
2322ex 449 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (𝑛𝐴 → ∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘)))
24 csbeq1a 3575 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = (𝑓𝑘) → 𝐵 = (𝑓𝑘) / 𝑛𝐵)
2524adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑓:ℕ–onto𝐴𝑛 = (𝑓𝑘)) → 𝐵 = (𝑓𝑘) / 𝑛𝐵)
2625eleq2d 2716 . . . . . . . . . . . . . . . . . 18 ((𝑓:ℕ–onto𝐴𝑛 = (𝑓𝑘)) → (𝑥𝐵𝑥(𝑓𝑘) / 𝑛𝐵))
2726biimpd 219 . . . . . . . . . . . . . . . . 17 ((𝑓:ℕ–onto𝐴𝑛 = (𝑓𝑘)) → (𝑥𝐵𝑥(𝑓𝑘) / 𝑛𝐵))
2827impancom 455 . . . . . . . . . . . . . . . 16 ((𝑓:ℕ–onto𝐴𝑥𝐵) → (𝑛 = (𝑓𝑘) → 𝑥(𝑓𝑘) / 𝑛𝐵))
2928reximdv 3045 . . . . . . . . . . . . . . 15 ((𝑓:ℕ–onto𝐴𝑥𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘) → ∃𝑘 ∈ ℕ 𝑥(𝑓𝑘) / 𝑛𝐵))
30 eliun 4556 . . . . . . . . . . . . . . 15 (𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ↔ ∃𝑘 ∈ ℕ 𝑥(𝑓𝑘) / 𝑛𝐵)
3129, 30syl6ibr 242 . . . . . . . . . . . . . 14 ((𝑓:ℕ–onto𝐴𝑥𝐵) → (∃𝑘 ∈ ℕ 𝑛 = (𝑓𝑘) → 𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
3231ex 449 . . . . . . . . . . . . 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 3055 . . . . . . . . . 10 (𝑓:ℕ–onto𝐴 → (∃𝑛𝐴 𝑥𝐵𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
3616, 35syl5bi 232 . . . . . . . . 9 (𝑓:ℕ–onto𝐴 → (𝑥 𝑛𝐴 𝐵𝑥 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
3736ssrdv 3642 . . . . . . . 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 6153 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴𝑓:ℕ⟶𝐴)
4039adantl 481 . . . . . . . . . . . 12 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → 𝑓:ℕ⟶𝐴)
4140ffvelrnda 6399 . . . . . . . . . . 11 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) ∈ 𝐴)
42 simpllr 815 . . . . . . . . . . 11 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0))
43 nfcv 2793 . . . . . . . . . . . . . 14 𝑛
4419, 43nfss 3629 . . . . . . . . . . . . 13 𝑛(𝑓𝑘) / 𝑛𝐵 ⊆ ℝ
45 nfcv 2793 . . . . . . . . . . . . . . 15 𝑛vol*
4645, 19nffv 6236 . . . . . . . . . . . . . 14 𝑛(vol*‘(𝑓𝑘) / 𝑛𝐵)
4746nfeq1 2807 . . . . . . . . . . . . 13 𝑛(vol*‘(𝑓𝑘) / 𝑛𝐵) = 0
4844, 47nfan 1868 . . . . . . . . . . . 12 𝑛((𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)
4924sseq1d 3665 . . . . . . . . . . . . 13 (𝑛 = (𝑓𝑘) → (𝐵 ⊆ ℝ ↔ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ))
5024fveq2d 6233 . . . . . . . . . . . . . 14 (𝑛 = (𝑓𝑘) → (vol*‘𝐵) = (vol*‘(𝑓𝑘) / 𝑛𝐵))
5150eqeq1d 2653 . . . . . . . . . . . . 13 (𝑛 = (𝑓𝑘) → ((vol*‘𝐵) = 0 ↔ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0))
5249, 51anbi12d 747 . . . . . . . . . . . 12 (𝑛 = (𝑓𝑘) → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) ↔ ((𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)))
5348, 52rspc 3334 . . . . . . . . . . 11 ((𝑓𝑘) ∈ 𝐴 → (∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0) → ((𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)))
5441, 42, 53sylc 65 . . . . . . . . . 10 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → ((𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0))
5554simpld 474 . . . . . . . . 9 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
5655ralrimiva 2995 . . . . . . . 8 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → ∀𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
57 iunss 4593 . . . . . . . 8 ( 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ↔ ∀𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
5856, 57sylibr 224 . . . . . . 7 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ)
59 eqid 2651 . . . . . . . . . 10 seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))) = seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)))
60 eqid 2651 . . . . . . . . . 10 (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)) = (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))
6154simprd 478 . . . . . . . . . . 11 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)
62 0re 10078 . . . . . . . . . . 11 0 ∈ ℝ
6361, 62syl6eqel 2738 . . . . . . . . . 10 ((((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) ∧ 𝑘 ∈ ℕ) → (vol*‘(𝑓𝑘) / 𝑛𝐵) ∈ ℝ)
6461mpteq2dva 4777 . . . . . . . . . . . . 13 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)) = (𝑘 ∈ ℕ ↦ 0))
65 fconstmpt 5197 . . . . . . . . . . . . . 14 (ℕ × {0}) = (𝑘 ∈ ℕ ↦ 0)
66 nnuz 11761 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6766xpeq1i 5169 . . . . . . . . . . . . . 14 (ℕ × {0}) = ((ℤ‘1) × {0})
6865, 67eqtr3i 2675 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ ↦ 0) = ((ℤ‘1) × {0})
6964, 68syl6eq 2701 . . . . . . . . . . . 12 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵)) = ((ℤ‘1) × {0}))
7069seqeq3d 12849 . . . . . . . . . . 11 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))) = seq1( + , ((ℤ‘1) × {0})))
71 1z 11445 . . . . . . . . . . . 12 1 ∈ ℤ
72 serclim0 14352 . . . . . . . . . . . 12 (1 ∈ ℤ → seq1( + , ((ℤ‘1) × {0})) ⇝ 0)
73 seqex 12843 . . . . . . . . . . . . 13 seq1( + , ((ℤ‘1) × {0})) ∈ V
74 c0ex 10072 . . . . . . . . . . . . 13 0 ∈ V
7573, 74breldm 5361 . . . . . . . . . . . 12 (seq1( + , ((ℤ‘1) × {0})) ⇝ 0 → seq1( + , ((ℤ‘1) × {0})) ∈ dom ⇝ )
7671, 72, 75mp2b 10 . . . . . . . . . . 11 seq1( + , ((ℤ‘1) × {0})) ∈ dom ⇝
7770, 76syl6eqel 2738 . . . . . . . . . 10 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → seq1( + , (𝑘 ∈ ℕ ↦ (vol*‘(𝑓𝑘) / 𝑛𝐵))) ∈ dom ⇝ )
7859, 60, 55, 63, 77ovoliun2 23320 . . . . . . . . 9 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ≤ Σ𝑘 ∈ ℕ (vol*‘(𝑓𝑘) / 𝑛𝐵))
7961sumeq2dv 14477 . . . . . . . . . 10 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → Σ𝑘 ∈ ℕ (vol*‘(𝑓𝑘) / 𝑛𝐵) = Σ𝑘 ∈ ℕ 0)
8066eqimssi 3692 . . . . . . . . . . . 12 ℕ ⊆ (ℤ‘1)
8180orci 404 . . . . . . . . . . 11 (ℕ ⊆ (ℤ‘1) ∨ ℕ ∈ Fin)
82 sumz 14497 . . . . . . . . . . 11 ((ℕ ⊆ (ℤ‘1) ∨ ℕ ∈ Fin) → Σ𝑘 ∈ ℕ 0 = 0)
8381, 82ax-mp 5 . . . . . . . . . 10 Σ𝑘 ∈ ℕ 0 = 0
8479, 83syl6eq 2701 . . . . . . . . 9 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → Σ𝑘 ∈ ℕ (vol*‘(𝑓𝑘) / 𝑛𝐵) = 0)
8578, 84breqtrd 4711 . . . . . . . 8 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ≤ 0)
86 ovolge0 23295 . . . . . . . . 9 ( 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ → 0 ≤ (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
8758, 86syl 17 . . . . . . . 8 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → 0 ≤ (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵))
88 ovolcl 23292 . . . . . . . . . 10 ( 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ∈ ℝ*)
8958, 88syl 17 . . . . . . . . 9 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) ∈ ℝ*)
90 0xr 10124 . . . . . . . . 9 0 ∈ ℝ*
91 xrletri3 12023 . . . . . . . . 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))))
9289, 90, 91sylancl 695 . . . . . . . 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))))
9385, 87, 92mpbir2and 977 . . . . . . 7 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) = 0)
94 ovolssnul 23301 . . . . . . 7 (( 𝑛𝐴 𝐵 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵 ⊆ ℝ ∧ (vol*‘ 𝑘 ∈ ℕ (𝑓𝑘) / 𝑛𝐵) = 0) → (vol*‘ 𝑛𝐴 𝐵) = 0)
9538, 58, 93, 94syl3anc 1366 . . . . . 6 (((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) ∧ 𝑓:ℕ–onto𝐴) → (vol*‘ 𝑛𝐴 𝐵) = 0)
9695ex 449 . . . . 5 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝑓:ℕ–onto𝐴 → (vol*‘ 𝑛𝐴 𝐵) = 0))
9796exlimdv 1901 . . . 4 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∃𝑓 𝑓:ℕ–onto𝐴 → (vol*‘ 𝑛𝐴 𝐵) = 0))
9815, 97syld 47 . . 3 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (∅ ≺ 𝐴 → (vol*‘ 𝑛𝐴 𝐵) = 0))
9912, 98sylbird 250 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (𝐴 ≠ ∅ → (vol*‘ 𝑛𝐴 𝐵) = 0))
1007, 99pm2.61dne 2909 1 ((𝐴 ≼ ℕ ∧ ∀𝑛𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) = 0)) → (vol*‘ 𝑛𝐴 𝐵) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  csb 3566  wss 3607  c0 3948  {csn 4210   ciun 4552   class class class wbr 4685  cmpt 4762   × cxp 5141  dom cdm 5143  wf 5922  ontowfo 5924  cfv 5926  cdom 7995  csdm 7996  Fincfn 7997  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  *cxr 10111  cle 10113  cn 11058  cz 11415  cuz 11725  seqcseq 12841  cli 14259  Σcsu 14460  vol*covol 23277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279
This theorem is referenced by: (None)
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