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Theorem ovoliunnfl 36120
Description: ovoliun 24869 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 21-Nov-2017.)
Hypothesis
Ref Expression
ovoliunnfl.0 ((𝑓 Fn ℕ ∧ ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ)) → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ))
Assertion
Ref Expression
ovoliunnfl ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
Distinct variable group:   𝑓,𝑛,𝑚,𝑥,𝐴

Proof of Theorem ovoliunnfl
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 unieq 4876 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4896 . . . . . . . . 9 ∅ = ∅
31, 2eqtrdi 2792 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6846 . . . . . . 7 (𝐴 = ∅ → (vol*‘ 𝐴) = (vol*‘∅))
5 ovol0 24857 . . . . . . 7 (vol*‘∅) = 0
64, 5eqtr2di 2793 . . . . . 6 (𝐴 = ∅ → 0 = (vol*‘ 𝐴))
76a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴)))
8 ovolge0 24845 . . . . . . . 8 ( 𝐴 ⊆ ℝ → 0 ≤ (vol*‘ 𝐴))
98ad2antll 727 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 ≤ (vol*‘ 𝐴))
10 reldom 8889 . . . . . . . . . . . 12 Rel ≼
1110brrelex1i 5688 . . . . . . . . . . 11 (𝐴 ≼ ℕ → 𝐴 ∈ V)
12 0sdomg 9048 . . . . . . . . . . 11 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1311, 12syl 17 . . . . . . . . . 10 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1413biimparc 480 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
15 fodomr 9072 . . . . . . . . 9 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
1614, 15sylancom 588 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
17 unissb 4900 . . . . . . . . . . . 12 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
1817anbi1i 624 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
19 r19.26 3114 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2018, 19bitr4i 277 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
21 brdom2 8922 . . . . . . . . . . . . . 14 (𝑥 ≼ ℕ ↔ (𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ))
22 nnenom 13885 . . . . . . . . . . . . . . . . 17 ℕ ≈ ω
23 sdomen2 9066 . . . . . . . . . . . . . . . . 17 (ℕ ≈ ω → (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω))
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω)
25 isfinite 9588 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
2624, 25bitr4i 277 . . . . . . . . . . . . . . 15 (𝑥 ≺ ℕ ↔ 𝑥 ∈ Fin)
2726orbi1i 912 . . . . . . . . . . . . . 14 ((𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ) ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
2821, 27bitri 274 . . . . . . . . . . . . 13 (𝑥 ≼ ℕ ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
29 ovolfi 24858 . . . . . . . . . . . . . . 15 ((𝑥 ∈ Fin ∧ 𝑥 ⊆ ℝ) → (vol*‘𝑥) = 0)
3029expcom 414 . . . . . . . . . . . . . 14 (𝑥 ⊆ ℝ → (𝑥 ∈ Fin → (vol*‘𝑥) = 0))
31 ovolctb 24854 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ ℝ ∧ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0)
3231ex 413 . . . . . . . . . . . . . 14 (𝑥 ⊆ ℝ → (𝑥 ≈ ℕ → (vol*‘𝑥) = 0))
3330, 32jaod 857 . . . . . . . . . . . . 13 (𝑥 ⊆ ℝ → ((𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0))
3428, 33biimtrid 241 . . . . . . . . . . . 12 (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
3534imdistani 569 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3635ralimi 3086 . . . . . . . . . 10 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3720, 36sylbi 216 . . . . . . . . 9 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3837ancoms 459 . . . . . . . 8 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
39 foima 6761 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴 → (𝑓 “ ℕ) = 𝐴)
4039raleqdv 3313 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
41 fofn 6758 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴𝑓 Fn ℕ)
42 ssid 3966 . . . . . . . . . . . . 13 ℕ ⊆ ℕ
43 sseq1 3969 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑙) → (𝑥 ⊆ ℝ ↔ (𝑓𝑙) ⊆ ℝ))
44 fveqeq2 6851 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑙) → ((vol*‘𝑥) = 0 ↔ (vol*‘(𝑓𝑙)) = 0))
4543, 44anbi12d 631 . . . . . . . . . . . . . 14 (𝑥 = (𝑓𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4645ralima 7188 . . . . . . . . . . . . 13 ((𝑓 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4741, 42, 46sylancl 586 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4840, 47bitr3d 280 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
49 fveq2 6842 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑛 → (𝑓𝑙) = (𝑓𝑛))
5049sseq1d 3975 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑛 → ((𝑓𝑙) ⊆ ℝ ↔ (𝑓𝑛) ⊆ ℝ))
51 2fveq3 6847 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑛 → (vol*‘(𝑓𝑙)) = (vol*‘(𝑓𝑛)))
5251eqeq1d 2738 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑛 → ((vol*‘(𝑓𝑙)) = 0 ↔ (vol*‘(𝑓𝑛)) = 0))
5350, 52anbi12d 631 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑛 → (((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0)))
5453cbvralvw 3225 . . . . . . . . . . . . . . 15 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0))
55 0re 11157 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
56 eleq1a 2833 . . . . . . . . . . . . . . . . . 18 (0 ∈ ℝ → ((vol*‘(𝑓𝑛)) = 0 → (vol*‘(𝑓𝑛)) ∈ ℝ))
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . 17 ((vol*‘(𝑓𝑛)) = 0 → (vol*‘(𝑓𝑛)) ∈ ℝ)
5857anim2i 617 . . . . . . . . . . . . . . . 16 (((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0) → ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
5958ralimi 3086 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0) → ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
6054, 59sylbi 216 . . . . . . . . . . . . . 14 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
61 ovoliunnfl.0 . . . . . . . . . . . . . 14 ((𝑓 Fn ℕ ∧ ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ)) → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ))
6241, 60, 61syl2an 596 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ))
63 fofun 6757 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–onto𝐴 → Fun 𝑓)
64 funiunfv 7195 . . . . . . . . . . . . . . . . 17 (Fun 𝑓 𝑚 ∈ ℕ (𝑓𝑚) = (𝑓 “ ℕ))
6563, 64syl 17 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto𝐴 𝑚 ∈ ℕ (𝑓𝑚) = (𝑓 “ ℕ))
6639unieqd 4879 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto𝐴 (𝑓 “ ℕ) = 𝐴)
6765, 66eqtrd 2776 . . . . . . . . . . . . . . 15 (𝑓:ℕ–onto𝐴 𝑚 ∈ ℕ (𝑓𝑚) = 𝐴)
6867fveq2d 6846 . . . . . . . . . . . . . 14 (𝑓:ℕ–onto𝐴 → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) = (vol*‘ 𝐴))
6968adantr 481 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) = (vol*‘ 𝐴))
70 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑚 → (𝑓𝑙) = (𝑓𝑚))
7170sseq1d 3975 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑚 → ((𝑓𝑙) ⊆ ℝ ↔ (𝑓𝑚) ⊆ ℝ))
72 2fveq3 6847 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑚 → (vol*‘(𝑓𝑙)) = (vol*‘(𝑓𝑚)))
7372eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑚 → ((vol*‘(𝑓𝑙)) = 0 ↔ (vol*‘(𝑓𝑚)) = 0))
7471, 73anbi12d 631 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑚 → (((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ((𝑓𝑚) ⊆ ℝ ∧ (vol*‘(𝑓𝑚)) = 0)))
7574rspccva 3580 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → ((𝑓𝑚) ⊆ ℝ ∧ (vol*‘(𝑓𝑚)) = 0))
7675simprd 496 . . . . . . . . . . . . . . . . . . 19 ((∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓𝑚)) = 0)
7776mpteq2dva 5205 . . . . . . . . . . . . . . . . . 18 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚))) = (𝑚 ∈ ℕ ↦ 0))
7877seqeq3d 13914 . . . . . . . . . . . . . . . . 17 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
7978rneqd 5893 . . . . . . . . . . . . . . . 16 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))) = ran seq1( + , (𝑚 ∈ ℕ ↦ 0)))
8079supeq1d 9382 . . . . . . . . . . . . . . 15 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ))
81 0cn 11147 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
82 ser1const 13964 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℂ ∧ 𝑙 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
8381, 82mpan 688 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
84 nncn 12161 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 ∈ ℕ → 𝑙 ∈ ℂ)
8584mul01d 11354 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 ∈ ℕ → (𝑙 · 0) = 0)
8683, 85eqtrd 2776 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = 0)
8786mpteq2ia 5208 . . . . . . . . . . . . . . . . . . . 20 (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)) = (𝑙 ∈ ℕ ↦ 0)
88 fconstmpt 5694 . . . . . . . . . . . . . . . . . . . . . 22 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
89 seqeq3 13911 . . . . . . . . . . . . . . . . . . . . . 22 ((ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0) → seq1( + , (ℕ × {0})) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
9088, 89ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 seq1( + , (ℕ × {0})) = seq1( + , (𝑚 ∈ ℕ ↦ 0))
91 1z 12533 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℤ
92 seqfn 13918 . . . . . . . . . . . . . . . . . . . . . . 23 (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ‘1))
9391, 92ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 seq1( + , (ℕ × {0})) Fn (ℤ‘1)
94 nnuz 12806 . . . . . . . . . . . . . . . . . . . . . . . 24 ℕ = (ℤ‘1)
9594fneq2i 6600 . . . . . . . . . . . . . . . . . . . . . . 23 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ‘1))
96 dffn5 6901 . . . . . . . . . . . . . . . . . . . . . . 23 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
9795, 96bitr3i 276 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn (ℤ‘1) ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
9893, 97mpbi 229 . . . . . . . . . . . . . . . . . . . . 21 seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
9990, 98eqtr3i 2766 . . . . . . . . . . . . . . . . . . . 20 seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
100 fconstmpt 5694 . . . . . . . . . . . . . . . . . . . 20 (ℕ × {0}) = (𝑙 ∈ ℕ ↦ 0)
10187, 99, 1003eqtr4i 2774 . . . . . . . . . . . . . . . . . . 19 seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (ℕ × {0})
102101rneqi 5892 . . . . . . . . . . . . . . . . . 18 ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
103 1nn 12164 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
104 ne0i 4294 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℕ → ℕ ≠ ∅)
105 rnxp 6122 . . . . . . . . . . . . . . . . . . 19 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
106103, 104, 105mp2b 10 . . . . . . . . . . . . . . . . . 18 ran (ℕ × {0}) = {0}
107102, 106eqtri 2764 . . . . . . . . . . . . . . . . 17 ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = {0}
108107supeq1i 9383 . . . . . . . . . . . . . . . 16 sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
109 xrltso 13060 . . . . . . . . . . . . . . . . 17 < Or ℝ*
110 0xr 11202 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ*
111 supsn 9408 . . . . . . . . . . . . . . . . 17 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
112109, 110, 111mp2an 690 . . . . . . . . . . . . . . . 16 sup({0}, ℝ*, < ) = 0
113108, 112eqtri 2764 . . . . . . . . . . . . . . 15 sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
11480, 113eqtrdi 2792 . . . . . . . . . . . . . 14 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = 0)
115114adantl 482 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = 0)
11662, 69, 1153brtr3d 5136 . . . . . . . . . . . 12 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝐴) ≤ 0)
117116ex 413 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → (vol*‘ 𝐴) ≤ 0))
11848, 117sylbid 239 . . . . . . . . . 10 (𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘ 𝐴) ≤ 0))
119118exlimiv 1933 . . . . . . . . 9 (∃𝑓 𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘ 𝐴) ≤ 0))
120119imp 407 . . . . . . . 8 ((∃𝑓 𝑓:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)) → (vol*‘ 𝐴) ≤ 0)
12116, 38, 120syl2an 596 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → (vol*‘ 𝐴) ≤ 0)
122 ovolcl 24842 . . . . . . . . 9 ( 𝐴 ⊆ ℝ → (vol*‘ 𝐴) ∈ ℝ*)
123 xrletri3 13073 . . . . . . . . 9 ((0 ∈ ℝ* ∧ (vol*‘ 𝐴) ∈ ℝ*) → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
124110, 122, 123sylancr 587 . . . . . . . 8 ( 𝐴 ⊆ ℝ → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
125124ad2antll 727 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
1269, 121, 125mpbir2and 711 . . . . . 6 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴))
127126expl 458 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴)))
1287, 127pm2.61ine 3028 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴))
129 renepnf 11203 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
13055, 129mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
131 fveq2 6842 . . . . . . 7 ( 𝐴 = ℝ → (vol*‘ 𝐴) = (vol*‘ℝ))
132 ovolre 24889 . . . . . . 7 (vol*‘ℝ) = +∞
133131, 132eqtrdi 2792 . . . . . 6 ( 𝐴 = ℝ → (vol*‘ 𝐴) = +∞)
134130, 133neeqtrrd 3018 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol*‘ 𝐴))
135134necon2i 2978 . . . 4 (0 = (vol*‘ 𝐴) → 𝐴 ≠ ℝ)
136128, 135syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
137136expr 457 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
138 eqimss 4000 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
139138necon3bi 2970 . 2 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ)
140137, 139pm2.61d1 180 1 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
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  Vcvv 3445  wss 3910  c0 4282  {csn 4586   cuni 4865   ciun 4954   class class class wbr 5105  cmpt 5188   Or wor 5544   × cxp 5631  ran crn 5634  cima 5636  Fun wfun 6490   Fn wfn 6491  ontowfo 6494  cfv 6496  (class class class)co 7357  ωcom 7802  cen 8880  cdom 8881  csdm 8882  Fincfn 8883  supcsup 9376  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  +∞cpnf 11186  *cxr 11188   < clt 11189  cle 11190  cn 12153  cz 12499  cuz 12763  seqcseq 13906  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-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-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  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-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  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-sum 15571  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-bases 22296  df-cmp 22738  df-ovol 24828
This theorem is referenced by:  ex-ovoliunnfl  36121
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