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Theorem ovoliunnfl 34926
Description: ovoliun 24098 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 4838 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4857 . . . . . . . . 9 ∅ = ∅
31, 2syl6eq 2870 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6667 . . . . . . 7 (𝐴 = ∅ → (vol*‘ 𝐴) = (vol*‘∅))
5 ovol0 24086 . . . . . . 7 (vol*‘∅) = 0
64, 5syl6req 2871 . . . . . 6 (𝐴 = ∅ → 0 = (vol*‘ 𝐴))
76a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴)))
8 ovolge0 24074 . . . . . . . 8 ( 𝐴 ⊆ ℝ → 0 ≤ (vol*‘ 𝐴))
98ad2antll 727 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 ≤ (vol*‘ 𝐴))
10 reldom 8507 . . . . . . . . . . . 12 Rel ≼
1110brrelex1i 5601 . . . . . . . . . . 11 (𝐴 ≼ ℕ → 𝐴 ∈ V)
12 0sdomg 8638 . . . . . . . . . . 11 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1311, 12syl 17 . . . . . . . . . 10 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1413biimparc 482 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
15 fodomr 8660 . . . . . . . . 9 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
1614, 15sylancom 590 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
17 unissb 4861 . . . . . . . . . . . 12 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
1817anbi1i 625 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
19 r19.26 3168 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2018, 19bitr4i 280 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
21 brdom2 8531 . . . . . . . . . . . . . 14 (𝑥 ≼ ℕ ↔ (𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ))
22 nnenom 13340 . . . . . . . . . . . . . . . . 17 ℕ ≈ ω
23 sdomen2 8654 . . . . . . . . . . . . . . . . 17 (ℕ ≈ ω → (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω))
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω)
25 isfinite 9107 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
2624, 25bitr4i 280 . . . . . . . . . . . . . . 15 (𝑥 ≺ ℕ ↔ 𝑥 ∈ Fin)
2726orbi1i 910 . . . . . . . . . . . . . 14 ((𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ) ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
2821, 27bitri 277 . . . . . . . . . . . . 13 (𝑥 ≼ ℕ ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
29 ovolfi 24087 . . . . . . . . . . . . . . 15 ((𝑥 ∈ Fin ∧ 𝑥 ⊆ ℝ) → (vol*‘𝑥) = 0)
3029expcom 416 . . . . . . . . . . . . . 14 (𝑥 ⊆ ℝ → (𝑥 ∈ Fin → (vol*‘𝑥) = 0))
31 ovolctb 24083 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ ℝ ∧ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0)
3231ex 415 . . . . . . . . . . . . . 14 (𝑥 ⊆ ℝ → (𝑥 ≈ ℕ → (vol*‘𝑥) = 0))
3330, 32jaod 855 . . . . . . . . . . . . 13 (𝑥 ⊆ ℝ → ((𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0))
3428, 33syl5bi 244 . . . . . . . . . . . 12 (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
3534imdistani 571 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3635ralimi 3158 . . . . . . . . . 10 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3720, 36sylbi 219 . . . . . . . . 9 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3837ancoms 461 . . . . . . . 8 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
39 foima 6588 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴 → (𝑓 “ ℕ) = 𝐴)
4039raleqdv 3414 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
41 fofn 6585 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴𝑓 Fn ℕ)
42 ssid 3987 . . . . . . . . . . . . 13 ℕ ⊆ ℕ
43 sseq1 3990 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑙) → (𝑥 ⊆ ℝ ↔ (𝑓𝑙) ⊆ ℝ))
44 fveqeq2 6672 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑙) → ((vol*‘𝑥) = 0 ↔ (vol*‘(𝑓𝑙)) = 0))
4543, 44anbi12d 632 . . . . . . . . . . . . . 14 (𝑥 = (𝑓𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4645ralima 6992 . . . . . . . . . . . . 13 ((𝑓 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4741, 42, 46sylancl 588 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4840, 47bitr3d 283 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
49 fveq2 6663 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑛 → (𝑓𝑙) = (𝑓𝑛))
5049sseq1d 3996 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑛 → ((𝑓𝑙) ⊆ ℝ ↔ (𝑓𝑛) ⊆ ℝ))
51 2fveq3 6668 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑛 → (vol*‘(𝑓𝑙)) = (vol*‘(𝑓𝑛)))
5251eqeq1d 2821 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑛 → ((vol*‘(𝑓𝑙)) = 0 ↔ (vol*‘(𝑓𝑛)) = 0))
5350, 52anbi12d 632 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑛 → (((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0)))
5453cbvralvw 3448 . . . . . . . . . . . . . . 15 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0))
55 0re 10635 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
56 eleq1a 2906 . . . . . . . . . . . . . . . . . 18 (0 ∈ ℝ → ((vol*‘(𝑓𝑛)) = 0 → (vol*‘(𝑓𝑛)) ∈ ℝ))
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . 17 ((vol*‘(𝑓𝑛)) = 0 → (vol*‘(𝑓𝑛)) ∈ ℝ)
5857anim2i 618 . . . . . . . . . . . . . . . 16 (((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0) → ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
5958ralimi 3158 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0) → ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
6054, 59sylbi 219 . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . 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 6584 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–onto𝐴 → Fun 𝑓)
64 funiunfv 6999 . . . . . . . . . . . . . . . . 17 (Fun 𝑓 𝑚 ∈ ℕ (𝑓𝑚) = (𝑓 “ ℕ))
6563, 64syl 17 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto𝐴 𝑚 ∈ ℕ (𝑓𝑚) = (𝑓 “ ℕ))
6639unieqd 4840 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto𝐴 (𝑓 “ ℕ) = 𝐴)
6765, 66eqtrd 2854 . . . . . . . . . . . . . . 15 (𝑓:ℕ–onto𝐴 𝑚 ∈ ℕ (𝑓𝑚) = 𝐴)
6867fveq2d 6667 . . . . . . . . . . . . . 14 (𝑓:ℕ–onto𝐴 → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) = (vol*‘ 𝐴))
6968adantr 483 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) = (vol*‘ 𝐴))
70 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑚 → (𝑓𝑙) = (𝑓𝑚))
7170sseq1d 3996 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑚 → ((𝑓𝑙) ⊆ ℝ ↔ (𝑓𝑚) ⊆ ℝ))
72 2fveq3 6668 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑚 → (vol*‘(𝑓𝑙)) = (vol*‘(𝑓𝑚)))
7372eqeq1d 2821 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑚 → ((vol*‘(𝑓𝑙)) = 0 ↔ (vol*‘(𝑓𝑚)) = 0))
7471, 73anbi12d 632 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑚 → (((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ((𝑓𝑚) ⊆ ℝ ∧ (vol*‘(𝑓𝑚)) = 0)))
7574rspccva 3620 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → ((𝑓𝑚) ⊆ ℝ ∧ (vol*‘(𝑓𝑚)) = 0))
7675simprd 498 . . . . . . . . . . . . . . . . . . 19 ((∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓𝑚)) = 0)
7776mpteq2dva 5152 . . . . . . . . . . . . . . . . . 18 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚))) = (𝑚 ∈ ℕ ↦ 0))
7877seqeq3d 13369 . . . . . . . . . . . . . . . . 17 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
7978rneqd 5801 . . . . . . . . . . . . . . . 16 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))) = ran seq1( + , (𝑚 ∈ ℕ ↦ 0)))
8079supeq1d 8902 . . . . . . . . . . . . . . 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 10625 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
82 ser1const 13418 . . . . . . . . . . . . . . . . . . . . . . 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 11638 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 ∈ ℕ → 𝑙 ∈ ℂ)
8584mul01d 10831 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 ∈ ℕ → (𝑙 · 0) = 0)
8683, 85eqtrd 2854 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = 0)
8786mpteq2ia 5148 . . . . . . . . . . . . . . . . . . . 20 (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)) = (𝑙 ∈ ℕ ↦ 0)
88 fconstmpt 5607 . . . . . . . . . . . . . . . . . . . . . 22 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
89 seqeq3 13366 . . . . . . . . . . . . . . . . . . . . . 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 12004 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℤ
92 seqfn 13373 . . . . . . . . . . . . . . . . . . . . . . 23 (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ‘1))
9391, 92ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 seq1( + , (ℕ × {0})) Fn (ℤ‘1)
94 nnuz 12273 . . . . . . . . . . . . . . . . . . . . . . . 24 ℕ = (ℤ‘1)
9594fneq2i 6444 . . . . . . . . . . . . . . . . . . . . . . 23 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ‘1))
96 dffn5 6717 . . . . . . . . . . . . . . . . . . . . . . 23 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
9795, 96bitr3i 279 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn (ℤ‘1) ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
9893, 97mpbi 232 . . . . . . . . . . . . . . . . . . . . 21 seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
9990, 98eqtr3i 2844 . . . . . . . . . . . . . . . . . . . 20 seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
100 fconstmpt 5607 . . . . . . . . . . . . . . . . . . . 20 (ℕ × {0}) = (𝑙 ∈ ℕ ↦ 0)
10187, 99, 1003eqtr4i 2852 . . . . . . . . . . . . . . . . . . 19 seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (ℕ × {0})
102101rneqi 5800 . . . . . . . . . . . . . . . . . 18 ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
103 1nn 11641 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
104 ne0i 4298 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℕ → ℕ ≠ ∅)
105 rnxp 6020 . . . . . . . . . . . . . . . . . . 19 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
106103, 104, 105mp2b 10 . . . . . . . . . . . . . . . . . 18 ran (ℕ × {0}) = {0}
107102, 106eqtri 2842 . . . . . . . . . . . . . . . . 17 ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = {0}
108107supeq1i 8903 . . . . . . . . . . . . . . . 16 sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
109 xrltso 12526 . . . . . . . . . . . . . . . . 17 < Or ℝ*
110 0xr 10680 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ*
111 supsn 8928 . . . . . . . . . . . . . . . . 17 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
112109, 110, 111mp2an 690 . . . . . . . . . . . . . . . 16 sup({0}, ℝ*, < ) = 0
113108, 112eqtri 2842 . . . . . . . . . . . . . . 15 sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
11480, 113syl6eq 2870 . . . . . . . . . . . . . 14 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = 0)
115114adantl 484 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = 0)
11662, 69, 1153brtr3d 5088 . . . . . . . . . . . 12 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝐴) ≤ 0)
117116ex 415 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → (vol*‘ 𝐴) ≤ 0))
11848, 117sylbid 242 . . . . . . . . . 10 (𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘ 𝐴) ≤ 0))
119118exlimiv 1925 . . . . . . . . 9 (∃𝑓 𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘ 𝐴) ≤ 0))
120119imp 409 . . . . . . . 8 ((∃𝑓 𝑓:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)) → (vol*‘ 𝐴) ≤ 0)
12116, 38, 120syl2an 597 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → (vol*‘ 𝐴) ≤ 0)
122 ovolcl 24071 . . . . . . . . 9 ( 𝐴 ⊆ ℝ → (vol*‘ 𝐴) ∈ ℝ*)
123 xrletri3 12539 . . . . . . . . 9 ((0 ∈ ℝ* ∧ (vol*‘ 𝐴) ∈ ℝ*) → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
124110, 122, 123sylancr 589 . . . . . . . 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 460 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴)))
1287, 127pm2.61ine 3098 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴))
129 renepnf 10681 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
13055, 129mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
131 fveq2 6663 . . . . . . 7 ( 𝐴 = ℝ → (vol*‘ 𝐴) = (vol*‘ℝ))
132 ovolre 24118 . . . . . . 7 (vol*‘ℝ) = +∞
133131, 132syl6eq 2870 . . . . . 6 ( 𝐴 = ℝ → (vol*‘ 𝐴) = +∞)
134130, 133neeqtrrd 3088 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol*‘ 𝐴))
135134necon2i 3048 . . . 4 (0 = (vol*‘ 𝐴) → 𝐴 ≠ ℝ)
136128, 135syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
137136expr 459 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
138 eqimss 4021 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
139138necon3bi 3040 . 2 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ)
140137, 139pm2.61d1 182 1 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843   = wceq 1531  wex 1774  wcel 2108  wne 3014  wral 3136  Vcvv 3493  wss 3934  c0 4289  {csn 4559   cuni 4830   ciun 4910   class class class wbr 5057  cmpt 5137   Or wor 5466   × cxp 5546  ran crn 5549  cima 5551  Fun wfun 6342   Fn wfn 6343  ontowfo 6346  cfv 6348  (class class class)co 7148  ωcom 7572  cen 8498  cdom 8499  csdm 8500  Fincfn 8501  supcsup 8896  cc 10527  cr 10528  0cc0 10529  1c1 10530   + caddc 10532   · cmul 10534  +∞cpnf 10664  *cxr 10666   < clt 10667  cle 10668  cn 11630  cz 11973  cuz 12235  seqcseq 13361  vol*covol 24055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-seq 13362  df-exp 13422  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-clim 14837  df-sum 15035  df-rest 16688  df-topgen 16709  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-top 21494  df-topon 21511  df-bases 21546  df-cmp 21987  df-ovol 24057
This theorem is referenced by:  ex-ovoliunnfl  34927
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