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Theorem ovoliunnfl 33766
Description: ovoliun 23492 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 4645 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4666 . . . . . . . . 9 ∅ = ∅
31, 2syl6eq 2863 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6415 . . . . . . 7 (𝐴 = ∅ → (vol*‘ 𝐴) = (vol*‘∅))
5 ovol0 23480 . . . . . . 7 (vol*‘∅) = 0
64, 5syl6req 2864 . . . . . 6 (𝐴 = ∅ → 0 = (vol*‘ 𝐴))
76a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴)))
8 ovolge0 23468 . . . . . . . 8 ( 𝐴 ⊆ ℝ → 0 ≤ (vol*‘ 𝐴))
98ad2antll 711 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 ≤ (vol*‘ 𝐴))
10 reldom 8201 . . . . . . . . . . . 12 Rel ≼
1110brrelexi 5365 . . . . . . . . . . 11 (𝐴 ≼ ℕ → 𝐴 ∈ V)
12 0sdomg 8331 . . . . . . . . . . 11 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1311, 12syl 17 . . . . . . . . . 10 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1413biimparc 467 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
15 fodomr 8353 . . . . . . . . 9 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
1614, 15sylancom 578 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
17 unissb 4670 . . . . . . . . . . . 12 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
1817anbi1i 612 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
19 r19.26 3259 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2018, 19bitr4i 269 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
21 brdom2 8225 . . . . . . . . . . . . . 14 (𝑥 ≼ ℕ ↔ (𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ))
22 nnenom 13006 . . . . . . . . . . . . . . . . 17 ℕ ≈ ω
23 sdomen2 8347 . . . . . . . . . . . . . . . . 17 (ℕ ≈ ω → (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω))
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω)
25 isfinite 8799 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
2624, 25bitr4i 269 . . . . . . . . . . . . . . 15 (𝑥 ≺ ℕ ↔ 𝑥 ∈ Fin)
2726orbi1i 928 . . . . . . . . . . . . . 14 ((𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ) ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
2821, 27bitri 266 . . . . . . . . . . . . 13 (𝑥 ≼ ℕ ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
29 ovolfi 23481 . . . . . . . . . . . . . . 15 ((𝑥 ∈ Fin ∧ 𝑥 ⊆ ℝ) → (vol*‘𝑥) = 0)
3029expcom 400 . . . . . . . . . . . . . 14 (𝑥 ⊆ ℝ → (𝑥 ∈ Fin → (vol*‘𝑥) = 0))
31 ovolctb 23477 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ ℝ ∧ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0)
3231ex 399 . . . . . . . . . . . . . 14 (𝑥 ⊆ ℝ → (𝑥 ≈ ℕ → (vol*‘𝑥) = 0))
3330, 32jaod 877 . . . . . . . . . . . . 13 (𝑥 ⊆ ℝ → ((𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0))
3428, 33syl5bi 233 . . . . . . . . . . . 12 (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
3534imdistani 560 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3635ralimi 3147 . . . . . . . . . 10 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3720, 36sylbi 208 . . . . . . . . 9 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3837ancoms 448 . . . . . . . 8 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
39 foima 6339 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴 → (𝑓 “ ℕ) = 𝐴)
4039raleqdv 3340 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
41 fofn 6336 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴𝑓 Fn ℕ)
42 ssid 3827 . . . . . . . . . . . . 13 ℕ ⊆ ℕ
43 sseq1 3830 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑙) → (𝑥 ⊆ ℝ ↔ (𝑓𝑙) ⊆ ℝ))
44 fveqeq2 6420 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑙) → ((vol*‘𝑥) = 0 ↔ (vol*‘(𝑓𝑙)) = 0))
4543, 44anbi12d 618 . . . . . . . . . . . . . 14 (𝑥 = (𝑓𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4645ralima 6726 . . . . . . . . . . . . 13 ((𝑓 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4741, 42, 46sylancl 576 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4840, 47bitr3d 272 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
49 fveq2 6411 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑛 → (𝑓𝑙) = (𝑓𝑛))
5049sseq1d 3836 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑛 → ((𝑓𝑙) ⊆ ℝ ↔ (𝑓𝑛) ⊆ ℝ))
51 2fveq3 6416 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑛 → (vol*‘(𝑓𝑙)) = (vol*‘(𝑓𝑛)))
5251eqeq1d 2815 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑛 → ((vol*‘(𝑓𝑙)) = 0 ↔ (vol*‘(𝑓𝑛)) = 0))
5350, 52anbi12d 618 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑛 → (((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0)))
5453cbvralv 3367 . . . . . . . . . . . . . . 15 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0))
55 0re 10330 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
56 eleq1a 2887 . . . . . . . . . . . . . . . . . 18 (0 ∈ ℝ → ((vol*‘(𝑓𝑛)) = 0 → (vol*‘(𝑓𝑛)) ∈ ℝ))
5755, 56ax-mp 5 . . . . . . . . . . . . . . . . 17 ((vol*‘(𝑓𝑛)) = 0 → (vol*‘(𝑓𝑛)) ∈ ℝ)
5857anim2i 605 . . . . . . . . . . . . . . . 16 (((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0) → ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
5958ralimi 3147 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0) → ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
6054, 59sylbi 208 . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . 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 6335 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–onto𝐴 → Fun 𝑓)
64 funiunfv 6733 . . . . . . . . . . . . . . . . 17 (Fun 𝑓 𝑚 ∈ ℕ (𝑓𝑚) = (𝑓 “ ℕ))
6563, 64syl 17 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto𝐴 𝑚 ∈ ℕ (𝑓𝑚) = (𝑓 “ ℕ))
6639unieqd 4647 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto𝐴 (𝑓 “ ℕ) = 𝐴)
6765, 66eqtrd 2847 . . . . . . . . . . . . . . 15 (𝑓:ℕ–onto𝐴 𝑚 ∈ ℕ (𝑓𝑚) = 𝐴)
6867fveq2d 6415 . . . . . . . . . . . . . 14 (𝑓:ℕ–onto𝐴 → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) = (vol*‘ 𝐴))
6968adantr 468 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) = (vol*‘ 𝐴))
70 fveq2 6411 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑚 → (𝑓𝑙) = (𝑓𝑚))
7170sseq1d 3836 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑚 → ((𝑓𝑙) ⊆ ℝ ↔ (𝑓𝑚) ⊆ ℝ))
72 2fveq3 6416 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑚 → (vol*‘(𝑓𝑙)) = (vol*‘(𝑓𝑚)))
7372eqeq1d 2815 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑚 → ((vol*‘(𝑓𝑙)) = 0 ↔ (vol*‘(𝑓𝑚)) = 0))
7471, 73anbi12d 618 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑚 → (((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ((𝑓𝑚) ⊆ ℝ ∧ (vol*‘(𝑓𝑚)) = 0)))
7574rspccva 3508 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → ((𝑓𝑚) ⊆ ℝ ∧ (vol*‘(𝑓𝑚)) = 0))
7675simprd 485 . . . . . . . . . . . . . . . . . . 19 ((∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓𝑚)) = 0)
7776mpteq2dva 4945 . . . . . . . . . . . . . . . . . 18 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚))) = (𝑚 ∈ ℕ ↦ 0))
7877seqeq3d 13035 . . . . . . . . . . . . . . . . 17 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
7978rneqd 5561 . . . . . . . . . . . . . . . 16 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))) = ran seq1( + , (𝑚 ∈ ℕ ↦ 0)))
8079supeq1d 8594 . . . . . . . . . . . . . . 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 10320 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
82 ser1const 13083 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℂ ∧ 𝑙 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
8381, 82mpan 673 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
84 nncn 11316 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 ∈ ℕ → 𝑙 ∈ ℂ)
8584mul01d 10523 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 ∈ ℕ → (𝑙 · 0) = 0)
8683, 85eqtrd 2847 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = 0)
8786mpteq2ia 4941 . . . . . . . . . . . . . . . . . . . 20 (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)) = (𝑙 ∈ ℕ ↦ 0)
88 fconstmpt 5370 . . . . . . . . . . . . . . . . . . . . . 22 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
89 seqeq3 13032 . . . . . . . . . . . . . . . . . . . . . 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 11676 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℤ
92 seqfn 13039 . . . . . . . . . . . . . . . . . . . . . . 23 (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ‘1))
9391, 92ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 seq1( + , (ℕ × {0})) Fn (ℤ‘1)
94 nnuz 11944 . . . . . . . . . . . . . . . . . . . . . . . 24 ℕ = (ℤ‘1)
9594fneq2i 6200 . . . . . . . . . . . . . . . . . . . . . . 23 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ‘1))
96 dffn5 6465 . . . . . . . . . . . . . . . . . . . . . . 23 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
9795, 96bitr3i 268 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn (ℤ‘1) ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
9893, 97mpbi 221 . . . . . . . . . . . . . . . . . . . . 21 seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
9990, 98eqtr3i 2837 . . . . . . . . . . . . . . . . . . . 20 seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
100 fconstmpt 5370 . . . . . . . . . . . . . . . . . . . 20 (ℕ × {0}) = (𝑙 ∈ ℕ ↦ 0)
10187, 99, 1003eqtr4i 2845 . . . . . . . . . . . . . . . . . . 19 seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (ℕ × {0})
102101rneqi 5560 . . . . . . . . . . . . . . . . . 18 ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
103 1nn 11319 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
104 ne0i 4129 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℕ → ℕ ≠ ∅)
105 rnxp 5782 . . . . . . . . . . . . . . . . . . 19 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
106103, 104, 105mp2b 10 . . . . . . . . . . . . . . . . . 18 ran (ℕ × {0}) = {0}
107102, 106eqtri 2835 . . . . . . . . . . . . . . . . 17 ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = {0}
108107supeq1i 8595 . . . . . . . . . . . . . . . 16 sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
109 xrltso 12193 . . . . . . . . . . . . . . . . 17 < Or ℝ*
110 0xr 10374 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ*
111 supsn 8620 . . . . . . . . . . . . . . . . 17 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
112109, 110, 111mp2an 675 . . . . . . . . . . . . . . . 16 sup({0}, ℝ*, < ) = 0
113108, 112eqtri 2835 . . . . . . . . . . . . . . 15 sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
11480, 113syl6eq 2863 . . . . . . . . . . . . . 14 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = 0)
115114adantl 469 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = 0)
11662, 69, 1153brtr3d 4882 . . . . . . . . . . . 12 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝐴) ≤ 0)
117116ex 399 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → (vol*‘ 𝐴) ≤ 0))
11848, 117sylbid 231 . . . . . . . . . 10 (𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘ 𝐴) ≤ 0))
119118exlimiv 2021 . . . . . . . . 9 (∃𝑓 𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘ 𝐴) ≤ 0))
120119imp 395 . . . . . . . 8 ((∃𝑓 𝑓:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)) → (vol*‘ 𝐴) ≤ 0)
12116, 38, 120syl2an 585 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → (vol*‘ 𝐴) ≤ 0)
122 ovolcl 23465 . . . . . . . . 9 ( 𝐴 ⊆ ℝ → (vol*‘ 𝐴) ∈ ℝ*)
123 xrletri3 12206 . . . . . . . . 9 ((0 ∈ ℝ* ∧ (vol*‘ 𝐴) ∈ ℝ*) → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
124110, 122, 123sylancr 577 . . . . . . . 8 ( 𝐴 ⊆ ℝ → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
125124ad2antll 711 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
1269, 121, 125mpbir2and 695 . . . . . 6 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴))
127126expl 447 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴)))
1287, 127pm2.61ine 3068 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴))
129 renepnf 10375 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
13055, 129mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
131 fveq2 6411 . . . . . . 7 ( 𝐴 = ℝ → (vol*‘ 𝐴) = (vol*‘ℝ))
132 ovolre 23512 . . . . . . 7 (vol*‘ℝ) = +∞
133131, 132syl6eq 2863 . . . . . 6 ( 𝐴 = ℝ → (vol*‘ 𝐴) = +∞)
134130, 133neeqtrrd 3059 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol*‘ 𝐴))
135134necon2i 3019 . . . 4 (0 = (vol*‘ 𝐴) → 𝐴 ≠ ℝ)
136128, 135syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
137136expr 446 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
138 eqimss 3861 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
139138necon3bi 3011 . 2 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ)
140137, 139pm2.61d1 172 1 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wex 1859  wcel 2157  wne 2985  wral 3103  Vcvv 3398  wss 3776  c0 4123  {csn 4377   cuni 4637   ciun 4719   class class class wbr 4851  cmpt 4930   Or wor 5238   × cxp 5316  ran crn 5319  cima 5321  Fun wfun 6098   Fn wfn 6099  ontowfo 6102  cfv 6104  (class class class)co 6877  ωcom 7298  cen 8192  cdom 8193  csdm 8194  Fincfn 8195  supcsup 8588  cc 10222  cr 10223  0cc0 10224  1c1 10225   + caddc 10227   · cmul 10229  +∞cpnf 10359  *cxr 10361   < clt 10362  cle 10363  cn 11308  cz 11646  cuz 11907  seqcseq 13027  vol*covol 23449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-inf2 8788  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-of 7130  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-fi 8559  df-sup 8590  df-inf 8591  df-oi 8657  df-card 9051  df-cda 9278  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-div 10973  df-nn 11309  df-2 11367  df-3 11368  df-n0 11563  df-z 11647  df-uz 11908  df-q 12011  df-rp 12050  df-xneg 12165  df-xadd 12166  df-xmul 12167  df-ioo 12400  df-ico 12402  df-icc 12403  df-fz 12553  df-fzo 12693  df-seq 13028  df-exp 13087  df-hash 13341  df-cj 14065  df-re 14066  df-im 14067  df-sqrt 14201  df-abs 14202  df-clim 14445  df-sum 14643  df-rest 16291  df-topgen 16312  df-psmet 19949  df-xmet 19950  df-met 19951  df-bl 19952  df-mopn 19953  df-top 20916  df-topon 20933  df-bases 20968  df-cmp 21408  df-ovol 23451
This theorem is referenced by:  ex-ovoliunnfl  33767
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