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Theorem ovoliunnfl 37669
Description: ovoliun 25540 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 4918 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4935 . . . . . . . . 9 ∅ = ∅
31, 2eqtrdi 2793 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6910 . . . . . . 7 (𝐴 = ∅ → (vol*‘ 𝐴) = (vol*‘∅))
5 ovol0 25528 . . . . . . 7 (vol*‘∅) = 0
64, 5eqtr2di 2794 . . . . . 6 (𝐴 = ∅ → 0 = (vol*‘ 𝐴))
76a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴)))
8 ovolge0 25516 . . . . . . . 8 ( 𝐴 ⊆ ℝ → 0 ≤ (vol*‘ 𝐴))
98ad2antll 729 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 ≤ (vol*‘ 𝐴))
10 reldom 8991 . . . . . . . . . . . 12 Rel ≼
1110brrelex1i 5741 . . . . . . . . . . 11 (𝐴 ≼ ℕ → 𝐴 ∈ V)
12 0sdomg 9144 . . . . . . . . . . 11 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1311, 12syl 17 . . . . . . . . . 10 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1413biimparc 479 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
15 fodomr 9168 . . . . . . . . 9 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
1614, 15sylancom 588 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
17 unissb 4939 . . . . . . . . . . . 12 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
1817anbi1i 624 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
19 r19.26 3111 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2018, 19bitr4i 278 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
21 brdom2 9022 . . . . . . . . . . . . . 14 (𝑥 ≼ ℕ ↔ (𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ))
22 nnenom 14021 . . . . . . . . . . . . . . . . 17 ℕ ≈ ω
23 sdomen2 9162 . . . . . . . . . . . . . . . . 17 (ℕ ≈ ω → (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω))
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω)
25 isfinite 9692 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
2624, 25bitr4i 278 . . . . . . . . . . . . . . 15 (𝑥 ≺ ℕ ↔ 𝑥 ∈ Fin)
2726orbi1i 914 . . . . . . . . . . . . . 14 ((𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ) ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
2821, 27bitri 275 . . . . . . . . . . . . 13 (𝑥 ≼ ℕ ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
29 ovolfi 25529 . . . . . . . . . . . . . . 15 ((𝑥 ∈ Fin ∧ 𝑥 ⊆ ℝ) → (vol*‘𝑥) = 0)
3029expcom 413 . . . . . . . . . . . . . 14 (𝑥 ⊆ ℝ → (𝑥 ∈ Fin → (vol*‘𝑥) = 0))
31 ovolctb 25525 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ ℝ ∧ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0)
3231ex 412 . . . . . . . . . . . . . 14 (𝑥 ⊆ ℝ → (𝑥 ≈ ℕ → (vol*‘𝑥) = 0))
3330, 32jaod 860 . . . . . . . . . . . . 13 (𝑥 ⊆ ℝ → ((𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0))
3428, 33biimtrid 242 . . . . . . . . . . . 12 (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
3534imdistani 568 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3635ralimi 3083 . . . . . . . . . 10 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3720, 36sylbi 217 . . . . . . . . 9 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3837ancoms 458 . . . . . . . 8 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
39 foima 6825 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴 → (𝑓 “ ℕ) = 𝐴)
4039raleqdv 3326 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
41 fofn 6822 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴𝑓 Fn ℕ)
42 ssid 4006 . . . . . . . . . . . . 13 ℕ ⊆ ℕ
43 sseq1 4009 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑙) → (𝑥 ⊆ ℝ ↔ (𝑓𝑙) ⊆ ℝ))
44 fveqeq2 6915 . . . . . . . . . . . . . . 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 7257 . . . . . . . . . . . . 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 281 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
49 fveq2 6906 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑛 → (𝑓𝑙) = (𝑓𝑛))
5049sseq1d 4015 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑛 → ((𝑓𝑙) ⊆ ℝ ↔ (𝑓𝑛) ⊆ ℝ))
51 2fveq3 6911 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑛 → (vol*‘(𝑓𝑙)) = (vol*‘(𝑓𝑛)))
5251eqeq1d 2739 . . . . . . . . . . . . . . . . 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 3237 . . . . . . . . . . . . . . 15 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0))
55 0re 11263 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
56 eleq1a 2836 . . . . . . . . . . . . . . . . . 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 3083 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0) → ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
6054, 59sylbi 217 . . . . . . . . . . . . . 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 6821 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–onto𝐴 → Fun 𝑓)
64 funiunfv 7268 . . . . . . . . . . . . . . . . 17 (Fun 𝑓 𝑚 ∈ ℕ (𝑓𝑚) = (𝑓 “ ℕ))
6563, 64syl 17 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto𝐴 𝑚 ∈ ℕ (𝑓𝑚) = (𝑓 “ ℕ))
6639unieqd 4920 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto𝐴 (𝑓 “ ℕ) = 𝐴)
6765, 66eqtrd 2777 . . . . . . . . . . . . . . 15 (𝑓:ℕ–onto𝐴 𝑚 ∈ ℕ (𝑓𝑚) = 𝐴)
6867fveq2d 6910 . . . . . . . . . . . . . 14 (𝑓:ℕ–onto𝐴 → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) = (vol*‘ 𝐴))
6968adantr 480 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) = (vol*‘ 𝐴))
70 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑚 → (𝑓𝑙) = (𝑓𝑚))
7170sseq1d 4015 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑚 → ((𝑓𝑙) ⊆ ℝ ↔ (𝑓𝑚) ⊆ ℝ))
72 2fveq3 6911 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑚 → (vol*‘(𝑓𝑙)) = (vol*‘(𝑓𝑚)))
7372eqeq1d 2739 . . . . . . . . . . . . . . . . . . . . . 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 3621 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → ((𝑓𝑚) ⊆ ℝ ∧ (vol*‘(𝑓𝑚)) = 0))
7675simprd 495 . . . . . . . . . . . . . . . . . . 19 ((∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓𝑚)) = 0)
7776mpteq2dva 5242 . . . . . . . . . . . . . . . . . 18 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚))) = (𝑚 ∈ ℕ ↦ 0))
7877seqeq3d 14050 . . . . . . . . . . . . . . . . 17 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
7978rneqd 5949 . . . . . . . . . . . . . . . 16 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))) = ran seq1( + , (𝑚 ∈ ℕ ↦ 0)))
8079supeq1d 9486 . . . . . . . . . . . . . . 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 11253 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
82 ser1const 14099 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℂ ∧ 𝑙 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
8381, 82mpan 690 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
84 nncn 12274 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 ∈ ℕ → 𝑙 ∈ ℂ)
8584mul01d 11460 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 ∈ ℕ → (𝑙 · 0) = 0)
8683, 85eqtrd 2777 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = 0)
8786mpteq2ia 5245 . . . . . . . . . . . . . . . . . . . 20 (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)) = (𝑙 ∈ ℕ ↦ 0)
88 fconstmpt 5747 . . . . . . . . . . . . . . . . . . . . . 22 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
89 seqeq3 14047 . . . . . . . . . . . . . . . . . . . . . 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 12647 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℤ
92 seqfn 14054 . . . . . . . . . . . . . . . . . . . . . . 23 (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ‘1))
9391, 92ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 seq1( + , (ℕ × {0})) Fn (ℤ‘1)
94 nnuz 12921 . . . . . . . . . . . . . . . . . . . . . . . 24 ℕ = (ℤ‘1)
9594fneq2i 6666 . . . . . . . . . . . . . . . . . . . . . . 23 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ‘1))
96 dffn5 6967 . . . . . . . . . . . . . . . . . . . . . . 23 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
9795, 96bitr3i 277 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn (ℤ‘1) ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
9893, 97mpbi 230 . . . . . . . . . . . . . . . . . . . . 21 seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
9990, 98eqtr3i 2767 . . . . . . . . . . . . . . . . . . . 20 seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
100 fconstmpt 5747 . . . . . . . . . . . . . . . . . . . 20 (ℕ × {0}) = (𝑙 ∈ ℕ ↦ 0)
10187, 99, 1003eqtr4i 2775 . . . . . . . . . . . . . . . . . . 19 seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (ℕ × {0})
102101rneqi 5948 . . . . . . . . . . . . . . . . . 18 ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
103 1nn 12277 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
104 ne0i 4341 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℕ → ℕ ≠ ∅)
105 rnxp 6190 . . . . . . . . . . . . . . . . . . 19 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
106103, 104, 105mp2b 10 . . . . . . . . . . . . . . . . . 18 ran (ℕ × {0}) = {0}
107102, 106eqtri 2765 . . . . . . . . . . . . . . . . 17 ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = {0}
108107supeq1i 9487 . . . . . . . . . . . . . . . 16 sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
109 xrltso 13183 . . . . . . . . . . . . . . . . 17 < Or ℝ*
110 0xr 11308 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ*
111 supsn 9512 . . . . . . . . . . . . . . . . 17 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
112109, 110, 111mp2an 692 . . . . . . . . . . . . . . . 16 sup({0}, ℝ*, < ) = 0
113108, 112eqtri 2765 . . . . . . . . . . . . . . 15 sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
11480, 113eqtrdi 2793 . . . . . . . . . . . . . 14 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = 0)
115114adantl 481 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = 0)
11662, 69, 1153brtr3d 5174 . . . . . . . . . . . 12 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝐴) ≤ 0)
117116ex 412 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → (vol*‘ 𝐴) ≤ 0))
11848, 117sylbid 240 . . . . . . . . . 10 (𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘ 𝐴) ≤ 0))
119118exlimiv 1930 . . . . . . . . 9 (∃𝑓 𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘ 𝐴) ≤ 0))
120119imp 406 . . . . . . . 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 25513 . . . . . . . . 9 ( 𝐴 ⊆ ℝ → (vol*‘ 𝐴) ∈ ℝ*)
123 xrletri3 13196 . . . . . . . . 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 729 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
1269, 121, 125mpbir2and 713 . . . . . 6 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴))
127126expl 457 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴)))
1287, 127pm2.61ine 3025 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴))
129 renepnf 11309 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
13055, 129mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
131 fveq2 6906 . . . . . . 7 ( 𝐴 = ℝ → (vol*‘ 𝐴) = (vol*‘ℝ))
132 ovolre 25560 . . . . . . 7 (vol*‘ℝ) = +∞
133131, 132eqtrdi 2793 . . . . . 6 ( 𝐴 = ℝ → (vol*‘ 𝐴) = +∞)
134130, 133neeqtrrd 3015 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol*‘ 𝐴))
135134necon2i 2975 . . . 4 (0 = (vol*‘ 𝐴) → 𝐴 ≠ ℝ)
136128, 135syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
137136expr 456 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
138 eqimss 4042 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
139138necon3bi 2967 . 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 206  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  Vcvv 3480  wss 3951  c0 4333  {csn 4626   cuni 4907   ciun 4991   class class class wbr 5143  cmpt 5225   Or wor 5591   × cxp 5683  ran crn 5686  cima 5688  Fun wfun 6555   Fn wfn 6556  ontowfo 6559  cfv 6561  (class class class)co 7431  ωcom 7887  cen 8982  cdom 8983  csdm 8984  Fincfn 8985  supcsup 9480  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cn 12266  cz 12613  cuz 12878  seqcseq 14042  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-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-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  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-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  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-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499
This theorem is referenced by:  ex-ovoliunnfl  37670
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