Mathbox for Brendan Leahy < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovoliunnfl Structured version   Visualization version   GIF version

Theorem ovoliunnfl 32621
 Description: ovoliun 23080 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 4380 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = ∅)
2 uni0 4401 . . . . . . . . 9 ∅ = ∅
31, 2syl6eq 2660 . . . . . . . 8 (𝐴 = ∅ → 𝐴 = ∅)
43fveq2d 6107 . . . . . . 7 (𝐴 = ∅ → (vol*‘ 𝐴) = (vol*‘∅))
5 ovol0 23068 . . . . . . 7 (vol*‘∅) = 0
64, 5syl6req 2661 . . . . . 6 (𝐴 = ∅ → 0 = (vol*‘ 𝐴))
76a1d 25 . . . . 5 (𝐴 = ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴)))
8 ovolge0 23056 . . . . . . . 8 ( 𝐴 ⊆ ℝ → 0 ≤ (vol*‘ 𝐴))
98ad2antll 761 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 ≤ (vol*‘ 𝐴))
10 reldom 7847 . . . . . . . . . . . 12 Rel ≼
1110brrelexi 5082 . . . . . . . . . . 11 (𝐴 ≼ ℕ → 𝐴 ∈ V)
12 0sdomg 7974 . . . . . . . . . . 11 (𝐴 ∈ V → (∅ ≺ 𝐴𝐴 ≠ ∅))
1311, 12syl 17 . . . . . . . . . 10 (𝐴 ≼ ℕ → (∅ ≺ 𝐴𝐴 ≠ ∅))
1413biimparc 503 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∅ ≺ 𝐴)
15 fodomr 7996 . . . . . . . . 9 ((∅ ≺ 𝐴𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
1614, 15sylancom 698 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) → ∃𝑓 𝑓:ℕ–onto𝐴)
17 unissb 4405 . . . . . . . . . . . 12 ( 𝐴 ⊆ ℝ ↔ ∀𝑥𝐴 𝑥 ⊆ ℝ)
1817anbi1i 727 . . . . . . . . . . 11 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
19 r19.26 3046 . . . . . . . . . . 11 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) ↔ (∀𝑥𝐴 𝑥 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ))
2018, 19bitr4i 266 . . . . . . . . . 10 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ))
21 brdom2 7871 . . . . . . . . . . . . . 14 (𝑥 ≼ ℕ ↔ (𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ))
22 nnenom 12641 . . . . . . . . . . . . . . . . 17 ℕ ≈ ω
23 sdomen2 7990 . . . . . . . . . . . . . . . . 17 (ℕ ≈ ω → (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω))
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑥 ≺ ℕ ↔ 𝑥 ≺ ω)
25 isfinite 8432 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
2624, 25bitr4i 266 . . . . . . . . . . . . . . 15 (𝑥 ≺ ℕ ↔ 𝑥 ∈ Fin)
2726orbi1i 541 . . . . . . . . . . . . . 14 ((𝑥 ≺ ℕ ∨ 𝑥 ≈ ℕ) ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
2821, 27bitri 263 . . . . . . . . . . . . 13 (𝑥 ≼ ℕ ↔ (𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ))
29 ovolfi 23069 . . . . . . . . . . . . . . 15 ((𝑥 ∈ Fin ∧ 𝑥 ⊆ ℝ) → (vol*‘𝑥) = 0)
3029expcom 450 . . . . . . . . . . . . . 14 (𝑥 ⊆ ℝ → (𝑥 ∈ Fin → (vol*‘𝑥) = 0))
31 ovolctb 23065 . . . . . . . . . . . . . . 15 ((𝑥 ⊆ ℝ ∧ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0)
3231ex 449 . . . . . . . . . . . . . 14 (𝑥 ⊆ ℝ → (𝑥 ≈ ℕ → (vol*‘𝑥) = 0))
3330, 32jaod 394 . . . . . . . . . . . . 13 (𝑥 ⊆ ℝ → ((𝑥 ∈ Fin ∨ 𝑥 ≈ ℕ) → (vol*‘𝑥) = 0))
3428, 33syl5bi 231 . . . . . . . . . . . 12 (𝑥 ⊆ ℝ → (𝑥 ≼ ℕ → (vol*‘𝑥) = 0))
3534imdistani 722 . . . . . . . . . . 11 ((𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3635ralimi 2936 . . . . . . . . . 10 (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3720, 36sylbi 206 . . . . . . . . 9 (( 𝐴 ⊆ ℝ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
3837ancoms 468 . . . . . . . 8 ((∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ) → ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0))
39 foima 6033 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴 → (𝑓 “ ℕ) = 𝐴)
4039raleqdv 3121 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)))
41 fofn 6030 . . . . . . . . . . . . 13 (𝑓:ℕ–onto𝐴𝑓 Fn ℕ)
42 ssid 3587 . . . . . . . . . . . . 13 ℕ ⊆ ℕ
43 sseq1 3589 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑙) → (𝑥 ⊆ ℝ ↔ (𝑓𝑙) ⊆ ℝ))
44 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑓𝑙) → (vol*‘𝑥) = (vol*‘(𝑓𝑙)))
4544eqeq1d 2612 . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑙) → ((vol*‘𝑥) = 0 ↔ (vol*‘(𝑓𝑙)) = 0))
4643, 45anbi12d 743 . . . . . . . . . . . . . 14 (𝑥 = (𝑓𝑙) → ((𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4746ralima 6402 . . . . . . . . . . . . 13 ((𝑓 Fn ℕ ∧ ℕ ⊆ ℕ) → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4841, 42, 47sylancl 693 . . . . . . . . . . . 12 (𝑓:ℕ–onto𝐴 → (∀𝑥 ∈ (𝑓 “ ℕ)(𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
4940, 48bitr3d 269 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) ↔ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)))
50 fveq2 6103 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑛 → (𝑓𝑙) = (𝑓𝑛))
5150sseq1d 3595 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑛 → ((𝑓𝑙) ⊆ ℝ ↔ (𝑓𝑛) ⊆ ℝ))
5250fveq2d 6107 . . . . . . . . . . . . . . . . . 18 (𝑙 = 𝑛 → (vol*‘(𝑓𝑙)) = (vol*‘(𝑓𝑛)))
5352eqeq1d 2612 . . . . . . . . . . . . . . . . 17 (𝑙 = 𝑛 → ((vol*‘(𝑓𝑙)) = 0 ↔ (vol*‘(𝑓𝑛)) = 0))
5451, 53anbi12d 743 . . . . . . . . . . . . . . . 16 (𝑙 = 𝑛 → (((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0)))
5554cbvralv 3147 . . . . . . . . . . . . . . 15 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0))
56 0re 9919 . . . . . . . . . . . . . . . . . 18 0 ∈ ℝ
57 eleq1a 2683 . . . . . . . . . . . . . . . . . 18 (0 ∈ ℝ → ((vol*‘(𝑓𝑛)) = 0 → (vol*‘(𝑓𝑛)) ∈ ℝ))
5856, 57ax-mp 5 . . . . . . . . . . . . . . . . 17 ((vol*‘(𝑓𝑛)) = 0 → (vol*‘(𝑓𝑛)) ∈ ℝ)
5958anim2i 591 . . . . . . . . . . . . . . . 16 (((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0) → ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
6059ralimi 2936 . . . . . . . . . . . . . . 15 (∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) = 0) → ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
6155, 60sylbi 206 . . . . . . . . . . . . . 14 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → ∀𝑛 ∈ ℕ ((𝑓𝑛) ⊆ ℝ ∧ (vol*‘(𝑓𝑛)) ∈ ℝ))
62 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*, < ))
6341, 61, 62syl2an 493 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) ≤ sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ))
64 fofun 6029 . . . . . . . . . . . . . . . . 17 (𝑓:ℕ–onto𝐴 → Fun 𝑓)
65 funiunfv 6410 . . . . . . . . . . . . . . . . 17 (Fun 𝑓 𝑚 ∈ ℕ (𝑓𝑚) = (𝑓 “ ℕ))
6664, 65syl 17 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto𝐴 𝑚 ∈ ℕ (𝑓𝑚) = (𝑓 “ ℕ))
6739unieqd 4382 . . . . . . . . . . . . . . . 16 (𝑓:ℕ–onto𝐴 (𝑓 “ ℕ) = 𝐴)
6866, 67eqtrd 2644 . . . . . . . . . . . . . . 15 (𝑓:ℕ–onto𝐴 𝑚 ∈ ℕ (𝑓𝑚) = 𝐴)
6968fveq2d 6107 . . . . . . . . . . . . . 14 (𝑓:ℕ–onto𝐴 → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) = (vol*‘ 𝐴))
7069adantr 480 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝑚 ∈ ℕ (𝑓𝑚)) = (vol*‘ 𝐴))
71 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑚 → (𝑓𝑙) = (𝑓𝑚))
7271sseq1d 3595 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑚 → ((𝑓𝑙) ⊆ ℝ ↔ (𝑓𝑚) ⊆ ℝ))
7371fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = 𝑚 → (vol*‘(𝑓𝑙)) = (vol*‘(𝑓𝑚)))
7473eqeq1d 2612 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 = 𝑚 → ((vol*‘(𝑓𝑙)) = 0 ↔ (vol*‘(𝑓𝑚)) = 0))
7572, 74anbi12d 743 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 = 𝑚 → (((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ↔ ((𝑓𝑚) ⊆ ℝ ∧ (vol*‘(𝑓𝑚)) = 0)))
7675rspccva 3281 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → ((𝑓𝑚) ⊆ ℝ ∧ (vol*‘(𝑓𝑚)) = 0))
7776simprd 478 . . . . . . . . . . . . . . . . . . 19 ((∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) ∧ 𝑚 ∈ ℕ) → (vol*‘(𝑓𝑚)) = 0)
7877mpteq2dva 4672 . . . . . . . . . . . . . . . . . 18 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚))) = (𝑚 ∈ ℕ ↦ 0))
7978seqeq3d 12671 . . . . . . . . . . . . . . . . 17 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
8079rneqd 5274 . . . . . . . . . . . . . . . 16 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))) = ran seq1( + , (𝑚 ∈ ℕ ↦ 0)))
8180supeq1d 8235 . . . . . . . . . . . . . . 15 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ))
82 0cn 9911 . . . . . . . . . . . . . . . . . . . . . . 23 0 ∈ ℂ
83 ser1const 12719 . . . . . . . . . . . . . . . . . . . . . . 23 ((0 ∈ ℂ ∧ 𝑙 ∈ ℕ) → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
8482, 83mpan 702 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = (𝑙 · 0))
85 nncn 10905 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 ∈ ℕ → 𝑙 ∈ ℂ)
8685mul01d 10114 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 ∈ ℕ → (𝑙 · 0) = 0)
8784, 86eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 (𝑙 ∈ ℕ → (seq1( + , (ℕ × {0}))‘𝑙) = 0)
8887mpteq2ia 4668 . . . . . . . . . . . . . . . . . . . 20 (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)) = (𝑙 ∈ ℕ ↦ 0)
89 fconstmpt 5085 . . . . . . . . . . . . . . . . . . . . . 22 (ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0)
90 seqeq3 12668 . . . . . . . . . . . . . . . . . . . . . 22 ((ℕ × {0}) = (𝑚 ∈ ℕ ↦ 0) → seq1( + , (ℕ × {0})) = seq1( + , (𝑚 ∈ ℕ ↦ 0)))
9189, 90ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 seq1( + , (ℕ × {0})) = seq1( + , (𝑚 ∈ ℕ ↦ 0))
92 1z 11284 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℤ
93 seqfn 12675 . . . . . . . . . . . . . . . . . . . . . . 23 (1 ∈ ℤ → seq1( + , (ℕ × {0})) Fn (ℤ‘1))
9492, 93ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 seq1( + , (ℕ × {0})) Fn (ℤ‘1)
95 nnuz 11599 . . . . . . . . . . . . . . . . . . . . . . . 24 ℕ = (ℤ‘1)
9695fneq2i 5900 . . . . . . . . . . . . . . . . . . . . . . 23 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) Fn (ℤ‘1))
97 dffn5 6151 . . . . . . . . . . . . . . . . . . . . . . 23 (seq1( + , (ℕ × {0})) Fn ℕ ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
9896, 97bitr3i 265 . . . . . . . . . . . . . . . . . . . . . 22 (seq1( + , (ℕ × {0})) Fn (ℤ‘1) ↔ seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙)))
9994, 98mpbi 219 . . . . . . . . . . . . . . . . . . . . 21 seq1( + , (ℕ × {0})) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
10091, 99eqtr3i 2634 . . . . . . . . . . . . . . . . . . . 20 seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (𝑙 ∈ ℕ ↦ (seq1( + , (ℕ × {0}))‘𝑙))
101 fconstmpt 5085 . . . . . . . . . . . . . . . . . . . 20 (ℕ × {0}) = (𝑙 ∈ ℕ ↦ 0)
10288, 100, 1013eqtr4i 2642 . . . . . . . . . . . . . . . . . . 19 seq1( + , (𝑚 ∈ ℕ ↦ 0)) = (ℕ × {0})
103102rneqi 5273 . . . . . . . . . . . . . . . . . 18 ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = ran (ℕ × {0})
104 1nn 10908 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℕ
105 ne0i 3880 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℕ → ℕ ≠ ∅)
106 rnxp 5483 . . . . . . . . . . . . . . . . . . 19 (ℕ ≠ ∅ → ran (ℕ × {0}) = {0})
107104, 105, 106mp2b 10 . . . . . . . . . . . . . . . . . 18 ran (ℕ × {0}) = {0}
108103, 107eqtri 2632 . . . . . . . . . . . . . . . . 17 ran seq1( + , (𝑚 ∈ ℕ ↦ 0)) = {0}
109108supeq1i 8236 . . . . . . . . . . . . . . . 16 sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = sup({0}, ℝ*, < )
110 xrltso 11850 . . . . . . . . . . . . . . . . 17 < Or ℝ*
111 0xr 9965 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ*
112 supsn 8261 . . . . . . . . . . . . . . . . 17 (( < Or ℝ* ∧ 0 ∈ ℝ*) → sup({0}, ℝ*, < ) = 0)
113110, 111, 112mp2an 704 . . . . . . . . . . . . . . . 16 sup({0}, ℝ*, < ) = 0
114109, 113eqtri 2632 . . . . . . . . . . . . . . 15 sup(ran seq1( + , (𝑚 ∈ ℕ ↦ 0)), ℝ*, < ) = 0
11581, 114syl6eq 2660 . . . . . . . . . . . . . 14 (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = 0)
116115adantl 481 . . . . . . . . . . . . 13 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → sup(ran seq1( + , (𝑚 ∈ ℕ ↦ (vol*‘(𝑓𝑚)))), ℝ*, < ) = 0)
11763, 70, 1163brtr3d 4614 . . . . . . . . . . . 12 ((𝑓:ℕ–onto𝐴 ∧ ∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0)) → (vol*‘ 𝐴) ≤ 0)
118117ex 449 . . . . . . . . . . 11 (𝑓:ℕ–onto𝐴 → (∀𝑙 ∈ ℕ ((𝑓𝑙) ⊆ ℝ ∧ (vol*‘(𝑓𝑙)) = 0) → (vol*‘ 𝐴) ≤ 0))
11949, 118sylbid 229 . . . . . . . . . 10 (𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘ 𝐴) ≤ 0))
120119exlimiv 1845 . . . . . . . . 9 (∃𝑓 𝑓:ℕ–onto𝐴 → (∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0) → (vol*‘ 𝐴) ≤ 0))
121120imp 444 . . . . . . . 8 ((∃𝑓 𝑓:ℕ–onto𝐴 ∧ ∀𝑥𝐴 (𝑥 ⊆ ℝ ∧ (vol*‘𝑥) = 0)) → (vol*‘ 𝐴) ≤ 0)
12216, 38, 121syl2an 493 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → (vol*‘ 𝐴) ≤ 0)
123 ovolcl 23053 . . . . . . . . 9 ( 𝐴 ⊆ ℝ → (vol*‘ 𝐴) ∈ ℝ*)
124 xrletri3 11861 . . . . . . . . 9 ((0 ∈ ℝ* ∧ (vol*‘ 𝐴) ∈ ℝ*) → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
125111, 123, 124sylancr 694 . . . . . . . 8 ( 𝐴 ⊆ ℝ → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
126125ad2antll 761 . . . . . . 7 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → (0 = (vol*‘ 𝐴) ↔ (0 ≤ (vol*‘ 𝐴) ∧ (vol*‘ 𝐴) ≤ 0)))
1279, 122, 126mpbir2and 959 . . . . . 6 (((𝐴 ≠ ∅ ∧ 𝐴 ≼ ℕ) ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴))
128127expl 646 . . . . 5 (𝐴 ≠ ∅ → ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴)))
1297, 128pm2.61ine 2865 . . . 4 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 0 = (vol*‘ 𝐴))
130 renepnf 9966 . . . . . . 7 (0 ∈ ℝ → 0 ≠ +∞)
13156, 130mp1i 13 . . . . . 6 ( 𝐴 = ℝ → 0 ≠ +∞)
132 fveq2 6103 . . . . . . 7 ( 𝐴 = ℝ → (vol*‘ 𝐴) = (vol*‘ℝ))
133 ovolre 23100 . . . . . . 7 (vol*‘ℝ) = +∞
134132, 133syl6eq 2660 . . . . . 6 ( 𝐴 = ℝ → (vol*‘ 𝐴) = +∞)
135131, 134neeqtrrd 2856 . . . . 5 ( 𝐴 = ℝ → 0 ≠ (vol*‘ 𝐴))
136135necon2i 2816 . . . 4 (0 = (vol*‘ 𝐴) → 𝐴 ≠ ℝ)
137129, 136syl 17 . . 3 ((𝐴 ≼ ℕ ∧ (∀𝑥𝐴 𝑥 ≼ ℕ ∧ 𝐴 ⊆ ℝ)) → 𝐴 ≠ ℝ)
138137expr 641 . 2 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → ( 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ))
139 eqimss 3620 . . 3 ( 𝐴 = ℝ → 𝐴 ⊆ ℝ)
140139necon3bi 2808 . 2 𝐴 ⊆ ℝ → 𝐴 ≠ ℝ)
141138, 140pm2.61d1 170 1 ((𝐴 ≼ ℕ ∧ ∀𝑥𝐴 𝑥 ≼ ℕ) → 𝐴 ≠ ℝ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   Or wor 4958   × cxp 5036  ran crn 5039   “ cima 5041  Fun wfun 5798   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549  ωcom 6957   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  Fincfn 7841  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  ℕcn 10897  ℤcz 11254  ℤ≥cuz 11563  seqcseq 12663  vol*covol 23038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040 This theorem is referenced by:  ex-ovoliunnfl  32622
 Copyright terms: Public domain W3C validator