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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cntnevol | Structured version Visualization version GIF version |
Description: Counting and Lebesgue measures are different. (Contributed by Thierry Arnoux, 27-Jan-2017.) |
Ref | Expression |
---|---|
cntnevol | ⊢ (♯ ↾ 𝒫 𝑂) ≠ vol |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1ne0 10595 | . . . . 5 ⊢ 1 ≠ 0 | |
2 | 1 | a1i 11 | . . . 4 ⊢ (1 ∈ 𝑂 → 1 ≠ 0) |
3 | snelpwi 5302 | . . . . . 6 ⊢ (1 ∈ 𝑂 → {1} ∈ 𝒫 𝑂) | |
4 | fvres 6664 | . . . . . 6 ⊢ ({1} ∈ 𝒫 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = (♯‘{1})) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = (♯‘{1})) |
6 | 1re 10630 | . . . . . 6 ⊢ 1 ∈ ℝ | |
7 | hashsng 13726 | . . . . . 6 ⊢ (1 ∈ ℝ → (♯‘{1}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . 5 ⊢ (♯‘{1}) = 1 |
9 | 5, 8 | eqtrdi 2849 | . . . 4 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = 1) |
10 | snssi 4701 | . . . . . . 7 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ) | |
11 | ovolsn 24099 | . . . . . . 7 ⊢ (1 ∈ ℝ → (vol*‘{1}) = 0) | |
12 | nulmbl 24139 | . . . . . . 7 ⊢ (({1} ⊆ ℝ ∧ (vol*‘{1}) = 0) → {1} ∈ dom vol) | |
13 | 10, 11, 12 | syl2anc 587 | . . . . . 6 ⊢ (1 ∈ ℝ → {1} ∈ dom vol) |
14 | mblvol 24134 | . . . . . . 7 ⊢ ({1} ∈ dom vol → (vol‘{1}) = (vol*‘{1})) | |
15 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (vol*‘{1}) = 0 |
16 | 14, 15 | eqtrdi 2849 | . . . . . 6 ⊢ ({1} ∈ dom vol → (vol‘{1}) = 0) |
17 | 6, 13, 16 | mp2b 10 | . . . . 5 ⊢ (vol‘{1}) = 0 |
18 | 17 | a1i 11 | . . . 4 ⊢ (1 ∈ 𝑂 → (vol‘{1}) = 0) |
19 | 2, 9, 18 | 3netr4d 3064 | . . 3 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) ≠ (vol‘{1})) |
20 | fveq1 6644 | . . . 4 ⊢ ((♯ ↾ 𝒫 𝑂) = vol → ((♯ ↾ 𝒫 𝑂)‘{1}) = (vol‘{1})) | |
21 | 20 | necon3i 3019 | . . 3 ⊢ (((♯ ↾ 𝒫 𝑂)‘{1}) ≠ (vol‘{1}) → (♯ ↾ 𝒫 𝑂) ≠ vol) |
22 | 19, 21 | syl 17 | . 2 ⊢ (1 ∈ 𝑂 → (♯ ↾ 𝒫 𝑂) ≠ vol) |
23 | 6, 13 | ax-mp 5 | . . . . . . 7 ⊢ {1} ∈ dom vol |
24 | 23 | biantrur 534 | . . . . . 6 ⊢ (¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ ({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂))) |
25 | snex 5297 | . . . . . . . . 9 ⊢ {1} ∈ V | |
26 | 25 | elpw 4501 | . . . . . . . 8 ⊢ ({1} ∈ 𝒫 𝑂 ↔ {1} ⊆ 𝑂) |
27 | dmhashres 13697 | . . . . . . . . 9 ⊢ dom (♯ ↾ 𝒫 𝑂) = 𝒫 𝑂 | |
28 | 27 | eleq2i 2881 | . . . . . . . 8 ⊢ ({1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ {1} ∈ 𝒫 𝑂) |
29 | 1ex 10626 | . . . . . . . . 9 ⊢ 1 ∈ V | |
30 | 29 | snss 4679 | . . . . . . . 8 ⊢ (1 ∈ 𝑂 ↔ {1} ⊆ 𝑂) |
31 | 26, 28, 30 | 3bitr4i 306 | . . . . . . 7 ⊢ ({1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ 1 ∈ 𝑂) |
32 | 31 | notbii 323 | . . . . . 6 ⊢ (¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ ¬ 1 ∈ 𝑂) |
33 | 24, 32 | bitr3i 280 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂)) ↔ ¬ 1 ∈ 𝑂) |
34 | nelne1 3083 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂)) → dom vol ≠ dom (♯ ↾ 𝒫 𝑂)) | |
35 | 33, 34 | sylbir 238 | . . . 4 ⊢ (¬ 1 ∈ 𝑂 → dom vol ≠ dom (♯ ↾ 𝒫 𝑂)) |
36 | 35 | necomd 3042 | . . 3 ⊢ (¬ 1 ∈ 𝑂 → dom (♯ ↾ 𝒫 𝑂) ≠ dom vol) |
37 | dmeq 5736 | . . . 4 ⊢ ((♯ ↾ 𝒫 𝑂) = vol → dom (♯ ↾ 𝒫 𝑂) = dom vol) | |
38 | 37 | necon3i 3019 | . . 3 ⊢ (dom (♯ ↾ 𝒫 𝑂) ≠ dom vol → (♯ ↾ 𝒫 𝑂) ≠ vol) |
39 | 36, 38 | syl 17 | . 2 ⊢ (¬ 1 ∈ 𝑂 → (♯ ↾ 𝒫 𝑂) ≠ vol) |
40 | 22, 39 | pm2.61i 185 | 1 ⊢ (♯ ↾ 𝒫 𝑂) ≠ vol |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 𝒫 cpw 4497 {csn 4525 dom cdm 5519 ↾ cres 5521 ‘cfv 6324 ℝcr 10525 0cc0 10526 1c1 10527 ♯chash 13686 vol*covol 24066 volcvol 24067 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-xmet 20084 df-met 20085 df-ovol 24068 df-vol 24069 |
This theorem is referenced by: (None) |
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