![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 11208 | . . . . 5 ⊢ 1 ≠ 0 | |
2 | 1 | a1i 11 | . . . 4 ⊢ (1 ∈ 𝑂 → 1 ≠ 0) |
3 | snelpwi 5445 | . . . . . 6 ⊢ (1 ∈ 𝑂 → {1} ∈ 𝒫 𝑂) | |
4 | fvres 6916 | . . . . . 6 ⊢ ({1} ∈ 𝒫 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = (♯‘{1})) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = (♯‘{1})) |
6 | 1re 11245 | . . . . . 6 ⊢ 1 ∈ ℝ | |
7 | hashsng 14361 | . . . . . 6 ⊢ (1 ∈ ℝ → (♯‘{1}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . 5 ⊢ (♯‘{1}) = 1 |
9 | 5, 8 | eqtrdi 2784 | . . . 4 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = 1) |
10 | snssi 4812 | . . . . . . 7 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ) | |
11 | ovolsn 25437 | . . . . . . 7 ⊢ (1 ∈ ℝ → (vol*‘{1}) = 0) | |
12 | nulmbl 25477 | . . . . . . 7 ⊢ (({1} ⊆ ℝ ∧ (vol*‘{1}) = 0) → {1} ∈ dom vol) | |
13 | 10, 11, 12 | syl2anc 583 | . . . . . 6 ⊢ (1 ∈ ℝ → {1} ∈ dom vol) |
14 | mblvol 25472 | . . . . . . 7 ⊢ ({1} ∈ dom vol → (vol‘{1}) = (vol*‘{1})) | |
15 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (vol*‘{1}) = 0 |
16 | 14, 15 | eqtrdi 2784 | . . . . . 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 3015 | . . 3 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) ≠ (vol‘{1})) |
20 | fveq1 6896 | . . . 4 ⊢ ((♯ ↾ 𝒫 𝑂) = vol → ((♯ ↾ 𝒫 𝑂)‘{1}) = (vol‘{1})) | |
21 | 20 | necon3i 2970 | . . 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 530 | . . . . . 6 ⊢ (¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ ({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂))) |
25 | snex 5433 | . . . . . . . . 9 ⊢ {1} ∈ V | |
26 | 25 | elpw 4607 | . . . . . . . 8 ⊢ ({1} ∈ 𝒫 𝑂 ↔ {1} ⊆ 𝑂) |
27 | dmhashres 14333 | . . . . . . . . 9 ⊢ dom (♯ ↾ 𝒫 𝑂) = 𝒫 𝑂 | |
28 | 27 | eleq2i 2821 | . . . . . . . 8 ⊢ ({1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ {1} ∈ 𝒫 𝑂) |
29 | 1ex 11241 | . . . . . . . . 9 ⊢ 1 ∈ V | |
30 | 29 | snss 4790 | . . . . . . . 8 ⊢ (1 ∈ 𝑂 ↔ {1} ⊆ 𝑂) |
31 | 26, 28, 30 | 3bitr4i 303 | . . . . . . 7 ⊢ ({1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ 1 ∈ 𝑂) |
32 | 31 | notbii 320 | . . . . . 6 ⊢ (¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ ¬ 1 ∈ 𝑂) |
33 | 24, 32 | bitr3i 277 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂)) ↔ ¬ 1 ∈ 𝑂) |
34 | nelne1 3036 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂)) → dom vol ≠ dom (♯ ↾ 𝒫 𝑂)) | |
35 | 33, 34 | sylbir 234 | . . . 4 ⊢ (¬ 1 ∈ 𝑂 → dom vol ≠ dom (♯ ↾ 𝒫 𝑂)) |
36 | 35 | necomd 2993 | . . 3 ⊢ (¬ 1 ∈ 𝑂 → dom (♯ ↾ 𝒫 𝑂) ≠ dom vol) |
37 | dmeq 5906 | . . . 4 ⊢ ((♯ ↾ 𝒫 𝑂) = vol → dom (♯ ↾ 𝒫 𝑂) = dom vol) | |
38 | 37 | necon3i 2970 | . . 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 182 | 1 ⊢ (♯ ↾ 𝒫 𝑂) ≠ vol |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ⊆ wss 3947 𝒫 cpw 4603 {csn 4629 dom cdm 5678 ↾ cres 5680 ‘cfv 6548 ℝcr 11138 0cc0 11139 1c1 11140 ♯chash 14322 vol*covol 25404 volcvol 25405 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-dju 9925 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-xnn0 12576 df-z 12590 df-uz 12854 df-q 12964 df-rp 13008 df-xadd 13126 df-ioo 13361 df-ico 13363 df-icc 13364 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-hash 14323 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-clim 15465 df-sum 15666 df-xmet 21272 df-met 21273 df-ovol 25406 df-vol 25407 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |