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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 11222 | . . . . 5 ⊢ 1 ≠ 0 | |
2 | 1 | a1i 11 | . . . 4 ⊢ (1 ∈ 𝑂 → 1 ≠ 0) |
3 | snelpwi 5454 | . . . . . 6 ⊢ (1 ∈ 𝑂 → {1} ∈ 𝒫 𝑂) | |
4 | fvres 6926 | . . . . . 6 ⊢ ({1} ∈ 𝒫 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = (♯‘{1})) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = (♯‘{1})) |
6 | 1re 11259 | . . . . . 6 ⊢ 1 ∈ ℝ | |
7 | hashsng 14405 | . . . . . 6 ⊢ (1 ∈ ℝ → (♯‘{1}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . 5 ⊢ (♯‘{1}) = 1 |
9 | 5, 8 | eqtrdi 2791 | . . . 4 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = 1) |
10 | snssi 4813 | . . . . . . 7 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ) | |
11 | ovolsn 25544 | . . . . . . 7 ⊢ (1 ∈ ℝ → (vol*‘{1}) = 0) | |
12 | nulmbl 25584 | . . . . . . 7 ⊢ (({1} ⊆ ℝ ∧ (vol*‘{1}) = 0) → {1} ∈ dom vol) | |
13 | 10, 11, 12 | syl2anc 584 | . . . . . 6 ⊢ (1 ∈ ℝ → {1} ∈ dom vol) |
14 | mblvol 25579 | . . . . . . 7 ⊢ ({1} ∈ dom vol → (vol‘{1}) = (vol*‘{1})) | |
15 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (vol*‘{1}) = 0 |
16 | 14, 15 | eqtrdi 2791 | . . . . . 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 3016 | . . 3 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) ≠ (vol‘{1})) |
20 | fveq1 6906 | . . . 4 ⊢ ((♯ ↾ 𝒫 𝑂) = vol → ((♯ ↾ 𝒫 𝑂)‘{1}) = (vol‘{1})) | |
21 | 20 | necon3i 2971 | . . 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 5442 | . . . . . . . . 9 ⊢ {1} ∈ V | |
26 | 25 | elpw 4609 | . . . . . . . 8 ⊢ ({1} ∈ 𝒫 𝑂 ↔ {1} ⊆ 𝑂) |
27 | dmhashres 14377 | . . . . . . . . 9 ⊢ dom (♯ ↾ 𝒫 𝑂) = 𝒫 𝑂 | |
28 | 27 | eleq2i 2831 | . . . . . . . 8 ⊢ ({1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ {1} ∈ 𝒫 𝑂) |
29 | 1ex 11255 | . . . . . . . . 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 3037 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂)) → dom vol ≠ dom (♯ ↾ 𝒫 𝑂)) | |
35 | 33, 34 | sylbir 235 | . . . 4 ⊢ (¬ 1 ∈ 𝑂 → dom vol ≠ dom (♯ ↾ 𝒫 𝑂)) |
36 | 35 | necomd 2994 | . . 3 ⊢ (¬ 1 ∈ 𝑂 → dom (♯ ↾ 𝒫 𝑂) ≠ dom vol) |
37 | dmeq 5917 | . . . 4 ⊢ ((♯ ↾ 𝒫 𝑂) = vol → dom (♯ ↾ 𝒫 𝑂) = dom vol) | |
38 | 37 | necon3i 2971 | . . 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 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 𝒫 cpw 4605 {csn 4631 dom cdm 5689 ↾ cres 5691 ‘cfv 6563 ℝcr 11152 0cc0 11153 1c1 11154 ♯chash 14366 vol*covol 25511 volcvol 25512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xadd 13153 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-xmet 21375 df-met 21376 df-ovol 25513 df-vol 25514 |
This theorem is referenced by: (None) |
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