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 10608 | . . . . 5 ⊢ 1 ≠ 0 | |
2 | 1 | a1i 11 | . . . 4 ⊢ (1 ∈ 𝑂 → 1 ≠ 0) |
3 | snelpwi 5339 | . . . . . 6 ⊢ (1 ∈ 𝑂 → {1} ∈ 𝒫 𝑂) | |
4 | fvres 6691 | . . . . . 6 ⊢ ({1} ∈ 𝒫 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = (♯‘{1})) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = (♯‘{1})) |
6 | 1re 10643 | . . . . . 6 ⊢ 1 ∈ ℝ | |
7 | hashsng 13733 | . . . . . 6 ⊢ (1 ∈ ℝ → (♯‘{1}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . 5 ⊢ (♯‘{1}) = 1 |
9 | 5, 8 | syl6eq 2874 | . . . 4 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) = 1) |
10 | snssi 4743 | . . . . . . 7 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ) | |
11 | ovolsn 24098 | . . . . . . 7 ⊢ (1 ∈ ℝ → (vol*‘{1}) = 0) | |
12 | nulmbl 24138 | . . . . . . 7 ⊢ (({1} ⊆ ℝ ∧ (vol*‘{1}) = 0) → {1} ∈ dom vol) | |
13 | 10, 11, 12 | syl2anc 586 | . . . . . 6 ⊢ (1 ∈ ℝ → {1} ∈ dom vol) |
14 | mblvol 24133 | . . . . . . 7 ⊢ ({1} ∈ dom vol → (vol‘{1}) = (vol*‘{1})) | |
15 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (vol*‘{1}) = 0 |
16 | 14, 15 | syl6eq 2874 | . . . . . 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 3095 | . . 3 ⊢ (1 ∈ 𝑂 → ((♯ ↾ 𝒫 𝑂)‘{1}) ≠ (vol‘{1})) |
20 | fveq1 6671 | . . . 4 ⊢ ((♯ ↾ 𝒫 𝑂) = vol → ((♯ ↾ 𝒫 𝑂)‘{1}) = (vol‘{1})) | |
21 | 20 | necon3i 3050 | . . 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 533 | . . . . . 6 ⊢ (¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ ({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂))) |
25 | snex 5334 | . . . . . . . . 9 ⊢ {1} ∈ V | |
26 | 25 | elpw 4545 | . . . . . . . 8 ⊢ ({1} ∈ 𝒫 𝑂 ↔ {1} ⊆ 𝑂) |
27 | dmhashres 13704 | . . . . . . . . 9 ⊢ dom (♯ ↾ 𝒫 𝑂) = 𝒫 𝑂 | |
28 | 27 | eleq2i 2906 | . . . . . . . 8 ⊢ ({1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ {1} ∈ 𝒫 𝑂) |
29 | 1ex 10639 | . . . . . . . . 9 ⊢ 1 ∈ V | |
30 | 29 | snss 4720 | . . . . . . . 8 ⊢ (1 ∈ 𝑂 ↔ {1} ⊆ 𝑂) |
31 | 26, 28, 30 | 3bitr4i 305 | . . . . . . 7 ⊢ ({1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ 1 ∈ 𝑂) |
32 | 31 | notbii 322 | . . . . . 6 ⊢ (¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂) ↔ ¬ 1 ∈ 𝑂) |
33 | 24, 32 | bitr3i 279 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂)) ↔ ¬ 1 ∈ 𝑂) |
34 | nelne1 3115 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (♯ ↾ 𝒫 𝑂)) → dom vol ≠ dom (♯ ↾ 𝒫 𝑂)) | |
35 | 33, 34 | sylbir 237 | . . . 4 ⊢ (¬ 1 ∈ 𝑂 → dom vol ≠ dom (♯ ↾ 𝒫 𝑂)) |
36 | 35 | necomd 3073 | . . 3 ⊢ (¬ 1 ∈ 𝑂 → dom (♯ ↾ 𝒫 𝑂) ≠ dom vol) |
37 | dmeq 5774 | . . . 4 ⊢ ((♯ ↾ 𝒫 𝑂) = vol → dom (♯ ↾ 𝒫 𝑂) = dom vol) | |
38 | 37 | necon3i 3050 | . . 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 184 | 1 ⊢ (♯ ↾ 𝒫 𝑂) ≠ vol |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ⊆ wss 3938 𝒫 cpw 4541 {csn 4569 dom cdm 5557 ↾ cres 5559 ‘cfv 6357 ℝcr 10538 0cc0 10539 1c1 10540 ♯chash 13693 vol*covol 24065 volcvol 24066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xadd 12511 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-xmet 20540 df-met 20541 df-ovol 24067 df-vol 24068 |
This theorem is referenced by: (None) |
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