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 9884 | . . . . 5 ⊢ 1 ≠ 0 | |
2 | 1 | a1i 11 | . . . 4 ⊢ (1 ∈ 𝑂 → 1 ≠ 0) |
3 | snelpwi 4839 | . . . . . 6 ⊢ (1 ∈ 𝑂 → {1} ∈ 𝒫 𝑂) | |
4 | fvres 6117 | . . . . . 6 ⊢ ({1} ∈ 𝒫 𝑂 → ((# ↾ 𝒫 𝑂)‘{1}) = (#‘{1})) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (1 ∈ 𝑂 → ((# ↾ 𝒫 𝑂)‘{1}) = (#‘{1})) |
6 | 1re 9918 | . . . . . 6 ⊢ 1 ∈ ℝ | |
7 | hashsng 13020 | . . . . . 6 ⊢ (1 ∈ ℝ → (#‘{1}) = 1) | |
8 | 6, 7 | ax-mp 5 | . . . . 5 ⊢ (#‘{1}) = 1 |
9 | 5, 8 | syl6eq 2660 | . . . 4 ⊢ (1 ∈ 𝑂 → ((# ↾ 𝒫 𝑂)‘{1}) = 1) |
10 | snssi 4280 | . . . . . . 7 ⊢ (1 ∈ ℝ → {1} ⊆ ℝ) | |
11 | ovolsn 23070 | . . . . . . 7 ⊢ (1 ∈ ℝ → (vol*‘{1}) = 0) | |
12 | nulmbl 23110 | . . . . . . 7 ⊢ (({1} ⊆ ℝ ∧ (vol*‘{1}) = 0) → {1} ∈ dom vol) | |
13 | 10, 11, 12 | syl2anc 691 | . . . . . 6 ⊢ (1 ∈ ℝ → {1} ∈ dom vol) |
14 | mblvol 23105 | . . . . . . 7 ⊢ ({1} ∈ dom vol → (vol‘{1}) = (vol*‘{1})) | |
15 | 6, 11 | ax-mp 5 | . . . . . . 7 ⊢ (vol*‘{1}) = 0 |
16 | 14, 15 | syl6eq 2660 | . . . . . 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 2859 | . . 3 ⊢ (1 ∈ 𝑂 → ((# ↾ 𝒫 𝑂)‘{1}) ≠ (vol‘{1})) |
20 | fveq1 6102 | . . . 4 ⊢ ((# ↾ 𝒫 𝑂) = vol → ((# ↾ 𝒫 𝑂)‘{1}) = (vol‘{1})) | |
21 | 20 | necon3i 2814 | . . 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 526 | . . . . . 6 ⊢ (¬ {1} ∈ dom (# ↾ 𝒫 𝑂) ↔ ({1} ∈ dom vol ∧ ¬ {1} ∈ dom (# ↾ 𝒫 𝑂))) |
25 | snex 4835 | . . . . . . . . 9 ⊢ {1} ∈ V | |
26 | 25 | elpw 4114 | . . . . . . . 8 ⊢ ({1} ∈ 𝒫 𝑂 ↔ {1} ⊆ 𝑂) |
27 | dmhashres 12991 | . . . . . . . . 9 ⊢ dom (# ↾ 𝒫 𝑂) = 𝒫 𝑂 | |
28 | 27 | eleq2i 2680 | . . . . . . . 8 ⊢ ({1} ∈ dom (# ↾ 𝒫 𝑂) ↔ {1} ∈ 𝒫 𝑂) |
29 | 1ex 9914 | . . . . . . . . 9 ⊢ 1 ∈ V | |
30 | 29 | snss 4259 | . . . . . . . 8 ⊢ (1 ∈ 𝑂 ↔ {1} ⊆ 𝑂) |
31 | 26, 28, 30 | 3bitr4i 291 | . . . . . . 7 ⊢ ({1} ∈ dom (# ↾ 𝒫 𝑂) ↔ 1 ∈ 𝑂) |
32 | 31 | notbii 309 | . . . . . 6 ⊢ (¬ {1} ∈ dom (# ↾ 𝒫 𝑂) ↔ ¬ 1 ∈ 𝑂) |
33 | 24, 32 | bitr3i 265 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (# ↾ 𝒫 𝑂)) ↔ ¬ 1 ∈ 𝑂) |
34 | nelne1 2878 | . . . . 5 ⊢ (({1} ∈ dom vol ∧ ¬ {1} ∈ dom (# ↾ 𝒫 𝑂)) → dom vol ≠ dom (# ↾ 𝒫 𝑂)) | |
35 | 33, 34 | sylbir 224 | . . . 4 ⊢ (¬ 1 ∈ 𝑂 → dom vol ≠ dom (# ↾ 𝒫 𝑂)) |
36 | 35 | necomd 2837 | . . 3 ⊢ (¬ 1 ∈ 𝑂 → dom (# ↾ 𝒫 𝑂) ≠ dom vol) |
37 | dmeq 5246 | . . . 4 ⊢ ((# ↾ 𝒫 𝑂) = vol → dom (# ↾ 𝒫 𝑂) = dom vol) | |
38 | 37 | necon3i 2814 | . . 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 175 | 1 ⊢ (# ↾ 𝒫 𝑂) ≠ vol |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 𝒫 cpw 4108 {csn 4125 dom cdm 5038 ↾ cres 5040 ‘cfv 5804 ℝcr 9814 0cc0 9815 1c1 9816 #chash 12979 vol*covol 23038 volcvol 23039 |
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-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-xnn0 11241 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 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-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041 |
This theorem is referenced by: (None) |
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