Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > vitali2 | Structured version Visualization version GIF version |
Description: There are non-measurable sets (the Axiom of Choice is used, in the invoked weth 10240). (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
vitali2 | ⊢ dom vol ⊊ 𝒫 ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reex 10951 | . . 3 ⊢ ℝ ∈ V | |
2 | weth 10240 | . . 3 ⊢ (ℝ ∈ V → ∃𝑜 𝑜 We ℝ) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ∃𝑜 𝑜 We ℝ |
4 | vitali 24766 | . . 3 ⊢ (𝑜 We ℝ → dom vol ⊊ 𝒫 ℝ) | |
5 | 4 | exlimiv 1933 | . 2 ⊢ (∃𝑜 𝑜 We ℝ → dom vol ⊊ 𝒫 ℝ) |
6 | 3, 5 | ax-mp 5 | 1 ⊢ dom vol ⊊ 𝒫 ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1782 ∈ wcel 2106 Vcvv 3431 ⊊ wpss 3889 𝒫 cpw 4535 We wwe 5540 dom cdm 5586 ℝcr 10859 volcvol 24616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-inf2 9388 ax-cc 10180 ax-ac2 10208 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 ax-pre-sup 10938 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-disj 5041 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-se 5542 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-isom 6437 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-oadd 8290 df-omul 8291 df-er 8487 df-ec 8489 df-qs 8493 df-map 8606 df-pm 8607 df-en 8723 df-dom 8724 df-sdom 8725 df-fin 8726 df-fi 9159 df-sup 9190 df-inf 9191 df-oi 9258 df-dju 9648 df-card 9686 df-acn 9689 df-ac 9861 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-div 11622 df-nn 11963 df-2 12025 df-3 12026 df-n0 12223 df-z 12309 df-uz 12572 df-q 12678 df-rp 12720 df-xneg 12837 df-xadd 12838 df-xmul 12839 df-ioo 13072 df-ico 13074 df-icc 13075 df-fz 13229 df-fzo 13372 df-fl 13501 df-seq 13711 df-exp 13772 df-hash 14034 df-cj 14799 df-re 14800 df-im 14801 df-sqrt 14935 df-abs 14936 df-clim 15186 df-rlim 15187 df-sum 15387 df-rest 17122 df-topgen 17143 df-psmet 20578 df-xmet 20579 df-met 20580 df-bl 20581 df-mopn 20582 df-top 22032 df-topon 22049 df-bases 22085 df-cmp 22527 df-ovol 24617 df-vol 24618 |
This theorem is referenced by: nsssmfmbf 44271 |
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