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Mirrors > Home > MPE Home > Th. List > rembl | Structured version Visualization version GIF version |
Description: The set of all real numbers is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
Ref | Expression |
---|---|
rembl | ⊢ ℝ ∈ dom vol |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif0 4330 | . 2 ⊢ (ℝ ∖ ∅) = ℝ | |
2 | 0mbl 24849 | . . 3 ⊢ ∅ ∈ dom vol | |
3 | cmmbl 24844 | . . 3 ⊢ (∅ ∈ dom vol → (ℝ ∖ ∅) ∈ dom vol) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (ℝ ∖ ∅) ∈ dom vol |
5 | 1, 4 | eqeltrri 2835 | 1 ⊢ ℝ ∈ dom vol |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 ∖ cdif 3905 ∅c0 4280 dom cdm 5631 ℝcr 11008 volcvol 24773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-inf2 9535 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-inf 9337 df-oi 9404 df-dju 9795 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-q 12828 df-rp 12870 df-xadd 12988 df-ioo 13222 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-fl 13651 df-seq 13861 df-exp 13922 df-hash 14185 df-cj 14938 df-re 14939 df-im 14940 df-sqrt 15074 df-abs 15075 df-clim 15324 df-sum 15525 df-xmet 20736 df-met 20737 df-ovol 24774 df-vol 24775 |
This theorem is referenced by: unidmvol 24851 ioombl1 24872 ioombl 24875 i1fd 24991 i1f0rn 24992 mbfi1fseqlem4 25029 mbfi1flim 25034 itg2monolem1 25061 itg2cnlem1 25072 ibladdlem 25130 itgaddlem1 25133 iblabslem 25138 itggt0 25154 itgcn 25155 dmvlsiga 32556 mblfinlem3 36049 mblfinlem4 36050 ismblfin 36051 voliunnfl 36054 volsupnfl 36055 ibladdnclem 36066 itgaddnclem1 36068 iblabsnclem 36073 ftc1anclem5 36087 ftc1anclem6 36088 ftc1anclem8 36090 areacirc 36103 arearect 41452 areaquad 41453 |
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