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Mirrors > Home > MPE Home > Th. List > mblss | Structured version Visualization version GIF version |
Description: A measurable set is a subset of the reals. (Contributed by Mario Carneiro, 17-Mar-2014.) |
Ref | Expression |
---|---|
mblss | ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismbl 24130 | . 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 𝒫 cpw 4497 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 + caddc 10529 vol*covol 24066 volcvol 24067 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-icc 12733 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-ovol 24068 df-vol 24069 |
This theorem is referenced by: volss 24137 nulmbl2 24140 unmbl 24141 shftmbl 24142 unidmvol 24145 inmbl 24146 difmbl 24147 volun 24149 volinun 24150 volfiniun 24151 voliunlem2 24155 voliunlem3 24156 volsup 24160 volsup2 24209 volcn 24210 vitalilem4 24215 vitalilem5 24216 vitali 24217 ismbf 24232 ismbfcn 24233 mbfconst 24237 mbfid 24239 cncombf 24262 cnmbf 24263 i1fima2 24283 i1fd 24285 itg1ge0 24290 i1f1lem 24293 itg11 24295 i1fadd 24299 i1fmul 24300 itg1addlem2 24301 itg1addlem5 24304 i1fres 24309 itg1ge0a 24315 itg1climres 24318 mbfi1fseqlem4 24322 mbfi1flim 24327 mbfmullem2 24328 itg2const2 24345 itg2splitlem 24352 itg2split 24353 itg2gt0 24364 itg2cnlem2 24366 ibladdlem 24423 itgaddlem1 24426 iblabslem 24431 itggt0 24447 itgcn 24448 ftc1lem4 24642 itgulm 25003 areaf 25547 dmvlsiga 31498 volsupnfl 35102 cnambfre 35105 itg2addnclem 35108 ibladdnclem 35113 itgaddnclem1 35115 iblabsnclem 35120 ftc1cnnclem 35128 volge0 42603 dmvolss 42627 vonvol 43301 |
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