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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 24056 | . 2 ⊢ (𝐴 ∈ dom vol ↔ (𝐴 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))))) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3138 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4537 dom cdm 5549 ‘cfv 6349 (class class class)co 7145 ℝcr 10525 + caddc 10529 vol*covol 23992 volcvol 23993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 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 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-sup 8895 df-inf 8896 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 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-rp 12380 df-ico 12734 df-icc 12735 df-fz 12883 df-seq 13360 df-exp 13420 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-ovol 23994 df-vol 23995 |
This theorem is referenced by: volss 24063 nulmbl2 24066 unmbl 24067 shftmbl 24068 unidmvol 24071 inmbl 24072 difmbl 24073 volun 24075 volinun 24076 volfiniun 24077 voliunlem2 24081 voliunlem3 24082 volsup 24086 volsup2 24135 volcn 24136 vitalilem4 24141 vitalilem5 24142 vitali 24143 ismbf 24158 ismbfcn 24159 mbfconst 24163 mbfid 24165 cncombf 24188 cnmbf 24189 i1fima2 24209 i1fd 24211 itg1ge0 24216 i1f1lem 24219 itg11 24221 i1fadd 24225 i1fmul 24226 itg1addlem2 24227 itg1addlem5 24230 i1fres 24235 itg1ge0a 24241 itg1climres 24244 mbfi1fseqlem4 24248 mbfi1flim 24253 mbfmullem2 24254 itg2const2 24271 itg2splitlem 24278 itg2split 24279 itg2gt0 24290 itg2cnlem2 24292 ibladdlem 24349 itgaddlem1 24352 iblabslem 24357 itggt0 24371 itgcn 24372 ftc1lem4 24565 itgulm 24925 areaf 25467 dmvlsiga 31288 volsupnfl 34819 cnambfre 34822 itg2addnclem 34825 ibladdnclem 34830 itgaddnclem1 34832 iblabsnclem 34837 ftc1cnnclem 34847 volge0 42126 dmvolss 42151 vonvol 42825 |
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