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Mirrors > Home > MPE Home > Th. List > 0mbl | Structured version Visualization version GIF version |
Description: The empty set is measurable. (Contributed by Mario Carneiro, 18-Mar-2014.) |
Ref | Expression |
---|---|
0mbl | ⊢ ∅ ∈ dom vol |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4349 | . 2 ⊢ ∅ ⊆ ℝ | |
2 | ovol0 24093 | . 2 ⊢ (vol*‘∅) = 0 | |
3 | nulmbl 24135 | . 2 ⊢ ((∅ ⊆ ℝ ∧ (vol*‘∅) = 0) → ∅ ∈ dom vol) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ ∅ ∈ dom vol |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∅c0 4290 dom cdm 5554 ‘cfv 6354 ℝcr 10535 0cc0 10536 vol*covol 24062 volcvol 24063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 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 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-oi 8973 df-dju 9329 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-rp 12389 df-xadd 12507 df-ioo 12741 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-xmet 20537 df-met 20538 df-ovol 24064 df-vol 24065 |
This theorem is referenced by: rembl 24140 finiunmbl 24144 volfiniun 24147 ioombl1 24162 icombl 24164 ioombl 24165 mbfconstlem 24227 mbfima 24230 mbf0 24234 ismbf2d 24240 itg1val2 24284 itg11 24291 itg1addlem4 24299 mblfinlem3 34930 ismblfin 34932 voliunnfl 34935 volsupnfl 34936 itg2addnclem2 34943 bddiblnc 34961 areacirc 34986 iocmbl 39817 arearect 39820 areaquad 39821 vol0 42242 dmvolsal 42628 |
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