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Mirrors > Home > MPE Home > Th. List > Mathboxes > volsn | Structured version Visualization version GIF version |
Description: A singleton has 0 Lebesgue measure. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
volsn | ⊢ (𝐴 ∈ ℝ → (vol‘{𝐴}) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snmbl 43087 | . . 3 ⊢ (𝐴 ∈ ℝ → {𝐴} ∈ dom vol) | |
2 | mblvol 24295 | . . 3 ⊢ ({𝐴} ∈ dom vol → (vol‘{𝐴}) = (vol*‘{𝐴})) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ → (vol‘{𝐴}) = (vol*‘{𝐴})) |
4 | ovolsn 24260 | . 2 ⊢ (𝐴 ∈ ℝ → (vol*‘{𝐴}) = 0) | |
5 | 3, 4 | eqtrd 2774 | 1 ⊢ (𝐴 ∈ ℝ → (vol‘{𝐴}) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 {csn 4526 dom cdm 5535 ‘cfv 6350 ℝcr 10627 0cc0 10628 vol*covol 24227 volcvol 24228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-inf2 9190 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 ax-pre-sup 10706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-of 7438 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-2o 8145 df-er 8333 df-map 8452 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-sup 8992 df-inf 8993 df-oi 9060 df-dju 9416 df-card 9454 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-div 11389 df-nn 11730 df-2 11792 df-3 11793 df-n0 11990 df-z 12076 df-uz 12338 df-q 12444 df-rp 12486 df-xadd 12604 df-ioo 12838 df-ico 12840 df-icc 12841 df-fz 12995 df-fzo 13138 df-fl 13266 df-seq 13474 df-exp 13535 df-hash 13796 df-cj 14561 df-re 14562 df-im 14563 df-sqrt 14697 df-abs 14698 df-clim 14948 df-sum 15149 df-xmet 20223 df-met 20224 df-ovol 24229 df-vol 24230 |
This theorem is referenced by: volioc 43096 volico 43107 vonsn 43812 |
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