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Theorem measvnul 33734
Description: The measure of the empty set is always zero. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
measvnul (𝑀 ∈ (measuresβ€˜π‘†) β†’ (π‘€β€˜βˆ…) = 0)

Proof of Theorem measvnul
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 measbase 33725 . . . 4 (𝑀 ∈ (measuresβ€˜π‘†) β†’ 𝑆 ∈ βˆͺ ran sigAlgebra)
2 ismeas 33727 . . . 4 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ (𝑀 ∈ (measuresβ€˜π‘†) ↔ (𝑀:π‘†βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯)))))
31, 2syl 17 . . 3 (𝑀 ∈ (measuresβ€˜π‘†) β†’ (𝑀 ∈ (measuresβ€˜π‘†) ↔ (𝑀:π‘†βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯)))))
43ibi 267 . 2 (𝑀 ∈ (measuresβ€˜π‘†) β†’ (𝑀:π‘†βŸΆ(0[,]+∞) ∧ (π‘€β€˜βˆ…) = 0 ∧ βˆ€π‘¦ ∈ 𝒫 𝑆((𝑦 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝑦 π‘₯) β†’ (π‘€β€˜βˆͺ 𝑦) = Ξ£*π‘₯ ∈ 𝑦(π‘€β€˜π‘₯))))
54simp2d 1140 1 (𝑀 ∈ (measuresβ€˜π‘†) β†’ (π‘€β€˜βˆ…) = 0)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆ…c0 4317  π’« cpw 4597  βˆͺ cuni 4902  Disj wdisj 5106   class class class wbr 5141  ran crn 5670  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  Ο‰com 7852   β‰Ό cdom 8939  0cc0 11112  +∞cpnf 11249  [,]cicc 13333  Ξ£*cesum 33555  sigAlgebracsiga 33636  measurescmeas 33723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-esum 33556  df-meas 33724
This theorem is referenced by:  measxun2  33738  measvunilem0  33741  measssd  33743  measinb  33749  measres  33750  measdivcst  33752  measdivcstALTV  33753  truae  33771  probnul  33943
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