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Theorem measvnul 33858
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 33849 . . . 4 (𝑀 ∈ (measuresβ€˜π‘†) β†’ 𝑆 ∈ βˆͺ ran sigAlgebra)
2 ismeas 33851 . . . 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 266 . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  βˆ…c0 4326  π’« cpw 4606  βˆͺ cuni 4912  Disj wdisj 5117   class class class wbr 5152  ran crn 5683  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426  Ο‰com 7876   β‰Ό cdom 8968  0cc0 11146  +∞cpnf 11283  [,]cicc 13367  Ξ£*cesum 33679  sigAlgebracsiga 33760  measurescmeas 33847
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-esum 33680  df-meas 33848
This theorem is referenced by:  measxun2  33862  measvunilem0  33865  measssd  33867  measinb  33873  measres  33874  measdivcst  33876  measdivcstALTV  33877  truae  33895  probnul  34067
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