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Theorem mea0 39975
Description: The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
mea0.1 (𝜑𝑀 ∈ Meas)
Assertion
Ref Expression
mea0 (𝜑 → (𝑀‘∅) = 0)

Proof of Theorem mea0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mea0.1 . . 3 (𝜑𝑀 ∈ Meas)
2 ismea 39972 . . 3 (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))
31, 2sylib 208 . 2 (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → (𝑀 𝑥) = (Σ^‘(𝑀𝑥)))))
43simplrd 792 1 (𝜑 → (𝑀‘∅) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  c0 3891  𝒫 cpw 4130   cuni 4402  Disj wdisj 4583   class class class wbr 4613  dom cdm 5074  cres 5076  wf 5843  cfv 5847  (class class class)co 6604  ωcom 7012  cdom 7897  0cc0 9880  +∞cpnf 10015  [,]cicc 12120  SAlgcsalg 39832  Σ^csumge0 39883  Meascmea 39970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-mea 39971
This theorem is referenced by:  meadjun  39983  meadjiunlem  39986  vonioo  40200  vonicc  40203
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