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Theorem mea0 43033
 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 43030 . . 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 221 . 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 769 1 (𝜑 → (𝑀‘∅) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∀wral 3130  ∅c0 4265  𝒫 cpw 4511  ∪ cuni 4813  Disj wdisj 5007   class class class wbr 5042  dom cdm 5532   ↾ cres 5534  ⟶wf 6330  ‘cfv 6334  (class class class)co 7140  ωcom 7565   ≼ cdom 8494  0cc0 10526  +∞cpnf 10661  [,]cicc 12729  SAlgcsalg 42890  Σ^csumge0 42941  Meascmea 43028 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-mea 43029 This theorem is referenced by:  meadjun  43041  meadjiunlem  43044  vonioo  43261  vonicc  43264
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