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Theorem probcun 33909
Description: The probability of the union of a countable disjoint set of events is the sum of their probabilities. (Third axiom of Kolmogorov) Here, the Σ construct cannot be used as it can handle infinite indexing set only if they are subsets of , which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
probcun ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑃 𝐴) = Σ*𝑥𝐴(𝑃𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑃

Proof of Theorem probcun
StepHypRef Expression
1 domprobmeas 33901 . 2 (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2 measvun 33699 . 2 ((𝑃 ∈ (measures‘dom 𝑃) ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑃 𝐴) = Σ*𝑥𝐴(𝑃𝑥))
31, 2syl3an1 1160 1 ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 ≼ ω ∧ Disj 𝑥𝐴 𝑥)) → (𝑃 𝐴) = Σ*𝑥𝐴(𝑃𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  𝒫 cpw 4595   cuni 4900  Disj wdisj 5104   class class class wbr 5139  dom cdm 5667  cfv 6534  ωcom 7849  cdom 8934  Σ*cesum 33517  measurescmeas 33685  Probcprb 33898
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 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-disj 5105  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-esum 33518  df-meas 33686  df-prob 33899
This theorem is referenced by:  probun  33910
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