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Theorem probcun 33946
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 Ξ£ construct cannot be used as it can handle infinite indexing set only if they are subsets of β„€, which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
probcun ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (π‘ƒβ€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘ƒβ€˜π‘₯))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑃

Proof of Theorem probcun
StepHypRef Expression
1 domprobmeas 33938 . 2 (𝑃 ∈ Prob β†’ 𝑃 ∈ (measuresβ€˜dom 𝑃))
2 measvun 33736 . 2 ((𝑃 ∈ (measuresβ€˜dom 𝑃) ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (π‘ƒβ€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘ƒβ€˜π‘₯))
31, 2syl3an1 1160 1 ((𝑃 ∈ Prob ∧ 𝐴 ∈ 𝒫 dom 𝑃 ∧ (𝐴 β‰Ό Ο‰ ∧ Disj π‘₯ ∈ 𝐴 π‘₯)) β†’ (π‘ƒβ€˜βˆͺ 𝐴) = Ξ£*π‘₯ ∈ 𝐴(π‘ƒβ€˜π‘₯))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  π’« cpw 4597  βˆͺ cuni 4902  Disj wdisj 5106   class class class wbr 5141  dom cdm 5669  β€˜cfv 6536  Ο‰com 7851   β‰Ό cdom 8936  Ξ£*cesum 33554  measurescmeas 33722  Probcprb 33935
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 7721
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-rmo 3370  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-disj 5107  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-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-esum 33555  df-meas 33723  df-prob 33936
This theorem is referenced by:  probun  33947
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