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Theorem probtot 33697
Description: The probability of the universe set is 1. Second axiom of Kolmogorov. (Contributed by Thierry Arnoux, 8-Dec-2016.)
Assertion
Ref Expression
probtot (𝑃 ∈ Prob → (𝑃 dom 𝑃) = 1)

Proof of Theorem probtot
StepHypRef Expression
1 elprob 33694 . 2 (𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))
21simprbi 497 1 (𝑃 ∈ Prob → (𝑃 dom 𝑃) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106   cuni 4908  dom cdm 5676  ran crn 5677  cfv 6543  1c1 11113  measurescmeas 33479  Probcprb 33692
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-dm 5686  df-iota 6495  df-fv 6551  df-prob 33693
This theorem is referenced by:  prob01  33698  probdsb  33707  dstrvprob  33756  dstfrvclim1  33762
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