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Theorem probtot 34414
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 34411 . 2 (𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))
21simprbi 496 1 (𝑃 ∈ Prob → (𝑃 dom 𝑃) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108   cuni 4907  dom cdm 5685  ran crn 5686  cfv 6561  1c1 11156  measurescmeas 34196  Probcprb 34409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-dm 5695  df-iota 6514  df-fv 6569  df-prob 34410
This theorem is referenced by:  prob01  34415  probdsb  34424  dstrvprob  34474  dstfrvclim1  34480
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