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Theorem probtot 29801
 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 29798 . 2 (𝑃 ∈ Prob ↔ (𝑃 ran measures ∧ (𝑃 dom 𝑃) = 1))
21simprbi 479 1 (𝑃 ∈ Prob → (𝑃 dom 𝑃) = 1)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∪ cuni 4372  dom cdm 5038  ran crn 5039  ‘cfv 5804  1c1 9816  measurescmeas 29585  Probcprb 29796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-dm 5048  df-iota 5768  df-fv 5812  df-prob 29797 This theorem is referenced by:  prob01  29802  probdsb  29811  dstrvprob  29860  dstfrvclim1  29866
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