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Theorem probtot 34394
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 34391 . 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 1537  wcel 2106   cuni 4912  dom cdm 5689  ran crn 5690  cfv 6563  1c1 11154  measurescmeas 34176  Probcprb 34389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-dm 5699  df-iota 6516  df-fv 6571  df-prob 34390
This theorem is referenced by:  prob01  34395  probdsb  34404  dstrvprob  34453  dstfrvclim1  34459
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