Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  probnul Structured version   Visualization version   GIF version

Theorem probnul 29803
 Description: The probability of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
probnul (𝑃 ∈ Prob → (𝑃‘∅) = 0)

Proof of Theorem probnul
StepHypRef Expression
1 domprobmeas 29799 . 2 (𝑃 ∈ Prob → 𝑃 ∈ (measures‘dom 𝑃))
2 measvnul 29596 . 2 (𝑃 ∈ (measures‘dom 𝑃) → (𝑃‘∅) = 0)
31, 2syl 17 1 (𝑃 ∈ Prob → (𝑃‘∅) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∅c0 3874  dom cdm 5038  ‘cfv 5804  0cc0 9815  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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-esum 29417  df-meas 29586  df-prob 29797 This theorem is referenced by:  probun  29808  cndprobnul  29826  dstrvprob  29860
 Copyright terms: Public domain W3C validator