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Theorem probnul 33054
Description: The probability of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Assertion
Ref Expression
probnul (𝑃 ∈ Prob β†’ (π‘ƒβ€˜βˆ…) = 0)

Proof of Theorem probnul
StepHypRef Expression
1 domprobmeas 33050 . 2 (𝑃 ∈ Prob β†’ 𝑃 ∈ (measuresβ€˜dom 𝑃))
2 measvnul 32845 . 2 (𝑃 ∈ (measuresβ€˜dom 𝑃) β†’ (π‘ƒβ€˜βˆ…) = 0)
31, 2syl 17 1 (𝑃 ∈ Prob β†’ (π‘ƒβ€˜βˆ…) = 0)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ…c0 4287  dom cdm 5638  β€˜cfv 6501  0cc0 11058  measurescmeas 32834  Probcprb 33047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-esum 32667  df-meas 32835  df-prob 33048
This theorem is referenced by:  probun  33059  cndprobnul  33077  dstrvprob  33111
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