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Theorem nuleldmp 34397
Description: The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
Assertion
Ref Expression
nuleldmp (𝑃 ∈ Prob → ∅ ∈ dom 𝑃)

Proof of Theorem nuleldmp
StepHypRef Expression
1 domprobsiga 34391 . 2 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
2 0elsiga 34093 . 2 (dom 𝑃 ran sigAlgebra → ∅ ∈ dom 𝑃)
31, 2syl 17 1 (𝑃 ∈ Prob → ∅ ∈ dom 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  c0 4332   cuni 4905  dom cdm 5683  ran crn 5684  sigAlgebracsiga 34087  Probcprb 34387
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
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-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fv 6567  df-ov 7432  df-esum 34007  df-siga 34088  df-meas 34175  df-prob 34388
This theorem is referenced by:  cndprobnul  34417
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