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Theorem nuleldmp 34360
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 34354 . 2 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
2 0elsiga 34056 . 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 2104  c0 4339   cuni 4914  dom cdm 5683  ran crn 5684  sigAlgebracsiga 34050  Probcprb 34350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  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 6510  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-ov 7428  df-esum 33970  df-siga 34051  df-meas 34138  df-prob 34351
This theorem is referenced by:  cndprobnul  34380
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