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Theorem nuleldmp 30809
 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 30803 . 2 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
2 0elsiga 30507 . 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 2139  ∅c0 4058  ∪ cuni 4588  dom cdm 5266  ran crn 5267  sigAlgebracsiga 30500  Probcprb 30799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-esum 30420  df-siga 30501  df-meas 30589  df-prob 30800 This theorem is referenced by:  cndprobnul  30829
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