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Theorem nuleldmp 32429
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 32423 . 2 (𝑃 ∈ Prob → dom 𝑃 ran sigAlgebra)
2 0elsiga 32127 . 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 4262   cuni 4844  dom cdm 5600  ran crn 5601  sigAlgebracsiga 32121  Probcprb 32419
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fv 6466  df-ov 7310  df-esum 32041  df-siga 32122  df-meas 32209  df-prob 32420
This theorem is referenced by:  cndprobnul  32449
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