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Theorem nuleldmp 28742
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  e. Prob  ->  (/)  e.  dom  P
)

Proof of Theorem nuleldmp
StepHypRef Expression
1 domprobsiga 28736 . 2  |-  ( P  e. Prob  ->  dom  P  e.  U.
ran sigAlgebra )
2 0elsiga 28442 . 2  |-  ( dom 
P  e.  U. ran sigAlgebra  ->  (/)  e.  dom  P )
31, 2syl 17 1  |-  ( P  e. Prob  ->  (/)  e.  dom  P
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842   (/)c0 3737   U.cuni 4190   dom cdm 4942   ran crn 4943  sigAlgebracsiga 28435  Probcprb 28732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6237  df-esum 28355  df-siga 28436  df-meas 28524  df-prob 28733
This theorem is referenced by:  cndprobnul  28762
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