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Theorem nuleldmp 26967
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 26961 . 2  |-  ( P  e. Prob  ->  dom  P  e.  U.
ran sigAlgebra )
2 0elsiga 26725 . 2  |-  ( dom 
P  e.  U. ran sigAlgebra  ->  (/)  e.  dom  P )
31, 2syl 16 1  |-  ( P  e. Prob  ->  (/)  e.  dom  P
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   (/)c0 3748   U.cuni 4202   dom cdm 4951   ran crn 4952  sigAlgebracsiga 26718  Probcprb 26957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-esum 26652  df-siga 26719  df-meas 26778  df-prob 26958
This theorem is referenced by:  cndprobnul  26987
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