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Theorem unveldomd 34064
Description: The universe is an element of the domain of the probability, the universe (entire probability space) being βˆͺ dom 𝑃 in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
Hypothesis
Ref Expression
unveldomd.1 (πœ‘ β†’ 𝑃 ∈ Prob)
Assertion
Ref Expression
unveldomd (πœ‘ β†’ βˆͺ dom 𝑃 ∈ dom 𝑃)

Proof of Theorem unveldomd
StepHypRef Expression
1 unveldomd.1 . 2 (πœ‘ β†’ 𝑃 ∈ Prob)
2 domprobsiga 34060 . 2 (𝑃 ∈ Prob β†’ dom 𝑃 ∈ βˆͺ ran sigAlgebra)
3 sgon 33772 . 2 (dom 𝑃 ∈ βˆͺ ran sigAlgebra β†’ dom 𝑃 ∈ (sigAlgebraβ€˜βˆͺ dom 𝑃))
4 baselsiga 33763 . 2 (dom 𝑃 ∈ (sigAlgebraβ€˜βˆͺ dom 𝑃) β†’ βˆͺ dom 𝑃 ∈ dom 𝑃)
51, 2, 3, 44syl 19 1 (πœ‘ β†’ βˆͺ dom 𝑃 ∈ dom 𝑃)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098  βˆͺ cuni 4901  dom cdm 5670  ran crn 5671  β€˜cfv 6541  sigAlgebracsiga 33756  Probcprb 34056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7417  df-esum 33676  df-siga 33757  df-meas 33844  df-prob 34057
This theorem is referenced by:  unveldom  34065  probdsb  34071  probtotrnd  34074  cndprobtot  34085  0rrv  34100  rrvadd  34101  dstfrvclim1  34126
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