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Theorem unveldomd 33944
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 33940 . 2 (𝑃 ∈ Prob β†’ dom 𝑃 ∈ βˆͺ ran sigAlgebra)
3 sgon 33652 . 2 (dom 𝑃 ∈ βˆͺ ran sigAlgebra β†’ dom 𝑃 ∈ (sigAlgebraβ€˜βˆͺ dom 𝑃))
4 baselsiga 33643 . 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 4902  dom cdm 5669  ran crn 5670  β€˜cfv 6537  sigAlgebracsiga 33636  Probcprb 33936
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-esum 33556  df-siga 33637  df-meas 33724  df-prob 33937
This theorem is referenced by:  unveldom  33945  probdsb  33951  probtotrnd  33954  cndprobtot  33965  0rrv  33980  rrvadd  33981  dstfrvclim1  34006
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