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Theorem unielsiga 30165
Description: A sigma-algebra contains its universe set. (Contributed by Thierry Arnoux, 13-Feb-2017.) (Shortened by Thierry Arnoux, 6-Jun-2017.)
Assertion
Ref Expression
unielsiga (𝑆 ran sigAlgebra → 𝑆𝑆)

Proof of Theorem unielsiga
StepHypRef Expression
1 sgon 30161 . 2 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
2 baselsiga 30152 . 2 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆𝑆)
31, 2syl 17 1 (𝑆 ran sigAlgebra → 𝑆𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988   cuni 4427  ran crn 5105  cfv 5876  sigAlgebracsiga 30144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-siga 30145
This theorem is referenced by:  mbfmcst  30295  1stmbfm  30296  2ndmbfm  30297  imambfm  30298  mbfmco  30300  br2base  30305  prob01  30449  probfinmeasb  30465
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