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Theorem unielsiga 32096
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 32092 . 2 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
2 baselsiga 32083 . 2 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆𝑆)
31, 2syl 17 1 (𝑆 ran sigAlgebra → 𝑆𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   cuni 4839  ran crn 5590  cfv 6433  sigAlgebracsiga 32076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-siga 32077
This theorem is referenced by:  mbfmcst  32226  1stmbfm  32227  2ndmbfm  32228  imambfm  32229  mbfmco  32231  br2base  32236  prob01  32380  probfinmeasb  32395
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