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Theorem unielsiga 32767
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 ∈ βˆͺ ran sigAlgebra β†’ βˆͺ 𝑆 ∈ 𝑆)

Proof of Theorem unielsiga
StepHypRef Expression
1 sgon 32763 . 2 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ 𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝑆))
2 baselsiga 32754 . 2 (𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝑆) β†’ βˆͺ 𝑆 ∈ 𝑆)
31, 2syl 17 1 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ βˆͺ 𝑆 ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  βˆͺ cuni 4870  ran crn 5639  β€˜cfv 6501  sigAlgebracsiga 32747
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509  df-siga 32748
This theorem is referenced by:  mbfmcst  32899  1stmbfm  32900  2ndmbfm  32901  imambfm  32902  mbfmco  32904  br2base  32909  prob01  33053  probfinmeasb  33068
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