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Theorem unielsiga 33656
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 33652 . 2 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ 𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝑆))
2 baselsiga 33643 . 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 2098  βˆͺ cuni 4902  ran crn 5670  β€˜cfv 6537  sigAlgebracsiga 33636
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
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-fv 6545  df-siga 33637
This theorem is referenced by:  mbfmcst  33788  1stmbfm  33789  2ndmbfm  33790  imambfm  33791  mbfmco  33793  br2base  33798  prob01  33942  probfinmeasb  33957
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