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Theorem sgon 33738
Description: A sigma-algebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
Assertion
Ref Expression
sgon (𝑆 ∈ βˆͺ ran sigAlgebra β†’ 𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝑆))

Proof of Theorem sgon
StepHypRef Expression
1 eqid 2728 . 2 βˆͺ 𝑆 = βˆͺ 𝑆
2 issgon 33737 . . 3 (𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝑆) ↔ (𝑆 ∈ βˆͺ ran sigAlgebra ∧ βˆͺ 𝑆 = βˆͺ 𝑆))
32biimpri 227 . 2 ((𝑆 ∈ βˆͺ ran sigAlgebra ∧ βˆͺ 𝑆 = βˆͺ 𝑆) β†’ 𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝑆))
41, 3mpan2 690 1 (𝑆 ∈ βˆͺ ran sigAlgebra β†’ 𝑆 ∈ (sigAlgebraβ€˜βˆͺ 𝑆))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆͺ cuni 4904  ran crn 5674  β€˜cfv 6543  sigAlgebracsiga 33722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-siga 33723
This theorem is referenced by:  elsigass  33739  isrnsigau  33741  unielsiga  33742  sigagenid  33765  1stmbfm  33875  2ndmbfm  33876  unveldomd  34030  probmeasb  34045
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