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Theorem sgon 31390
 Description: A sigma-algebra is a sigma on its union set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
Assertion
Ref Expression
sgon (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))

Proof of Theorem sgon
StepHypRef Expression
1 eqid 2821 . 2 𝑆 = 𝑆
2 issgon 31389 . . 3 (𝑆 ∈ (sigAlgebra‘ 𝑆) ↔ (𝑆 ran sigAlgebra ∧ 𝑆 = 𝑆))
32biimpri 231 . 2 ((𝑆 ran sigAlgebra ∧ 𝑆 = 𝑆) → 𝑆 ∈ (sigAlgebra‘ 𝑆))
41, 3mpan2 690 1 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∪ cuni 4811  ran crn 5529  ‘cfv 6328  sigAlgebracsiga 31374 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-siga 31375 This theorem is referenced by:  elsigass  31391  isrnsigau  31393  unielsiga  31394  sigagenid  31417  1stmbfm  31525  2ndmbfm  31526  unveldomd  31680  probmeasb  31695
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