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Theorem isrnsigau 30788
Description: The property of being a sigma-algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016.)
Assertion
Ref Expression
isrnsigau (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
Distinct variable group:   𝑥,𝑆

Proof of Theorem isrnsigau
StepHypRef Expression
1 sgon 30785 . 2 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
2 elex 3414 . . 3 (𝑆 ran sigAlgebra → 𝑆 ∈ V)
3 issiga 30772 . . 3 (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘ 𝑆) ↔ (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
42, 3syl 17 . 2 (𝑆 ran sigAlgebra → (𝑆 ∈ (sigAlgebra‘ 𝑆) ↔ (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
51, 4mpbid 224 1 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071  wcel 2107  wral 3090  Vcvv 3398  cdif 3789  wss 3792  𝒫 cpw 4379   cuni 4671   class class class wbr 4886  ran crn 5356  cfv 6135  ωcom 7343  cdom 8239  sigAlgebracsiga 30768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143  df-siga 30769
This theorem is referenced by:  sigaclci  30793  difelsiga  30794  unelsiga  30795  cntmeas  30887  probfinmeasbOLD  31089
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