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Theorem sigaclcu 29953
Description: A sigma-algebra is closed under countable union. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
sigaclcu ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴𝑆)

Proof of Theorem sigaclcu
Dummy variables 𝑜 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1060 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴 ∈ 𝒫 𝑆)
2 isrnsiga 29949 . . . . 5 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
32simprbi 480 . . . 4 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
4 simpr3 1067 . . . . 5 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
54exlimiv 1860 . . . 4 (∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
63, 5syl 17 . . 3 (𝑆 ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
763ad2ant1 1080 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
8 simp3 1061 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴 ≼ ω)
9 breq1 4621 . . . 4 (𝑥 = 𝐴 → (𝑥 ≼ ω ↔ 𝐴 ≼ ω))
10 unieq 4415 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
1110eleq1d 2688 . . . 4 (𝑥 = 𝐴 → ( 𝑥𝑆 𝐴𝑆))
129, 11imbi12d 334 . . 3 (𝑥 = 𝐴 → ((𝑥 ≼ ω → 𝑥𝑆) ↔ (𝐴 ≼ ω → 𝐴𝑆)))
1312rspcv 3296 . 2 (𝐴 ∈ 𝒫 𝑆 → (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆) → (𝐴 ≼ ω → 𝐴𝑆)))
141, 7, 8, 13syl3c 66 1 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1992  wral 2912  Vcvv 3191  cdif 3557  wss 3560  𝒫 cpw 4135   cuni 4407   class class class wbr 4618  ran crn 5080  ωcom 7013  cdom 7898  sigAlgebracsiga 29943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-fv 5858  df-siga 29944
This theorem is referenced by:  sigaclcuni  29954  sigaclfu  29955  sigaclcu2  29956  sigainb  29972  elsigagen2  29984  sigaldsys  29995  measinb  30057  measres  30058  measdivcstOLD  30060  measdivcst  30061  imambfm  30097  totprobd  30261  dstrvprob  30306
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