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Theorem sigaclcu 31275
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 1129 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴 ∈ 𝒫 𝑆)
2 isrnsiga 31271 . . . . 5 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
32simprbi 497 . . . 4 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
4 simpr3 1188 . . . . 5 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
54exlimiv 1922 . . . 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 1125 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
8 simp3 1130 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴 ≼ ω)
9 breq1 5060 . . . 4 (𝑥 = 𝐴 → (𝑥 ≼ ω ↔ 𝐴 ≼ ω))
10 unieq 4838 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
1110eleq1d 2894 . . . 4 (𝑥 = 𝐴 → ( 𝑥𝑆 𝐴𝑆))
129, 11imbi12d 346 . . 3 (𝑥 = 𝐴 → ((𝑥 ≼ ω → 𝑥𝑆) ↔ (𝐴 ≼ ω → 𝐴𝑆)))
1312rspcv 3615 . 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 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wral 3135  Vcvv 3492  cdif 3930  wss 3933  𝒫 cpw 4535   cuni 4830   class class class wbr 5057  ran crn 5549  ωcom 7569  cdom 8495  sigAlgebracsiga 31266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-siga 31267
This theorem is referenced by:  sigaclcuni  31276  sigaclfu  31277  sigaclcu2  31278  sigainb  31294  elsigagen2  31306  sigaldsys  31317  measinb  31379  measres  31380  measdivcst  31382  measdivcstALTV  31383  imambfm  31419  totprobd  31583  dstrvprob  31628
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