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Theorem sigaclcu 29507
 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 1055 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴 ∈ 𝒫 𝑆)
2 isrnsiga 29503 . . . . 5 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
32simprbi 479 . . . 4 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
4 simpr3 1062 . . . . 5 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
54exlimiv 1845 . . . 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 1075 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
8 simp3 1056 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴 ≼ ω)
9 breq1 4586 . . . 4 (𝑥 = 𝐴 → (𝑥 ≼ ω ↔ 𝐴 ≼ ω))
10 unieq 4380 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
1110eleq1d 2672 . . . 4 (𝑥 = 𝐴 → ( 𝑥𝑆 𝐴𝑆))
129, 11imbi12d 333 . . 3 (𝑥 = 𝐴 → ((𝑥 ≼ ω → 𝑥𝑆) ↔ (𝐴 ≼ ω → 𝐴𝑆)))
1312rspcv 3278 . 2 (𝐴 ∈ 𝒫 𝑆 → (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆) → (𝐴 ≼ ω → 𝐴𝑆)))
141, 7, 8, 13syl3c 64 1 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  ran crn 5039  ωcom 6957   ≼ cdom 7839  sigAlgebracsiga 29497 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-siga 29498 This theorem is referenced by:  sigaclcuni  29508  sigaclfu  29509  sigaclcu2  29510  sigainb  29526  elsigagen2  29538  sigaldsys  29549  measinb  29611  measres  29612  measdivcstOLD  29614  measdivcst  29615  imambfm  29651  totprobd  29815  dstrvprob  29860
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