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Theorem sigaclcu 30510
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 1132 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴 ∈ 𝒫 𝑆)
2 isrnsiga 30506 . . . . 5 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
32simprbi 483 . . . 4 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
4 simpr3 1238 . . . . 5 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
54exlimiv 2007 . . . 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 1128 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
8 simp3 1133 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴 ≼ ω)
9 breq1 4807 . . . 4 (𝑥 = 𝐴 → (𝑥 ≼ ω ↔ 𝐴 ≼ ω))
10 unieq 4596 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
1110eleq1d 2824 . . . 4 (𝑥 = 𝐴 → ( 𝑥𝑆 𝐴𝑆))
129, 11imbi12d 333 . . 3 (𝑥 = 𝐴 → ((𝑥 ≼ ω → 𝑥𝑆) ↔ (𝐴 ≼ ω → 𝐴𝑆)))
1312rspcv 3445 . 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 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  wral 3050  Vcvv 3340  cdif 3712  wss 3715  𝒫 cpw 4302   cuni 4588   class class class wbr 4804  ran crn 5267  ωcom 7231  cdom 8121  sigAlgebracsiga 30500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-siga 30501
This theorem is referenced by:  sigaclcuni  30511  sigaclfu  30512  sigaclcu2  30513  sigainb  30529  elsigagen2  30541  sigaldsys  30552  measinb  30614  measres  30615  measdivcstOLD  30617  measdivcst  30618  imambfm  30654  totprobd  30818  dstrvprob  30863
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