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Theorem sigaclcu 31618
 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 1135 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴 ∈ 𝒫 𝑆)
2 isrnsiga 31614 . . . . 5 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
32simprbi 500 . . . 4 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
4 simpr3 1194 . . . . 5 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
54exlimiv 1932 . . . 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 1131 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
8 simp3 1136 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆𝐴 ≼ ω) → 𝐴 ≼ ω)
9 breq1 5040 . . . 4 (𝑥 = 𝐴 → (𝑥 ≼ ω ↔ 𝐴 ≼ ω))
10 unieq 4813 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
1110eleq1d 2837 . . . 4 (𝑥 = 𝐴 → ( 𝑥𝑆 𝐴𝑆))
129, 11imbi12d 348 . . 3 (𝑥 = 𝐴 → ((𝑥 ≼ ω → 𝑥𝑆) ↔ (𝐴 ≼ ω → 𝐴𝑆)))
1312rspcv 3539 . 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 399   ∧ w3a 1085   = wceq 1539  ∃wex 1782   ∈ wcel 2112  ∀wral 3071  Vcvv 3410   ∖ cdif 3858   ⊆ wss 3861  𝒫 cpw 4498  ∪ cuni 4802   class class class wbr 5037  ran crn 5530  ωcom 7586   ≼ cdom 8539  sigAlgebracsiga 31609 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-fv 6349  df-siga 31610 This theorem is referenced by:  sigaclcuni  31619  sigaclfu  31620  sigaclcu2  31621  sigainb  31637  elsigagen2  31649  sigaldsys  31660  measinb  31722  measres  31723  measdivcst  31725  measdivcstALTV  31726  imambfm  31762  totprobd  31926  dstrvprob  31971
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