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Theorem elsigagen2 31464
Description: Any countable union of elements of a set is also in the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.)
Assertion
Ref Expression
elsigagen2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))

Proof of Theorem elsigagen2
StepHypRef Expression
1 simp1 1133 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐴𝑉)
21sgsiga 31458 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → (sigaGen‘𝐴) ∈ ran sigAlgebra)
3 sssigagen 31461 . . . 4 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
4 sspw 4535 . . . 4 (𝐴 ⊆ (sigaGen‘𝐴) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
51, 3, 43syl 18 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
6 simp2 1134 . . . 4 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵𝐴)
7 simp3 1135 . . . . 5 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ≼ ω)
8 ctex 8520 . . . . 5 (𝐵 ≼ ω → 𝐵 ∈ V)
9 elpwg 4525 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
107, 8, 93syl 18 . . . 4 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
116, 10mpbird 260 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ 𝒫 𝐴)
125, 11sseldd 3954 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ 𝒫 (sigaGen‘𝐴))
13 sigaclcu 31433 . 2 (((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 (sigaGen‘𝐴) ∧ 𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
142, 12, 7, 13syl3anc 1368 1 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084  wcel 2115  Vcvv 3480  wss 3919  𝒫 cpw 4522   cuni 4824   class class class wbr 5052  ran crn 5543  cfv 6343  ωcom 7574  cdom 8503  sigAlgebracsiga 31424  sigaGencsigagen 31454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-dom 8507  df-siga 31425  df-sigagen 31455
This theorem is referenced by:  sxbrsigalem1  31600
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