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Theorem elsigagen2 30016
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 1059 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐴𝑉)
21sgsiga 30010 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → (sigaGen‘𝐴) ∈ ran sigAlgebra)
3 sssigagen 30013 . . . 4 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
4 sspwb 4883 . . . . 5 (𝐴 ⊆ (sigaGen‘𝐴) ↔ 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
54biimpi 206 . . . 4 (𝐴 ⊆ (sigaGen‘𝐴) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
61, 3, 53syl 18 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
7 simp2 1060 . . . 4 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵𝐴)
8 simp3 1061 . . . . 5 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ≼ ω)
9 ctex 7922 . . . . 5 (𝐵 ≼ ω → 𝐵 ∈ V)
10 elpwg 4143 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
118, 9, 103syl 18 . . . 4 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
127, 11mpbird 247 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ 𝒫 𝐴)
136, 12sseldd 3588 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ 𝒫 (sigaGen‘𝐴))
14 sigaclcu 29985 . 2 (((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 (sigaGen‘𝐴) ∧ 𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
152, 13, 8, 14syl3anc 1323 1 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036  wcel 1987  Vcvv 3189  wss 3559  𝒫 cpw 4135   cuni 4407   class class class wbr 4618  ran crn 5080  cfv 5852  ωcom 7019  cdom 7905  sigAlgebracsiga 29975  sigaGencsigagen 30006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fv 5860  df-dom 7909  df-siga 29976  df-sigagen 30007
This theorem is referenced by:  sxbrsigalem1  30152
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