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Theorem elsigagen2 30809
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 1127 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐴𝑉)
21sgsiga 30803 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → (sigaGen‘𝐴) ∈ ran sigAlgebra)
3 sssigagen 30806 . . . 4 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
4 sspwb 5149 . . . . 5 (𝐴 ⊆ (sigaGen‘𝐴) ↔ 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
54biimpi 208 . . . 4 (𝐴 ⊆ (sigaGen‘𝐴) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
61, 3, 53syl 18 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
7 simp2 1128 . . . 4 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵𝐴)
8 simp3 1129 . . . . 5 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ≼ ω)
9 ctex 8256 . . . . 5 (𝐵 ≼ ω → 𝐵 ∈ V)
10 elpwg 4387 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
118, 9, 103syl 18 . . . 4 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
127, 11mpbird 249 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ 𝒫 𝐴)
136, 12sseldd 3822 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ 𝒫 (sigaGen‘𝐴))
14 sigaclcu 30778 . 2 (((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 (sigaGen‘𝐴) ∧ 𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
152, 13, 8, 14syl3anc 1439 1 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1071  wcel 2107  Vcvv 3398  wss 3792  𝒫 cpw 4379   cuni 4671   class class class wbr 4886  ran crn 5356  cfv 6135  ωcom 7343  cdom 8239  sigAlgebracsiga 30768  sigaGencsigagen 30799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143  df-dom 8243  df-siga 30769  df-sigagen 30800
This theorem is referenced by:  sxbrsigalem1  30945
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