Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elsigagen2 Structured version   Visualization version   GIF version

Theorem elsigagen2 30745
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 1170 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐴𝑉)
21sgsiga 30739 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → (sigaGen‘𝐴) ∈ ran sigAlgebra)
3 sssigagen 30742 . . . 4 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
4 sspwb 5138 . . . . 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 1171 . . . 4 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵𝐴)
8 simp3 1172 . . . . 5 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ≼ ω)
9 ctex 8237 . . . . 5 (𝐵 ≼ ω → 𝐵 ∈ V)
10 elpwg 4386 . . . . 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 3828 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ 𝒫 (sigaGen‘𝐴))
14 sigaclcu 30714 . 2 (((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 (sigaGen‘𝐴) ∧ 𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
152, 13, 8, 14syl3anc 1494 1 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1111  wcel 2164  Vcvv 3414  wss 3798  𝒫 cpw 4378   cuni 4658   class class class wbr 4873  ran crn 5343  cfv 6123  ωcom 7326  cdom 8220  sigAlgebracsiga 30704  sigaGencsigagen 30735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-int 4698  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-fv 6131  df-dom 8224  df-siga 30705  df-sigagen 30736
This theorem is referenced by:  sxbrsigalem1  30881
  Copyright terms: Public domain W3C validator