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 30541
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 1131 . . 3 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐴𝑉)
21sgsiga 30535 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → (sigaGen‘𝐴) ∈ ran sigAlgebra)
3 sssigagen 30538 . . . 4 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
4 sspwb 5066 . . . . 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 1132 . . . 4 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵𝐴)
8 simp3 1133 . . . . 5 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ≼ ω)
9 ctex 8138 . . . . 5 (𝐵 ≼ ω → 𝐵 ∈ V)
10 elpwg 4310 . . . . 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 3745 . 2 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ 𝒫 (sigaGen‘𝐴))
14 sigaclcu 30510 . 2 (((sigaGen‘𝐴) ∈ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 (sigaGen‘𝐴) ∧ 𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
152, 13, 8, 14syl3anc 1477 1 ((𝐴𝑉𝐵𝐴𝐵 ≼ ω) → 𝐵 ∈ (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072  wcel 2139  Vcvv 3340  wss 3715  𝒫 cpw 4302   cuni 4588   class class class wbr 4804  ran crn 5267  cfv 6049  ωcom 7231  cdom 8121  sigAlgebracsiga 30500  sigaGencsigagen 30531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fv 6057  df-dom 8125  df-siga 30501  df-sigagen 30532
This theorem is referenced by:  sxbrsigalem1  30677
  Copyright terms: Public domain W3C validator