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Theorem elsigagen2 24327
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  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A )
)

Proof of Theorem elsigagen2
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  A  e.  V )
21sgsiga 24321 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  (sigaGen `  A
)  e.  U. ran sigAlgebra )
3 sssigagen 24324 . . . 4  |-  ( A  e.  V  ->  A  C_  (sigaGen `  A )
)
4 sspwb 4354 . . . . 5  |-  ( A 
C_  (sigaGen `  A )  <->  ~P A  C_  ~P (sigaGen `  A ) )
54biimpi 187 . . . 4  |-  ( A 
C_  (sigaGen `  A )  ->  ~P A  C_  ~P (sigaGen `  A ) )
61, 3, 53syl 19 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  ~P A  C_ 
~P (sigaGen `  A )
)
7 simp2 958 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  C_  A )
8 simp3 959 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  ~<_  om )
9 ctex 23941 . . . . 5  |-  ( B  ~<_  om  ->  B  e.  _V )
10 elpwg 3749 . . . . 5  |-  ( B  e.  _V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
118, 9, 103syl 19 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  ( B  e.  ~P A  <->  B  C_  A
) )
127, 11mpbird 224 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  e. 
~P A )
136, 12sseldd 3292 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  e. 
~P (sigaGen `  A )
)
14 sigaclcu 24296 . 2  |-  ( ( (sigaGen `  A )  e.  U. ran sigAlgebra  /\  B  e. 
~P (sigaGen `  A )  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A ) )
152, 13, 8, 14syl3anc 1184 1  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    e. wcel 1717   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   class class class wbr 4153   omcom 4785   ran crn 4819   ` cfv 5394    ~<_ cdom 7043  sigAlgebracsiga 24286  sigaGencsigagen 24317
This theorem is referenced by:  sxbrsigalem1  24429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fv 5402  df-dom 7047  df-siga 24287  df-sigagen 24318
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