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Theorem elsigagen2 28922
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 1005 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  A  e.  V )
21sgsiga 28916 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  (sigaGen `  A
)  e.  U. ran sigAlgebra )
3 sssigagen 28919 . . . 4  |-  ( A  e.  V  ->  A  C_  (sigaGen `  A )
)
4 sspwb 4613 . . . . 5  |-  ( A 
C_  (sigaGen `  A )  <->  ~P A  C_  ~P (sigaGen `  A ) )
54biimpi 197 . . . 4  |-  ( A 
C_  (sigaGen `  A )  ->  ~P A  C_  ~P (sigaGen `  A ) )
61, 3, 53syl 18 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  ~P A  C_ 
~P (sigaGen `  A )
)
7 simp2 1006 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  C_  A )
8 simp3 1007 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  ~<_  om )
9 ctex 28242 . . . . 5  |-  ( B  ~<_  om  ->  B  e.  _V )
10 elpwg 3932 . . . . 5  |-  ( B  e.  _V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
118, 9, 103syl 18 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  ( B  e.  ~P A  <->  B  C_  A
) )
127, 11mpbird 235 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  e. 
~P A )
136, 12sseldd 3408 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  e. 
~P (sigaGen `  A )
)
14 sigaclcu 28891 . 2  |-  ( ( (sigaGen `  A )  e.  U. ran sigAlgebra  /\  B  e. 
~P (sigaGen `  A )  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A ) )
152, 13, 8, 14syl3anc 1264 1  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ w3a 982    e. wcel 1872   _Vcvv 3022    C_ wss 3379   ~Pcpw 3924   U.cuni 4162   class class class wbr 4366   ran crn 4797   ` cfv 5544   omcom 6650    ~<_ cdom 7522  sigAlgebracsiga 28881  sigaGencsigagen 28912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-int 4199  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fv 5552  df-dom 7526  df-siga 28882  df-sigagen 28913
This theorem is referenced by:  sxbrsigalem1  29059
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