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Theorem elsigagen2 27973
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 996 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  A  e.  V )
21sgsiga 27967 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  (sigaGen `  A
)  e.  U. ran sigAlgebra )
3 sssigagen 27970 . . . 4  |-  ( A  e.  V  ->  A  C_  (sigaGen `  A )
)
4 sspwb 4702 . . . . 5  |-  ( A 
C_  (sigaGen `  A )  <->  ~P A  C_  ~P (sigaGen `  A ) )
54biimpi 194 . . . 4  |-  ( A 
C_  (sigaGen `  A )  ->  ~P A  C_  ~P (sigaGen `  A ) )
61, 3, 53syl 20 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  ~P A  C_ 
~P (sigaGen `  A )
)
7 simp2 997 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  C_  A )
8 simp3 998 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  ~<_  om )
9 ctex 27354 . . . . 5  |-  ( B  ~<_  om  ->  B  e.  _V )
10 elpwg 4024 . . . . 5  |-  ( B  e.  _V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
118, 9, 103syl 20 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  ( B  e.  ~P A  <->  B  C_  A
) )
127, 11mpbird 232 . . 3  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  e. 
~P A )
136, 12sseldd 3510 . 2  |-  ( ( A  e.  V  /\  B  C_  A  /\  B  ~<_  om )  ->  B  e. 
~P (sigaGen `  A )
)
14 sigaclcu 27942 . 2  |-  ( ( (sigaGen `  A )  e.  U. ran sigAlgebra  /\  B  e. 
~P (sigaGen `  A )  /\  B  ~<_  om )  ->  U. B  e.  (sigaGen `  A ) )
152, 13, 8, 14syl3anc 1228 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 184    /\ w3a 973    e. wcel 1767   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016   U.cuni 4251   class class class wbr 4453   ran crn 5006   ` cfv 5594   omcom 6695    ~<_ cdom 7526  sigAlgebracsiga 27932  sigaGencsigagen 27963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fv 5602  df-dom 7530  df-siga 27933  df-sigagen 27964
This theorem is referenced by:  sxbrsigalem1  28081
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