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Theorem sigaclcu 27983
Description: A sigma-algebra is closed under countable union. (Contributed by Thierry Arnoux, 26-Dec-2016.)
Assertion
Ref Expression
sigaclcu  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  ~<_  om )  ->  U. A  e.  S
)

Proof of Theorem sigaclcu
Dummy variables  o  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 996 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  ~<_  om )  ->  A  e.  ~P S
)
2 isrnsiga 27979 . . . . 5  |-  ( S  e.  U. ran sigAlgebra  <->  ( S  e.  _V  /\  E. o
( S  C_  ~P o  /\  ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om 
->  U. x  e.  S
) ) ) ) )
32simprbi 464 . . . 4  |-  ( S  e.  U. ran sigAlgebra  ->  E. o
( S  C_  ~P o  /\  ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om 
->  U. x  e.  S
) ) ) )
4 simpr3 1003 . . . . 5  |-  ( ( S  C_  ~P o  /\  ( o  e.  S  /\  A. x  e.  S  ( o  \  x
)  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) )  ->  A. x  e.  ~P  S ( x  ~<_  om  ->  U. x  e.  S ) )
54exlimiv 1707 . . . 4  |-  ( E. o ( S  C_  ~P o  /\  (
o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) )  ->  A. x  e.  ~P  S ( x  ~<_  om  ->  U. x  e.  S ) )
63, 5syl 16 . . 3  |-  ( S  e.  U. ran sigAlgebra  ->  A. x  e.  ~P  S ( x  ~<_  om  ->  U. x  e.  S ) )
763ad2ant1 1016 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  ~<_  om )  ->  A. x  e.  ~P  S ( x  ~<_  om 
->  U. x  e.  S
) )
8 simp3 997 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  ~<_  om )  ->  A  ~<_  om )
9 breq1 4436 . . . 4  |-  ( x  =  A  ->  (
x  ~<_  om  <->  A  ~<_  om )
)
10 unieq 4238 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
1110eleq1d 2510 . . . 4  |-  ( x  =  A  ->  ( U. x  e.  S  <->  U. A  e.  S ) )
129, 11imbi12d 320 . . 3  |-  ( x  =  A  ->  (
( x  ~<_  om  ->  U. x  e.  S )  <-> 
( A  ~<_  om  ->  U. A  e.  S ) ) )
1312rspcv 3190 . 2  |-  ( A  e.  ~P S  -> 
( A. x  e. 
~P  S ( x  ~<_  om  ->  U. x  e.  S )  ->  ( A  ~<_  om  ->  U. A  e.  S ) ) )
141, 7, 8, 13syl3c 61 1  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  ~<_  om )  ->  U. A  e.  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381   E.wex 1597    e. wcel 1802   A.wral 2791   _Vcvv 3093    \ cdif 3455    C_ wss 3458   ~Pcpw 3993   U.cuni 4230   class class class wbr 4433   ran crn 4986   omcom 6681    ~<_ cdom 7512  sigAlgebracsiga 27973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-fv 5582  df-siga 27974
This theorem is referenced by:  sigaclcuni  27984  sigaclfu  27985  sigaclcu2  27986  sigainb  28002  elsigagen2  28014  measinb  28058  measres  28059  measdivcstOLD  28061  measdivcst  28062  imambfm  28099  totprobd  28231  dstrvprob  28276
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