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Theorem sigaclcu 28932
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 1006 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  ~<_  om )  ->  A  e.  ~P S
)
2 isrnsiga 28928 . . . . 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 465 . . . 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 1013 . . . . 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 1766 . . . 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 17 . . 3  |-  ( S  e.  U. ran sigAlgebra  ->  A. x  e.  ~P  S ( x  ~<_  om  ->  U. x  e.  S ) )
763ad2ant1 1026 . 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 1007 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  ~<_  om )  ->  A  ~<_  om )
9 breq1 4423 . . . 4  |-  ( x  =  A  ->  (
x  ~<_  om  <->  A  ~<_  om )
)
10 unieq 4224 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
1110eleq1d 2491 . . . 4  |-  ( x  =  A  ->  ( U. x  e.  S  <->  U. A  e.  S ) )
129, 11imbi12d 321 . . 3  |-  ( x  =  A  ->  (
( x  ~<_  om  ->  U. x  e.  S )  <-> 
( A  ~<_  om  ->  U. A  e.  S ) ) )
1312rspcv 3178 . 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 63 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 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1868   A.wral 2775   _Vcvv 3081    \ cdif 3433    C_ wss 3436   ~Pcpw 3979   U.cuni 4216   class class class wbr 4420   ran crn 4850   omcom 6702    ~<_ cdom 7571  sigAlgebracsiga 28922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-fv 5605  df-siga 28923
This theorem is referenced by:  sigaclcuni  28933  sigaclfu  28934  sigaclcu2  28935  sigainb  28951  elsigagen2  28963  sigaldsys  28974  measinb  29036  measres  29037  measdivcstOLD  29039  measdivcst  29040  imambfm  29077  totprobd  29252  dstrvprob  29297
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