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Theorem sigaclcu 26565
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 989 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  ~<_  om )  ->  A  e.  ~P S
)
2 isrnsiga 26561 . . . . 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 996 . . . . 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 1688 . . . 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 1009 . 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 990 . 2  |-  ( ( S  e.  U. ran sigAlgebra  /\  A  e.  ~P S  /\  A  ~<_  om )  ->  A  ~<_  om )
9 breq1 4300 . . . 4  |-  ( x  =  A  ->  (
x  ~<_  om  <->  A  ~<_  om )
)
10 unieq 4104 . . . . 5  |-  ( x  =  A  ->  U. x  =  U. A )
1110eleq1d 2509 . . . 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 3074 . 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 965    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2720   _Vcvv 2977    \ cdif 3330    C_ wss 3333   ~Pcpw 3865   U.cuni 4096   class class class wbr 4297   ran crn 4846   omcom 6481    ~<_ cdom 7313  sigAlgebracsiga 26555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431  df-siga 26556
This theorem is referenced by:  sigaclcuni  26566  sigaclfu  26567  sigaclcu2  26568  sigainb  26584  elsigagen2  26596  measinb  26640  measres  26641  measdivcstOLD  26643  measdivcst  26644  imambfm  26682  totprobd  26814  dstrvprob  26859
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