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Theorem pwsiga 27767
Description: Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
Assertion
Ref Expression
pwsiga  |-  ( O  e.  V  ->  ~P O  e.  (sigAlgebra `  O
) )

Proof of Theorem pwsiga
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssid 3523 . . 3  |-  ~P O  C_ 
~P O
21a1i 11 . 2  |-  ( O  e.  V  ->  ~P O  C_  ~P O )
3 pwidg 4023 . . 3  |-  ( O  e.  V  ->  O  e.  ~P O )
4 difss 3631 . . . . . 6  |-  ( O 
\  x )  C_  O
5 elpw2g 4610 . . . . . 6  |-  ( O  e.  V  ->  (
( O  \  x
)  e.  ~P O  <->  ( O  \  x ) 
C_  O ) )
64, 5mpbiri 233 . . . . 5  |-  ( O  e.  V  ->  ( O  \  x )  e. 
~P O )
76a1d 25 . . . 4  |-  ( O  e.  V  ->  (
x  e.  ~P O  ->  ( O  \  x
)  e.  ~P O
) )
87ralrimiv 2876 . . 3  |-  ( O  e.  V  ->  A. x  e.  ~P  O ( O 
\  x )  e. 
~P O )
9 sspwuni 4411 . . . . . . . 8  |-  ( x 
C_  ~P O  <->  U. x  C_  O )
10 vex 3116 . . . . . . . . . 10  |-  x  e. 
_V
1110uniex 6578 . . . . . . . . 9  |-  U. x  e.  _V
1211elpw 4016 . . . . . . . 8  |-  ( U. x  e.  ~P O  <->  U. x  C_  O )
139, 12bitr4i 252 . . . . . . 7  |-  ( x 
C_  ~P O  <->  U. x  e.  ~P O )
1413biimpi 194 . . . . . 6  |-  ( x 
C_  ~P O  ->  U. x  e.  ~P O )
1514a1d 25 . . . . 5  |-  ( x 
C_  ~P O  ->  (
x  ~<_  om  ->  U. x  e.  ~P O ) )
16 elpwi 4019 . . . . . 6  |-  ( x  e.  ~P ~P O  ->  x  C_  ~P O
)
1716imim1i 58 . . . . 5  |-  ( ( x  C_  ~P O  ->  ( x  ~<_  om  ->  U. x  e.  ~P O
) )  ->  (
x  e.  ~P ~P O  ->  ( x  ~<_  om 
->  U. x  e.  ~P O ) ) )
1815, 17mp1i 12 . . . 4  |-  ( O  e.  V  ->  (
x  e.  ~P ~P O  ->  ( x  ~<_  om 
->  U. x  e.  ~P O ) ) )
1918ralrimiv 2876 . . 3  |-  ( O  e.  V  ->  A. x  e.  ~P  ~P O ( x  ~<_  om  ->  U. x  e.  ~P O ) )
203, 8, 193jca 1176 . 2  |-  ( O  e.  V  ->  ( O  e.  ~P O  /\  A. x  e.  ~P  O ( O  \  x )  e.  ~P O  /\  A. x  e. 
~P  ~P O ( x  ~<_  om  ->  U. x  e.  ~P O ) ) )
21 pwexg 4631 . . 3  |-  ( O  e.  V  ->  ~P O  e.  _V )
22 issiga 27748 . . 3  |-  ( ~P O  e.  _V  ->  ( ~P O  e.  (sigAlgebra `  O )  <->  ( ~P O  C_  ~P O  /\  ( O  e.  ~P O  /\  A. x  e. 
~P  O ( O 
\  x )  e. 
~P O  /\  A. x  e.  ~P  ~P O
( x  ~<_  om  ->  U. x  e.  ~P O
) ) ) ) )
2321, 22syl 16 . 2  |-  ( O  e.  V  ->  ( ~P O  e.  (sigAlgebra `  O )  <->  ( ~P O  C_  ~P O  /\  ( O  e.  ~P O  /\  A. x  e. 
~P  O ( O 
\  x )  e. 
~P O  /\  A. x  e.  ~P  ~P O
( x  ~<_  om  ->  U. x  e.  ~P O
) ) ) ) )
242, 20, 23mpbir2and 920 1  |-  ( O  e.  V  ->  ~P O  e.  (sigAlgebra `  O
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767   A.wral 2814   _Vcvv 3113    \ cdif 3473    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447   ` cfv 5586   omcom 6678    ~<_ cdom 7511  sigAlgebracsiga 27744
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-siga 27745
This theorem is referenced by:  sigagenval  27777  dmsigagen  27781  pwcntmeas  27835  ddemeas  27845  mbfmcnt  27876
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