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Theorem pwsiga 28578
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 3461 . . 3  |-  ~P O  C_ 
~P O
21a1i 11 . 2  |-  ( O  e.  V  ->  ~P O  C_  ~P O )
3 pwidg 3968 . . 3  |-  ( O  e.  V  ->  O  e.  ~P O )
4 difss 3570 . . . . . 6  |-  ( O 
\  x )  C_  O
5 elpw2g 4557 . . . . . 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 2816 . . 3  |-  ( O  e.  V  ->  A. x  e.  ~P  O ( O 
\  x )  e. 
~P O )
9 sspwuni 4360 . . . . . . . 8  |-  ( x 
C_  ~P O  <->  U. x  C_  O )
10 vex 3062 . . . . . . . . . 10  |-  x  e. 
_V
1110uniex 6578 . . . . . . . . 9  |-  U. x  e.  _V
1211elpw 3961 . . . . . . . 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 3964 . . . . . 6  |-  ( x  e.  ~P ~P O  ->  x  C_  ~P O
)
1716imim1i 57 . . . . 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 13 . . . 4  |-  ( O  e.  V  ->  (
x  e.  ~P ~P O  ->  ( x  ~<_  om 
->  U. x  e.  ~P O ) ) )
1918ralrimiv 2816 . . 3  |-  ( O  e.  V  ->  A. x  e.  ~P  ~P O ( x  ~<_  om  ->  U. x  e.  ~P O ) )
203, 8, 193jca 1177 . 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 4578 . . 3  |-  ( O  e.  V  ->  ~P O  e.  _V )
22 issiga 28559 . . 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 17 . 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 923 1  |-  ( O  e.  V  ->  ~P O  e.  (sigAlgebra `  O
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    e. wcel 1842   A.wral 2754   _Vcvv 3059    \ cdif 3411    C_ wss 3414   ~Pcpw 3955   U.cuni 4191   class class class wbr 4395   ` cfv 5569   omcom 6683    ~<_ cdom 7552  sigAlgebracsiga 28555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-siga 28556
This theorem is referenced by:  sigagenval  28588  dmsigagen  28592  ldsysgenld  28608  pwcntmeas  28675  ddemeas  28685  mbfmcnt  28716
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