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Theorem pwsiga 26578
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 3380 . . 3  |-  ~P O  C_ 
~P O
21a1i 11 . 2  |-  ( O  e.  V  ->  ~P O  C_  ~P O )
3 pwidg 3878 . . 3  |-  ( O  e.  V  ->  O  e.  ~P O )
4 difss 3488 . . . . . 6  |-  ( O 
\  x )  C_  O
5 elpw2g 4460 . . . . . 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 2803 . . 3  |-  ( O  e.  V  ->  A. x  e.  ~P  O ( O 
\  x )  e. 
~P O )
9 sspwuni 4261 . . . . . . . 8  |-  ( x 
C_  ~P O  <->  U. x  C_  O )
10 vex 2980 . . . . . . . . . 10  |-  x  e. 
_V
1110uniex 6381 . . . . . . . . 9  |-  U. x  e.  _V
1211elpw 3871 . . . . . . . 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 3874 . . . . . 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 2803 . . 3  |-  ( O  e.  V  ->  A. x  e.  ~P  ~P O ( x  ~<_  om  ->  U. x  e.  ~P O ) )
203, 8, 193jca 1168 . 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 4481 . . 3  |-  ( O  e.  V  ->  ~P O  e.  _V )
22 issiga 26559 . . 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 913 1  |-  ( O  e.  V  ->  ~P O  e.  (sigAlgebra `  O
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1756   A.wral 2720   _Vcvv 2977    \ cdif 3330    C_ wss 3333   ~Pcpw 3865   U.cuni 4096   class class class wbr 4297   ` cfv 5423   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  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-fal 1375  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-iota 5386  df-fun 5425  df-fv 5431  df-siga 26556
This theorem is referenced by:  sigagenval  26588  dmsigagen  26592  pwcntmeas  26646  ddemeas  26657  mbfmcnt  26688
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