Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwsiga Structured version   Visualization version   Unicode version

Theorem pwsiga 28945
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 3450 . . 3  |-  ~P O  C_ 
~P O
21a1i 11 . 2  |-  ( O  e.  V  ->  ~P O  C_  ~P O )
3 pwidg 3963 . . 3  |-  ( O  e.  V  ->  O  e.  ~P O )
4 difss 3559 . . . . . 6  |-  ( O 
\  x )  C_  O
5 elpw2g 4565 . . . . . 6  |-  ( O  e.  V  ->  (
( O  \  x
)  e.  ~P O  <->  ( O  \  x ) 
C_  O ) )
64, 5mpbiri 237 . . . . 5  |-  ( O  e.  V  ->  ( O  \  x )  e. 
~P O )
76a1d 26 . . . 4  |-  ( O  e.  V  ->  (
x  e.  ~P O  ->  ( O  \  x
)  e.  ~P O
) )
87ralrimiv 2799 . . 3  |-  ( O  e.  V  ->  A. x  e.  ~P  O ( O 
\  x )  e. 
~P O )
9 sspwuni 4366 . . . . . . . 8  |-  ( x 
C_  ~P O  <->  U. x  C_  O )
10 vex 3047 . . . . . . . . . 10  |-  x  e. 
_V
1110uniex 6584 . . . . . . . . 9  |-  U. x  e.  _V
1211elpw 3956 . . . . . . . 8  |-  ( U. x  e.  ~P O  <->  U. x  C_  O )
139, 12bitr4i 256 . . . . . . 7  |-  ( x 
C_  ~P O  <->  U. x  e.  ~P O )
1413biimpi 198 . . . . . 6  |-  ( x 
C_  ~P O  ->  U. x  e.  ~P O )
1514a1d 26 . . . . 5  |-  ( x 
C_  ~P O  ->  (
x  ~<_  om  ->  U. x  e.  ~P O ) )
16 elpwi 3959 . . . . . 6  |-  ( x  e.  ~P ~P O  ->  x  C_  ~P O
)
1716imim1i 60 . . . . 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 2799 . . 3  |-  ( O  e.  V  ->  A. x  e.  ~P  ~P O ( x  ~<_  om  ->  U. x  e.  ~P O ) )
203, 8, 193jca 1187 . 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 4586 . . 3  |-  ( O  e.  V  ->  ~P O  e.  _V )
22 issiga 28926 . . 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 932 1  |-  ( O  e.  V  ->  ~P O  e.  (sigAlgebra `  O
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    e. wcel 1886   A.wral 2736   _Vcvv 3044    \ cdif 3400    C_ wss 3403   ~Pcpw 3950   U.cuni 4197   class class class wbr 4401   ` cfv 5581   omcom 6689    ~<_ cdom 7564  sigAlgebracsiga 28922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-siga 28923
This theorem is referenced by:  sigagenval  28955  dmsigagen  28959  ldsysgenld  28975  pwcntmeas  29042  ddemeas  29052  mbfmcnt  29083
  Copyright terms: Public domain W3C validator