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Theorem sigasspw 28774
Description: A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.)
Assertion
Ref Expression
sigasspw  |-  ( S  e.  (sigAlgebra `  A )  ->  S  C_  ~P A )

Proof of Theorem sigasspw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 3087 . . 3  |-  ( S  e.  (sigAlgebra `  A )  ->  S  e.  _V )
2 issiga 28769 . . . 4  |-  ( S  e.  _V  ->  ( S  e.  (sigAlgebra `  A
)  <->  ( S  C_  ~P A  /\  ( A  e.  S  /\  A. x  e.  S  ( A  \  x )  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) ) ) )
32biimpa 486 . . 3  |-  ( ( S  e.  _V  /\  S  e.  (sigAlgebra `  A
) )  ->  ( S  C_  ~P A  /\  ( A  e.  S  /\  A. x  e.  S  ( A  \  x
)  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) ) )
41, 3mpancom 673 . 2  |-  ( S  e.  (sigAlgebra `  A )  -> 
( S  C_  ~P A  /\  ( A  e.  S  /\  A. x  e.  S  ( A  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om 
->  U. x  e.  S
) ) ) )
54simpld 460 1  |-  ( S  e.  (sigAlgebra `  A )  ->  S  C_  ~P A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    e. wcel 1867   A.wral 2773   _Vcvv 3078    \ cdif 3430    C_ wss 3433   ~Pcpw 3976   U.cuni 4213   class class class wbr 4417   ` cfv 5592   omcom 6697    ~<_ cdom 7566  sigAlgebracsiga 28765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fv 5600  df-siga 28766
This theorem is referenced by:  elsigass  28783  insiga  28795  sigapisys  28813  sigaldsys  28817  brsigasspwrn  28843  1stmbfm  28918  2ndmbfm  28919
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