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Theorem sigasspw 26558
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 2980 . . 3  |-  ( S  e.  (sigAlgebra `  A )  ->  S  e.  _V )
2 issiga 26553 . . . 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 484 . . 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 669 . 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 459 1  |-  ( S  e.  (sigAlgebra `  A )  ->  S  C_  ~P A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1756   A.wral 2714   _Vcvv 2971    \ cdif 3324    C_ wss 3327   ~Pcpw 3859   U.cuni 4090   class class class wbr 4291   ` cfv 5417   omcom 6475    ~<_ cdom 7307  sigAlgebracsiga 26549
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 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-siga 26550
This theorem is referenced by:  elsigass  26567  insiga  26579  brsigasspwrn  26598  1stmbfm  26674  2ndmbfm  26675
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