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Theorem baselsiga 27755
Description: A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.)
Assertion
Ref Expression
baselsiga  |-  ( S  e.  (sigAlgebra `  A )  ->  A  e.  S )

Proof of Theorem baselsiga
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elex 3122 . 2  |-  ( S  e.  (sigAlgebra `  A )  ->  S  e.  _V )
2 issiga 27751 . . . 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 ) ) ) ) )
32simplbda 624 . . 3  |-  ( ( S  e.  _V  /\  S  e.  (sigAlgebra `  A
) )  ->  ( A  e.  S  /\  A. x  e.  S  ( A  \  x )  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) )
43simp1d 1008 . 2  |-  ( ( S  e.  _V  /\  S  e.  (sigAlgebra `  A
) )  ->  A  e.  S )
51, 4mpancom 669 1  |-  ( S  e.  (sigAlgebra `  A )  ->  A  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767   A.wral 2814   _Vcvv 3113    \ cdif 3473    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447   ` cfv 5586   omcom 6678    ~<_ cdom 7511  sigAlgebracsiga 27747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-siga 27748
This theorem is referenced by:  unielsiga  27768  cldssbrsiga  27798  1stmbfm  27871  2ndmbfm  27872  unveldomd  27994  probmeasb  28009  dstrvprob  28050
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