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Theorem baselsiga 26510
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 2976 . 2  |-  ( S  e.  (sigAlgebra `  A )  ->  S  e.  _V )
2 issiga 26506 . . . 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 1000 . 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 965    e. wcel 1756   A.wral 2710   _Vcvv 2967    \ cdif 3320    C_ wss 3323   ~Pcpw 3855   U.cuni 4086   class class class wbr 4287   ` cfv 5413   omcom 6471    ~<_ cdom 7300  sigAlgebracsiga 26502
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-siga 26503
This theorem is referenced by:  unielsiga  26523  cldssbrsiga  26553  1stmbfm  26627  2ndmbfm  26628  unveldomd  26750  probmeasb  26765  dstrvprob  26806
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