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Theorem 0elsiga 24499
Description: A sigma-algebra contains the empty set. (Contributed by Thierry Arnoux, 4-Sep-2016.)
Assertion
Ref Expression
0elsiga  |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S
)

Proof of Theorem 0elsiga
Dummy variables  o  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isrnsiga 24498 . . 3  |-  ( S  e.  U. ran sigAlgebra  <->  ( S  e.  _V  /\  E. o
( S  C_  ~P o  /\  ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om 
->  U. x  e.  S
) ) ) ) )
21simprbi 452 . 2  |-  ( S  e.  U. ran sigAlgebra  ->  E. o
( S  C_  ~P o  /\  ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om 
->  U. x  e.  S
) ) ) )
3 3simpa 955 . . . 4  |-  ( ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) )  ->  ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S
) )
43adantl 454 . . 3  |-  ( ( S  C_  ~P o  /\  ( o  e.  S  /\  A. x  e.  S  ( o  \  x
)  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) )  ->  (
o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S ) )
54eximi 1586 . 2  |-  ( E. o ( S  C_  ~P o  /\  (
o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) )  ->  E. o
( o  e.  S  /\  A. x  e.  S  ( o  \  x
)  e.  S ) )
6 difeq2 3461 . . . . . 6  |-  ( x  =  o  ->  (
o  \  x )  =  ( o  \ 
o ) )
7 difid 3698 . . . . . 6  |-  ( o 
\  o )  =  (/)
86, 7syl6eq 2486 . . . . 5  |-  ( x  =  o  ->  (
o  \  x )  =  (/) )
98eleq1d 2504 . . . 4  |-  ( x  =  o  ->  (
( o  \  x
)  e.  S  <->  (/)  e.  S
) )
109rspcva 3052 . . 3  |-  ( ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S )  ->  (/) 
e.  S )
1110exlimiv 1645 . 2  |-  ( E. o ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S
)  ->  (/)  e.  S
)
122, 5, 113syl 19 1  |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    e. wcel 1726   A.wral 2707   _Vcvv 2958    \ cdif 3319    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   U.cuni 4017   class class class wbr 4214   omcom 4847   ran crn 4881    ~<_ cdom 7109  sigAlgebracsiga 24492
This theorem is referenced by:  sigaclfu2  24506  brsiga  24539  measvuni  24570  measinb  24577  measres  24578  measdivcstOLD  24580  measdivcst  24581  cntmeas  24582  mbfmcst  24611  sibfof  24656  nuleldmp  24677  0rrv  24711  dstrvprob  24731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464  df-siga 24493
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