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Theorem 0elsiga 26557
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 26556 . . 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 464 . 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 985 . . . 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 466 . . 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 1625 . 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 3468 . . . . . 6  |-  ( x  =  o  ->  (
o  \  x )  =  ( o  \ 
o ) )
7 difid 3747 . . . . . 6  |-  ( o 
\  o )  =  (/)
86, 7syl6eq 2491 . . . . 5  |-  ( x  =  o  ->  (
o  \  x )  =  (/) )
98eleq1d 2509 . . . 4  |-  ( x  =  o  ->  (
( o  \  x
)  e.  S  <->  (/)  e.  S
) )
109rspcva 3071 . . 3  |-  ( ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S )  ->  (/) 
e.  S )
1110exlimiv 1688 . 2  |-  ( E. o ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S
)  ->  (/)  e.  S
)
122, 5, 113syl 20 1  |-  ( S  e.  U. ran sigAlgebra  ->  (/)  e.  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965   E.wex 1586    e. wcel 1756   A.wral 2715   _Vcvv 2972    \ cdif 3325    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   U.cuni 4091   class class class wbr 4292   ran crn 4841   omcom 6476    ~<_ cdom 7308  sigAlgebracsiga 26550
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 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-fv 5426  df-siga 26551
This theorem is referenced by:  sigaclfu2  26564  brsiga  26597  measvuni  26628  measinb  26635  measres  26636  measdivcstOLD  26638  measdivcst  26639  cntmeas  26640  mbfmcst  26674  sibfof  26726  nuleldmp  26800  0rrv  26834  dstrvprob  26854
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