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Theorem 0elsiga 28443
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 28442 . . 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 462 . 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 992 . . . 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 464 . . 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 1675 . 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 3552 . . . . . 6  |-  ( x  =  o  ->  (
o  \  x )  =  ( o  \ 
o ) )
7 difid 3837 . . . . . 6  |-  ( o 
\  o )  =  (/)
86, 7syl6eq 2457 . . . . 5  |-  ( x  =  o  ->  (
o  \  x )  =  (/) )
98eleq1d 2469 . . . 4  |-  ( x  =  o  ->  (
( o  \  x
)  e.  S  <->  (/)  e.  S
) )
109rspcva 3155 . . 3  |-  ( ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S )  ->  (/) 
e.  S )
1110exlimiv 1741 . 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 367    /\ w3a 972   E.wex 1631    e. wcel 1840   A.wral 2751   _Vcvv 3056    \ cdif 3408    C_ wss 3411   (/)c0 3735   ~Pcpw 3952   U.cuni 4188   class class class wbr 4392   ran crn 4941   omcom 6636    ~<_ cdom 7470  sigAlgebracsiga 28436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-fv 5531  df-siga 28437
This theorem is referenced by:  sigaclfu2  28450  sigaldsys  28488  brsiga  28512  measvuni  28543  measinb  28550  measres  28551  measdivcstOLD  28553  measdivcst  28554  cntmeas  28555  volmeas  28561  mbfmcst  28588  sibfof  28669  nuleldmp  28743  0rrv  28777  dstrvprob  28797
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