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Theorem 0elsiga 27740
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 27739 . . 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 988 . . . 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 1630 . 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 3609 . . . . . 6  |-  ( x  =  o  ->  (
o  \  x )  =  ( o  \ 
o ) )
7 difid 3888 . . . . . 6  |-  ( o 
\  o )  =  (/)
86, 7syl6eq 2517 . . . . 5  |-  ( x  =  o  ->  (
o  \  x )  =  (/) )
98eleq1d 2529 . . . 4  |-  ( x  =  o  ->  (
( o  \  x
)  e.  S  <->  (/)  e.  S
) )
109rspcva 3205 . . 3  |-  ( ( o  e.  S  /\  A. x  e.  S  ( o  \  x )  e.  S )  ->  (/) 
e.  S )
1110exlimiv 1693 . 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 968   E.wex 1591    e. wcel 1762   A.wral 2807   _Vcvv 3106    \ cdif 3466    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   U.cuni 4238   class class class wbr 4440   ran crn 4993   omcom 6671    ~<_ cdom 7504  sigAlgebracsiga 27733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587  df-siga 27734
This theorem is referenced by:  sigaclfu2  27747  brsiga  27780  measvuni  27811  measinb  27818  measres  27819  measdivcstOLD  27821  measdivcst  27822  cntmeas  27823  mbfmcst  27856  sibfof  27908  nuleldmp  27982  0rrv  28016  dstrvprob  28036
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