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Theorem isrnsigau 28561
Description: The property of being a sigma algebra, universe is the union set. (Contributed by Thierry Arnoux, 11-Nov-2016.)
Assertion
Ref Expression
isrnsigau  |-  ( S  e.  U. ran sigAlgebra  ->  ( S  C_  ~P U. S  /\  ( U. S  e.  S  /\  A. x  e.  S  ( U. S  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om  ->  U. x  e.  S ) ) ) )
Distinct variable group:    x, S

Proof of Theorem isrnsigau
StepHypRef Expression
1 sgon 28558 . 2  |-  ( S  e.  U. ran sigAlgebra  ->  S  e.  (sigAlgebra `  U. S ) )
2 elex 3067 . . 3  |-  ( S  e.  U. ran sigAlgebra  ->  S  e.  _V )
3 issiga 28545 . . 3  |-  ( S  e.  _V  ->  ( S  e.  (sigAlgebra `  U. S )  <->  ( S  C_ 
~P U. S  /\  ( U. S  e.  S  /\  A. x  e.  S  ( U. S  \  x
)  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) ) ) )
42, 3syl 17 . 2  |-  ( S  e.  U. ran sigAlgebra  ->  ( S  e.  (sigAlgebra `  U. S )  <->  ( S  C_ 
~P U. S  /\  ( U. S  e.  S  /\  A. x  e.  S  ( U. S  \  x
)  e.  S  /\  A. x  e.  ~P  S
( x  ~<_  om  ->  U. x  e.  S ) ) ) ) )
51, 4mpbid 210 1  |-  ( S  e.  U. ran sigAlgebra  ->  ( S  C_  ~P U. S  /\  ( U. S  e.  S  /\  A. x  e.  S  ( U. S  \  x )  e.  S  /\  A. x  e.  ~P  S ( x  ~<_  om  ->  U. x  e.  S ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    e. wcel 1842   A.wral 2753   _Vcvv 3058    \ cdif 3410    C_ wss 3413   ~Pcpw 3954   U.cuni 4190   class class class wbr 4394   ran crn 4823   ` cfv 5568   omcom 6682    ~<_ cdom 7551  sigAlgebracsiga 28541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-fv 5576  df-siga 28542
This theorem is referenced by:  sigaclci  28566  difelsiga  28567  unelsiga  28568  cntmeas  28660  probfinmeasbOLD  28859
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