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Theorem sigagensiga 26722
Description: A generated sigma algebra is a sigma algebra. (Contributed by Thierry Arnoux, 27-Dec-2016.)
Assertion
Ref Expression
sigagensiga  |-  ( A  e.  V  ->  (sigaGen `  A )  e.  (sigAlgebra ` 
U. A ) )

Proof of Theorem sigagensiga
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 sigagenval 26721 . 2  |-  ( A  e.  V  ->  (sigaGen `  A )  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
2 fvex 5802 . . . . 5  |-  (sigaGen `  A
)  e.  _V
31, 2syl6eqelr 2548 . . . 4  |-  ( A  e.  V  ->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  _V )
4 intex 4549 . . . 4  |-  ( { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  =/=  (/)  <->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  _V )
53, 4sylibr 212 . . 3  |-  ( A  e.  V  ->  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  =/=  (/) )
6 ssrab2 3538 . . . . 5  |-  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  C_  (sigAlgebra `  U. A )
76a1i 11 . . . 4  |-  ( A  e.  V  ->  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  C_  (sigAlgebra `  U. A ) )
8 fvex 5802 . . . . 5  |-  (sigAlgebra `  U. A )  e.  _V
98elpw2 4557 . . . 4  |-  ( { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  ~P (sigAlgebra `
 U. A )  <->  { s  e.  (sigAlgebra ` 
U. A )  |  A  C_  s }  C_  (sigAlgebra `  U. A ) )
107, 9sylibr 212 . . 3  |-  ( A  e.  V  ->  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  ~P (sigAlgebra ` 
U. A ) )
11 insiga 26718 . . 3  |-  ( ( { s  e.  (sigAlgebra ` 
U. A )  |  A  C_  s }  =/=  (/)  /\  { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  ~P (sigAlgebra ` 
U. A ) )  ->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  (sigAlgebra `  U. A ) )
125, 10, 11syl2anc 661 . 2  |-  ( A  e.  V  ->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  (sigAlgebra `  U. A ) )
131, 12eqeltrd 2539 1  |-  ( A  e.  V  ->  (sigaGen `  A )  e.  (sigAlgebra ` 
U. A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758    =/= wne 2644   {crab 2799   _Vcvv 3071    C_ wss 3429   (/)c0 3738   ~Pcpw 3961   U.cuni 4192   |^|cint 4229   ` cfv 5519  sigAlgebracsiga 26688  sigaGencsigagen 26719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-int 4230  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-siga 26689  df-sigagen 26720
This theorem is referenced by:  sgsiga  26723  unisg  26724  sigagenss2  26731  brsiga  26735  brsigarn  26736  cldssbrsiga  26739  sxsiga  26743  cnmbfm  26815  sxbrsiga  26842
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