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Theorem sigagensiga 28371
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 28370 . 2  |-  ( A  e.  V  ->  (sigaGen `  A )  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
2 fvex 5858 . . . . 5  |-  (sigaGen `  A
)  e.  _V
31, 2syl6eqelr 2551 . . . 4  |-  ( A  e.  V  ->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  _V )
4 intex 4593 . . . 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 3571 . . . . 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 5858 . . . . 5  |-  (sigAlgebra `  U. A )  e.  _V
98elpw2 4601 . . . 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 28367 . . 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 659 . 2  |-  ( A  e.  V  ->  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s }  e.  (sigAlgebra `  U. A ) )
131, 12eqeltrd 2542 1  |-  ( A  e.  V  ->  (sigaGen `  A )  e.  (sigAlgebra ` 
U. A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235   |^|cint 4271   ` cfv 5570  sigAlgebracsiga 28337  sigaGencsigagen 28368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-siga 28338  df-sigagen 28369
This theorem is referenced by:  sgsiga  28372  unisg  28373  sigagenss2  28380  brsiga  28391  brsigarn  28392  cldssbrsiga  28395  sxsiga  28399  cnmbfm  28471  sxbrsiga  28498
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