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Theorem sssigagen 28967
Description: A set is a subset of the sigma-algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017.)
Assertion
Ref Expression
sssigagen  |-  ( A  e.  V  ->  A  C_  (sigaGen `  A )
)

Proof of Theorem sssigagen
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 ssintub 4252 . 2  |-  A  C_  |^|
{ s  e.  (sigAlgebra ` 
U. A )  |  A  C_  s }
2 sigagenval 28962 . 2  |-  ( A  e.  V  ->  (sigaGen `  A )  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
31, 2syl5sseqr 3481 1  |-  ( A  e.  V  ->  A  C_  (sigaGen `  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1887   {crab 2741    C_ wss 3404   U.cuni 4198   |^|cint 4234   ` cfv 5582  sigAlgebracsiga 28929  sigaGencsigagen 28960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-iota 5546  df-fun 5584  df-fv 5590  df-siga 28930  df-sigagen 28961
This theorem is referenced by:  sssigagen2  28968  elsigagen  28969  elsigagen2  28970  sigagenid  28973  elsx  29016  imambfm  29084  cnmbfm  29085  elmbfmvol2  29089  sxbrsigalem3  29094  orvcoel  29294
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