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Theorem sssigagen 28963
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 4270 . 2  |-  A  C_  |^|
{ s  e.  (sigAlgebra ` 
U. A )  |  A  C_  s }
2 sigagenval 28958 . 2  |-  ( A  e.  V  ->  (sigaGen `  A )  =  |^| { s  e.  (sigAlgebra `  U. A )  |  A  C_  s } )
31, 2syl5sseqr 3513 1  |-  ( A  e.  V  ->  A  C_  (sigaGen `  A )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1868   {crab 2779    C_ wss 3436   U.cuni 4216   |^|cint 4252   ` cfv 5598  sigAlgebracsiga 28925  sigaGencsigagen 28956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-int 4253  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-iota 5562  df-fun 5600  df-fv 5606  df-siga 28926  df-sigagen 28957
This theorem is referenced by:  sssigagen2  28964  elsigagen  28965  elsigagen2  28966  sigagenid  28969  elsx  29012  imambfm  29080  cnmbfm  29081  elmbfmvol2  29085  sxbrsigalem3  29090  orvcoel  29290
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