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Theorem sssigagen 29989
 Description: A set is a subset of the sigma-algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017.)
Assertion
Ref Expression
sssigagen (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))

Proof of Theorem sssigagen
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ssintub 4460 . 2 𝐴 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠}
2 sigagenval 29984 . 2 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
31, 2syl5sseqr 3633 1 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1987  {crab 2911   ⊆ wss 3555  ∪ cuni 4402  ∩ cint 4440  ‘cfv 5847  sigAlgebracsiga 29951  sigaGencsigagen 29982 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-siga 29952  df-sigagen 29983 This theorem is referenced by:  sssigagen2  29990  elsigagen  29991  elsigagen2  29992  sigagenid  29995  elsx  30038  imambfm  30105  cnmbfm  30106  elmbfmvol2  30110  sxbrsigalem3  30115  orvcoel  30304
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