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Theorem sssigagen 31404
 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 4893 . 2 𝐴 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠}
2 sigagenval 31399 . 2 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
31, 2sseqtrrid 4019 1 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2110  {crab 3142   ⊆ wss 3935  ∪ cuni 4837  ∩ cint 4875  ‘cfv 6354  sigAlgebracsiga 31367  sigaGencsigagen 31397 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-int 4876  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-siga 31368  df-sigagen 31398 This theorem is referenced by:  sssigagen2  31405  elsigagen  31406  elsigagen2  31407  sigagenid  31410  elsx  31453  imambfm  31520  cnmbfm  31521  elmbfmvol2  31525  sxbrsigalem3  31530  orvcoel  31719
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