Theorem sssigagen 30820

Description: A set is a subset of the sigma-algebra it generates. (Contributed by Thierry Arnoux, 24-Jan-2017.)

Assertion

Ref	Expression
sssigagen	⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))

Proof of Theorem sssigagen

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssintub 4728	. 2 ⊢ 𝐴 ⊆ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠}
2		sigagenval 30815	. 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
3	1, 2	syl5sseqr 3872	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))

Syntax hints:   → wi 4   ∈ wcel 2106  {crab 3093   ⊆ wss 3791  ∪ cuni 4671  ∩ cint 4710  ‘cfv 6135  sigAlgebracsiga 30782  sigaGencsigagen 30813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-siga 30783  df-sigagen 30814

This theorem is referenced by:  sssigagen2  30821  elsigagen  30822  elsigagen2  30823  sigagenid  30826  elsx  30869  imambfm  30936  cnmbfm  30937  elmbfmvol2  30941  sxbrsigalem3  30946  orvcoel  31136