Theorem sssigagen 32013

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 4894	. 2 ⊢ 𝐴 ⊆ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠}
2		sigagenval 32008	. 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
3	1, 2	sseqtrrid 3970	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))

Syntax hints:   → wi 4   ∈ wcel 2108  {crab 3067   ⊆ wss 3883  ∪ cuni 4836  ∩ cint 4876  ‘cfv 6418  sigAlgebracsiga 31976  sigaGencsigagen 32006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-siga 31977  df-sigagen 32007

This theorem is referenced by:  sssigagen2  32014  elsigagen  32015  elsigagen2  32016  sigagenid  32019  elsx  32062  imambfm  32129  cnmbfm  32130  elmbfmvol2  32134  sxbrsigalem3  32139  orvcoel  32328