Theorem sssigagen 34112

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 4916	. 2 ⊢ 𝐴 ⊆ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠}
2		sigagenval 34107	. 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) = ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝐴) ∣ 𝐴 ⊆ 𝑠})
3	1, 2	sseqtrrid 3979	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))

Syntax hints:   → wi 4   ∈ wcel 2109  {crab 3394   ⊆ wss 3903  ∪ cuni 4858  ∩ cint 4896  ‘cfv 6482  sigAlgebracsiga 34075  sigaGencsigagen 34105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-siga 34076  df-sigagen 34106

This theorem is referenced by:  sssigagen2  34113  elsigagen  34114  elsigagen2  34115  sigagenid  34118  elsx  34161  imambfm  34230  cnmbfm  34231  elmbfmvol2  34235  sxbrsigalem3  34240  orvcoel  34430