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Theorem sssigagen 30548
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 4630 . 2 𝐴 {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠}
2 sigagenval 30543 . 2 (𝐴𝑉 → (sigaGen‘𝐴) = {𝑠 ∈ (sigAlgebra‘ 𝐴) ∣ 𝐴𝑠})
31, 2syl5sseqr 3803 1 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  {crab 3065  wss 3723   cuni 4575   cint 4612  cfv 6030  sigAlgebracsiga 30510  sigaGencsigagen 30541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-int 4613  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-siga 30511  df-sigagen 30542
This theorem is referenced by:  sssigagen2  30549  elsigagen  30550  elsigagen2  30551  sigagenid  30554  elsx  30597  imambfm  30664  cnmbfm  30665  elmbfmvol2  30669  sxbrsigalem3  30674  orvcoel  30863
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