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Theorem sssigagen 32808
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 β†’ 𝐴 βŠ† (sigaGenβ€˜π΄))

Proof of Theorem sssigagen
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ssintub 4931 . 2 𝐴 βŠ† ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠}
2 sigagenval 32803 . 2 (𝐴 ∈ 𝑉 β†’ (sigaGenβ€˜π΄) = ∩ {𝑠 ∈ (sigAlgebraβ€˜βˆͺ 𝐴) ∣ 𝐴 βŠ† 𝑠})
31, 2sseqtrrid 4001 1 (𝐴 ∈ 𝑉 β†’ 𝐴 βŠ† (sigaGenβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107  {crab 3406   βŠ† wss 3914  βˆͺ cuni 4869  βˆ© cint 4911  β€˜cfv 6500  sigAlgebracsiga 32771  sigaGencsigagen 32801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-siga 32772  df-sigagen 32802
This theorem is referenced by:  sssigagen2  32809  elsigagen  32810  elsigagen2  32811  sigagenid  32814  elsx  32857  imambfm  32926  cnmbfm  32927  elmbfmvol2  32931  sxbrsigalem3  32936  orvcoel  33125
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