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Theorem elsigagen 30812
 Description: Any element of a set is also an element of the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 27-Mar-2017.)
Assertion
Ref Expression
elsigagen ((𝐴𝑉𝐵𝐴) → 𝐵 ∈ (sigaGen‘𝐴))

Proof of Theorem elsigagen
StepHypRef Expression
1 sssigagen 30810 . 2 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
21sselda 3821 1 ((𝐴𝑉𝐵𝐴) → 𝐵 ∈ (sigaGen‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2107  ‘cfv 6137  sigaGencsigagen 30803 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 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228 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 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-int 4713  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-siga 30773  df-sigagen 30804 This theorem is referenced by:  cldssbrsiga  30852  dya2iocbrsiga  30939  dya2icobrsiga  30940  sxbrsigalem2  30950
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