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Theorem elsigagen 28971
Description: Any element of set is also an element of the sigma algebra that set generates. (Contributed by Thierry Arnoux, 27-Mar-2017.)
Assertion
Ref Expression
elsigagen  |-  ( ( A  e.  V  /\  B  e.  A )  ->  B  e.  (sigaGen `  A
) )

Proof of Theorem elsigagen
StepHypRef Expression
1 sssigagen 28969 . 2  |-  ( A  e.  V  ->  A  C_  (sigaGen `  A )
)
21sselda 3465 1  |-  ( ( A  e.  V  /\  B  e.  A )  ->  B  e.  (sigaGen `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1869   ` cfv 5599  sigaGencsigagen 28962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-int 4254  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-iota 5563  df-fun 5601  df-fv 5607  df-siga 28932  df-sigagen 28963
This theorem is referenced by:  cldssbrsiga  29011  dya2iocbrsiga  29099  dya2icobrsiga  29100  sxbrsigalem2  29110
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