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Theorem sssigagen2 28913
Description: A subset of the generating set is also a subset of the generated sigma-algebra. (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
sssigagen2  |-  ( ( A  e.  V  /\  B  C_  A )  ->  B  C_  (sigaGen `  A
) )

Proof of Theorem sssigagen2
StepHypRef Expression
1 simpr 462 . 2  |-  ( ( A  e.  V  /\  B  C_  A )  ->  B  C_  A )
2 sssigagen 28912 . . 3  |-  ( A  e.  V  ->  A  C_  (sigaGen `  A )
)
32adantr 466 . 2  |-  ( ( A  e.  V  /\  B  C_  A )  ->  A  C_  (sigaGen `  A
) )
41, 3sstrd 3410 1  |-  ( ( A  e.  V  /\  B  C_  A )  ->  B  C_  (sigaGen `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1872    C_ wss 3372   ` cfv 5537  sigaGencsigagen 28905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-int 4192  df-br 4360  df-opab 4419  df-mpt 4420  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-iota 5501  df-fun 5539  df-fv 5545  df-siga 28875  df-sigagen 28906
This theorem is referenced by:  sxbrsigalem5  29055
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