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Theorem sssigagen2 30032
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 ((𝐴𝑉𝐵𝐴) → 𝐵 ⊆ (sigaGen‘𝐴))

Proof of Theorem sssigagen2
StepHypRef Expression
1 simpr 477 . 2 ((𝐴𝑉𝐵𝐴) → 𝐵𝐴)
2 sssigagen 30031 . . 3 (𝐴𝑉𝐴 ⊆ (sigaGen‘𝐴))
32adantr 481 . 2 ((𝐴𝑉𝐵𝐴) → 𝐴 ⊆ (sigaGen‘𝐴))
41, 3sstrd 3598 1 ((𝐴𝑉𝐵𝐴) → 𝐵 ⊆ (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wss 3560  cfv 5857  sigaGencsigagen 30024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-siga 29994  df-sigagen 30025
This theorem is referenced by:  sxbrsigalem5  30173
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