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Theorem elsigass 31061
Description: An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.)
Assertion
Ref Expression
elsigass ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 𝑆)

Proof of Theorem elsigass
StepHypRef Expression
1 sgon 31060 . . . 4 (𝑆 ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘ 𝑆))
2 sigasspw 31052 . . . 4 (𝑆 ∈ (sigAlgebra‘ 𝑆) → 𝑆 ⊆ 𝒫 𝑆)
31, 2syl 17 . . 3 (𝑆 ran sigAlgebra → 𝑆 ⊆ 𝒫 𝑆)
43sselda 3851 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 ∈ 𝒫 𝑆)
54elpwid 4428 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆) → 𝐴 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wcel 2051  wss 3822  𝒫 cpw 4416   cuni 4708  ran crn 5404  cfv 6185  sigAlgebracsiga 31043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-fv 6193  df-siga 31044
This theorem is referenced by: (None)
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