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Theorem saldifcld 46367
Description: The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
saldifcld.1 (𝜑𝑆 ∈ SAlg)
saldifcld.2 (𝜑𝐸𝑆)
Assertion
Ref Expression
saldifcld (𝜑 → ( 𝑆𝐸) ∈ 𝑆)

Proof of Theorem saldifcld
StepHypRef Expression
1 saldifcld.1 . 2 (𝜑𝑆 ∈ SAlg)
2 saldifcld.2 . 2 (𝜑𝐸𝑆)
3 saldifcl 46339 . 2 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)
41, 2, 3syl2anc 584 1 (𝜑 → ( 𝑆𝐸) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cdif 3947   cuni 4906  SAlgcsalg 46328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-ss 3967  df-pw 4601  df-uni 4907  df-salg 46329
This theorem is referenced by:  subsalsal  46379  salpreimagelt  46727  salpreimalegt  46729  smfresal  46808
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