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Theorem saldifcld 45735
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 45707 . 2 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)
41, 2, 3syl2anc 583 1 (𝜑 → ( 𝑆𝐸) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  cdif 3944   cuni 4908  SAlgcsalg 45696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3430  df-v 3473  df-dif 3950  df-in 3954  df-ss 3964  df-pw 4605  df-uni 4909  df-salg 45697
This theorem is referenced by:  subsalsal  45747  salpreimagelt  46095  salpreimalegt  46097  smfresal  46176
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