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Theorem saldifcld 44558
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 44530 . 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 2106  cdif 3906   cuni 4864  SAlgcsalg 44519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3064  df-rab 3407  df-v 3446  df-dif 3912  df-in 3916  df-ss 3926  df-pw 4561  df-uni 4865  df-salg 44520
This theorem is referenced by:  subsalsal  44570  salpreimagelt  44918  salpreimalegt  44920  smfresal  44999
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