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Theorem saldifcld 41357
 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 41331 . 2 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)
41, 2, 3syl2anc 581 1 (𝜑 → ( 𝑆𝐸) ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2166   ∖ cdif 3796  ∪ cuni 4659  SAlgcsalg 41320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2804 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-in 3806  df-ss 3813  df-pw 4381  df-uni 4660  df-salg 41321 This theorem is referenced by:  subsalsal  41369  salpreimagelt  41713  salpreimalegt  41715  smfresal  41790
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