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Theorem saldifcld 41066
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 41040 . 2 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)
41, 2, 3syl2anc 696 1 (𝜑 → ( 𝑆𝐸) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2137  cdif 3710   cuni 4586  SAlgcsalg 41029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-in 3720  df-ss 3727  df-pw 4302  df-uni 4587  df-salg 41030
This theorem is referenced by:  subsalsal  41078  salpreimagelt  41422  salpreimalegt  41424  smfresal  41499
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