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Theorem saldifcl 39215
 Description: The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
saldifcl ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)

Proof of Theorem saldifcl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpr 476 . 2 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → 𝐸𝑆)
2 issal 39210 . . . . 5 (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
32ibi 255 . . . 4 (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
43simp2d 1067 . . 3 (𝑆 ∈ SAlg → ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆)
54adantr 480 . 2 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆)
6 difeq2 3684 . . . 4 (𝑦 = 𝐸 → ( 𝑆𝑦) = ( 𝑆𝐸))
76eleq1d 2672 . . 3 (𝑦 = 𝐸 → (( 𝑆𝑦) ∈ 𝑆 ↔ ( 𝑆𝐸) ∈ 𝑆))
87rspcva 3280 . 2 ((𝐸𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆) → ( 𝑆𝐸) ∈ 𝑆)
91, 5, 8syl2anc 691 1 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  ωcom 6957   ≼ cdom 7839  SAlgcsalg 39204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-salg 39205 This theorem is referenced by:  salincl  39219  saluni  39220  saliincl  39221  saldifcl2  39222  intsal  39224  saldifcld  39241
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