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Theorem saldifcl 41456
 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 difeq2 3944 . . 3 (𝑦 = 𝐸 → ( 𝑆𝑦) = ( 𝑆𝐸))
21eleq1d 2843 . 2 (𝑦 = 𝐸 → (( 𝑆𝑦) ∈ 𝑆 ↔ ( 𝑆𝐸) ∈ 𝑆))
3 issal 41451 . . . . 5 (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
43ibi 259 . . . 4 (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
54simp2d 1134 . . 3 (𝑆 ∈ SAlg → ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆)
65adantr 474 . 2 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆)
7 simpr 479 . 2 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → 𝐸𝑆)
82, 6, 7rspcdva 3516 1 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106  ∀wral 3089   ∖ cdif 3788  ∅c0 4140  𝒫 cpw 4378  ∪ cuni 4671   class class class wbr 4886  ωcom 7343   ≼ cdom 8239  SAlgcsalg 41445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-in 3798  df-ss 3805  df-pw 4380  df-uni 4672  df-salg 41446 This theorem is referenced by:  salincl  41460  saluni  41461  saliincl  41462  saldifcl2  41463  intsal  41465  saldifcld  41482
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