Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  saldifcl Structured version   Visualization version   GIF version

Theorem saldifcl 42904
 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 4068 . . 3 (𝑦 = 𝐸 → ( 𝑆𝑦) = ( 𝑆𝐸))
21eleq1d 2898 . 2 (𝑦 = 𝐸 → (( 𝑆𝑦) ∈ 𝑆 ↔ ( 𝑆𝐸) ∈ 𝑆))
3 issal 42899 . . . . 5 (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
43ibi 270 . . . 4 (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
54simp2d 1140 . . 3 (𝑆 ∈ SAlg → ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆)
65adantr 484 . 2 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆)
7 simpr 488 . 2 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → 𝐸𝑆)
82, 6, 7rspcdva 3600 1 ((𝑆 ∈ SAlg ∧ 𝐸𝑆) → ( 𝑆𝐸) ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2114  ∀wral 3130   ∖ cdif 3905  ∅c0 4265  𝒫 cpw 4511  ∪ cuni 4813   class class class wbr 5042  ωcom 7565   ≼ cdom 8494  SAlgcsalg 42893 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ral 3135  df-rab 3139  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-pw 4513  df-uni 4814  df-salg 42894 This theorem is referenced by:  salincl  42908  saluni  42909  saliincl  42910  saldifcl2  42911  intsal  42913  saldifcld  42930
 Copyright terms: Public domain W3C validator