Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isldsys Structured version   Visualization version   GIF version

Theorem isldsys 33673
Description: The property of being a lambda-system or Dynkin system. Lambda-systems contain the empty set, are closed under complement, and closed under countable disjoint union. (Contributed by Thierry Arnoux, 13-Jun-2020.)
Hypothesis
Ref Expression
isldsys.l 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
Assertion
Ref Expression
isldsys (𝑆𝐿 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑆))))
Distinct variable groups:   𝑦,𝑠   𝑂,𝑠,𝑥   𝑆,𝑠,𝑥
Allowed substitution hints:   𝑆(𝑦)   𝐿(𝑥,𝑦,𝑠)   𝑂(𝑦)

Proof of Theorem isldsys
StepHypRef Expression
1 eleq2 2814 . . 3 (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆))
2 eleq2 2814 . . . 4 (𝑠 = 𝑆 → ((𝑂𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑆))
32raleqbi1dv 3325 . . 3 (𝑠 = 𝑆 → (∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ↔ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆))
4 pweq 4609 . . . 4 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
5 eleq2 2814 . . . . 5 (𝑠 = 𝑆 → ( 𝑥𝑠 𝑥𝑆))
65imbi2d 340 . . . 4 (𝑠 = 𝑆 → (((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑆)))
74, 6raleqbidv 3334 . . 3 (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑆)))
81, 3, 73anbi123d 1432 . 2 (𝑠 = 𝑆 → ((∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑆))))
9 isldsys.l . 2 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
108, 9elrab2 3679 1 (𝑆𝐿 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  {crab 3424  cdif 3938  c0 4315  𝒫 cpw 4595   cuni 4900  Disj wdisj 5104   class class class wbr 5139  ωcom 7849  cdom 8934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-in 3948  df-ss 3958  df-pw 4597
This theorem is referenced by:  pwldsys  33674  unelldsys  33675  sigaldsys  33676  ldsysgenld  33677  sigapildsyslem  33678  sigapildsys  33679  ldgenpisyslem1  33680
  Copyright terms: Public domain W3C validator