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Theorem isldsys 34112
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 2827 . . 3 (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆))
2 eleq2 2827 . . . 4 (𝑠 = 𝑆 → ((𝑂𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑆))
32raleqbi1dv 3341 . . 3 (𝑠 = 𝑆 → (∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ↔ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆))
4 pweq 4636 . . . 4 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
5 eleq2 2827 . . . . 5 (𝑠 = 𝑆 → ( 𝑥𝑠 𝑥𝑆))
65imbi2d 340 . . . 4 (𝑠 = 𝑆 → (((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑆)))
74, 6raleqbidv 3349 . . 3 (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑆)))
81, 3, 73anbi123d 1436 . 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 3706 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 206  wa 395  w3a 1087   = wceq 1537  wcel 2103  wral 3063  {crab 3438  cdif 3967  c0 4347  𝒫 cpw 4622   cuni 4931  Disj wdisj 5136   class class class wbr 5169  ωcom 7899  cdom 8997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-ss 3987  df-pw 4624
This theorem is referenced by:  pwldsys  34113  unelldsys  34114  sigaldsys  34115  ldsysgenld  34116  sigapildsyslem  34117  sigapildsys  34118  ldgenpisyslem1  34119
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