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Theorem isldsys 31403
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 2899 . . 3 (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆))
2 eleq2 2899 . . . 4 (𝑠 = 𝑆 → ((𝑂𝑥) ∈ 𝑠 ↔ (𝑂𝑥) ∈ 𝑆))
32raleqbi1dv 3402 . . 3 (𝑠 = 𝑆 → (∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ↔ ∀𝑥𝑆 (𝑂𝑥) ∈ 𝑆))
4 pweq 4540 . . . 4 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
5 eleq2 2899 . . . . 5 (𝑠 = 𝑆 → ( 𝑥𝑠 𝑥𝑆))
65imbi2d 343 . . . 4 (𝑠 = 𝑆 → (((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑆)))
74, 6raleqbidv 3400 . . 3 (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑆)))
81, 3, 73anbi123d 1429 . 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 3681 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 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  {crab 3140  cdif 3931  c0 4289  𝒫 cpw 4537   cuni 4830  Disj wdisj 5022   class class class wbr 5057  ωcom 7572  cdom 8499
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rab 3145  df-v 3495  df-in 3941  df-ss 3950  df-pw 4539
This theorem is referenced by:  pwldsys  31404  unelldsys  31405  sigaldsys  31406  ldsysgenld  31407  sigapildsyslem  31408  sigapildsys  31409  ldgenpisyslem1  31410
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