Theorem isldsys 34116

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 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}

Assertion

Ref	Expression
isldsys	⊢ (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆))))

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

Step	Hyp	Ref	Expression
1		eleq2 2822	. . 3 ⊢ (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆))
2		eleq2 2822	. . . 4 ⊢ (𝑠 = 𝑆 → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
3	2	raleqbi1dv 3321	. . 3 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆))
4		pweq 4594	. . . 4 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
5		eleq2 2822	. . . . 5 ⊢ (𝑠 = 𝑆 → (∪ 𝑥 ∈ 𝑠 ↔ ∪ 𝑥 ∈ 𝑆))
6	5	imbi2d 340	. . . 4 ⊢ (𝑠 = 𝑆 → (((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆)))
7	4, 6	raleqbidv 3329	. . 3 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆)))
8	1, 3, 7	3anbi123d 1437	. 2 ⊢ (𝑠 = 𝑆 → ((∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆))))
9		isldsys.l	. 2 ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}
10	8, 9	elrab2 3678	1 ⊢ (𝑆 ∈ 𝐿 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3050  {crab 3419   ∖ cdif 3928  ∅c0 4313  𝒫 cpw 4580  ∪ cuni 4887  Disj wdisj 5090   class class class wbr 5123  ωcom 7869   ≼ cdom 8965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-ss 3948  df-pw 4582

This theorem is referenced by:  pwldsys  34117  unelldsys  34118  sigaldsys  34119  ldsysgenld  34120  sigapildsyslem  34121  sigapildsys  34122  ldgenpisyslem1  34123