Theorem isldsys 30520

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 2820	. . 3 ⊢ (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆))
2		eleq2 2820	. . . 4 ⊢ (𝑠 = 𝑆 → ((𝑂 ∖ 𝑥) ∈ 𝑠 ↔ (𝑂 ∖ 𝑥) ∈ 𝑆))
3	2	raleqbi1dv 3277	. . 3 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑆 (𝑂 ∖ 𝑥) ∈ 𝑆))
4		pweq 4297	. . . 4 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
5		eleq2 2820	. . . . 5 ⊢ (𝑠 = 𝑆 → (∪ 𝑥 ∈ 𝑠 ↔ ∪ 𝑥 ∈ 𝑆))
6	5	imbi2d 329	. . . 4 ⊢ (𝑠 = 𝑆 → (((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠) ↔ ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑆)))
7	4, 6	raleqbidv 3283	. . 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 1540	. 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 3499	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 196   ∧ wa 383   ∧ w3a 1072   = wceq 1624   ∈ wcel 2131  ∀wral 3042  {crab 3046   ∖ cdif 3704  ∅c0 4050  𝒫 cpw 4294  ∪ cuni 4580  Disj wdisj 4764   class class class wbr 4796  ωcom 7222   ≼ cdom 8111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rab 3051  df-v 3334  df-in 3714  df-ss 3721  df-pw 4296

This theorem is referenced by:  pwldsys  30521  unelldsys  30522  sigaldsys  30523  ldsysgenld  30524  sigapildsyslem  30525  sigapildsys  30526  ldgenpisyslem1  30527