Theorem pwldsys 29547

Description: The power set of the universe set 𝑂 is always a lambda-system. (Contributed by Thierry Arnoux, 21-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
pwldsys	⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝐿)

Distinct variable groups:   𝑦,𝑠   𝑂,𝑠,𝑥   𝑥,𝑉

Allowed substitution hints:   𝐿(𝑥,𝑦,𝑠)   𝑂(𝑦)   𝑉(𝑦,𝑠)

Proof of Theorem pwldsys

Step	Hyp	Ref	Expression
1		pwexg 4776	. . . 4 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V)
2		pwidg 4121	. . . 4 ⊢ (𝒫 𝑂 ∈ V → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
3	1, 2	syl 17	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
4		0elpw 4760	. . . . 5 ⊢ ∅ ∈ 𝒫 𝑂
5	4	a1i 11	. . . 4 ⊢ (𝑂 ∈ 𝑉 → ∅ ∈ 𝒫 𝑂)
6		pwidg 4121	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
7	6	adantr 480	. . . . . 6 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑂) → 𝑂 ∈ 𝒫 𝑂)
8	7	elpwdifcl 28742	. . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑂) → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
9	8	ralrimiva 2949	. . . 4 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
10		elpwi 4117	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → 𝑥 ⊆ 𝒫 𝑂)
11		pwuniss 28753	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝒫 𝑂 → ∪ 𝑥 ⊆ 𝑂)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → ∪ 𝑥 ⊆ 𝑂)
13	12	adantl 481	. . . . . . . 8 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ∪ 𝑥 ⊆ 𝑂)
14		vuniex 6852	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V
15	14	elpw 4114	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
16	13, 15	sylibr 223	. . . . . . 7 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ∪ 𝑥 ∈ 𝒫 𝑂)
17	16	adantr 480	. . . . . 6 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝒫 𝑂)
18	17	ex 449	. . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂))
19	18	ralrimiva 2949	. . . 4 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂))
20	5, 9, 19	3jca 1235	. . 3 ⊢ (𝑂 ∈ 𝑉 → (∅ ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂)))
21	3, 20	jca 553	. 2 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂))))
22		isldsys.l	. . 3 ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}
23	22	isldsys 29546	. 2 ⊢ (𝒫 𝑂 ∈ 𝐿 ↔ (𝒫 𝑂 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂))))
24	21, 23	sylibr 223	1 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  Disj wdisj 4553   class class class wbr 4583  ωcom 6957   ≼ cdom 7839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373

This theorem is referenced by:  ldgenpisyslem1  29553