Theorem pwldsys 34301

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 5320	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V)
2		pwidg 4561	. . 3 ⊢ (𝒫 𝑂 ∈ V → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
3	1, 2	syl 17	. 2 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
4		0elpw 5297	. . . 4 ⊢ ∅ ∈ 𝒫 𝑂
5	4	a1i 11	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∅ ∈ 𝒫 𝑂)
6		pwidg 4561	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
7	6	adantr 480	. . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑂) → 𝑂 ∈ 𝒫 𝑂)
8	7	elpwdifcl 32596	. . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑂) → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
9	8	ralrimiva 3129	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
10		elpwi 4548	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → 𝑥 ⊆ 𝒫 𝑂)
11		sspwuni 5042	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
12	10, 11	sylib 218	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → ∪ 𝑥 ⊆ 𝑂)
13	12	adantl 481	. . . . . 6 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ∪ 𝑥 ⊆ 𝑂)
14		vuniex 7693	. . . . . . 7 ⊢ ∪ 𝑥 ∈ V
15	14	elpw 4545	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
16	13, 15	sylibr 234	. . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ∪ 𝑥 ∈ 𝒫 𝑂)
17	16	a1d 25	. . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂))
18	17	ralrimiva 3129	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂))
19	5, 9, 18	3jca 1129	. 2 ⊢ (𝑂 ∈ 𝑉 → (∅ ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂)))
20		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))}
21	20	isldsys 34300	. 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))))
22	3, 19, 21	sylanbrc 584	1 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389  Vcvv 3429   ∖ cdif 3886   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  ∪ cuni 4850  Disj wdisj 5052   class class class wbr 5085  ωcom 7817   ≼ cdom 8891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-in 3896  df-ss 3906  df-nul 4274  df-pw 4543  df-uni 4851

This theorem is referenced by:  ldgenpisyslem1  34307