Theorem pwldsys 32125

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 5301	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V)
2		pwidg 4555	. . 3 ⊢ (𝒫 𝑂 ∈ V → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
3	1, 2	syl 17	. 2 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
4		0elpw 5278	. . . 4 ⊢ ∅ ∈ 𝒫 𝑂
5	4	a1i 11	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∅ ∈ 𝒫 𝑂)
6		pwidg 4555	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
7	6	adantr 481	. . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑂) → 𝑂 ∈ 𝒫 𝑂)
8	7	elpwdifcl 30875	. . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑂) → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
9	8	ralrimiva 3103	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
10		elpwi 4542	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → 𝑥 ⊆ 𝒫 𝑂)
11		sspwuni 5029	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
12	10, 11	sylib 217	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → ∪ 𝑥 ⊆ 𝑂)
13	12	adantl 482	. . . . . 6 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ∪ 𝑥 ⊆ 𝑂)
14		vuniex 7592	. . . . . . 7 ⊢ ∪ 𝑥 ∈ V
15	14	elpw 4537	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
16	13, 15	sylibr 233	. . . . 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 3103	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂))
19	5, 9, 18	3jca 1127	. 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 32124	. 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 583	1 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839  Disj wdisj 5039   class class class wbr 5074  ωcom 7712   ≼ cdom 8731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-uni 4840

This theorem is referenced by:  ldgenpisyslem1  32131