Theorem pwldsys 31416

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 5279	. . 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 5256	. . . 4 ⊢ ∅ ∈ 𝒫 𝑂
5	4	a1i 11	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∅ ∈ 𝒫 𝑂)
6		pwidg 4561	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
7	6	adantr 483	. . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑂) → 𝑂 ∈ 𝒫 𝑂)
8	7	elpwdifcl 30287	. . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑂) → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
9	8	ralrimiva 3182	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
10		elpwi 4548	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → 𝑥 ⊆ 𝒫 𝑂)
11		sspwuni 5022	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
12	10, 11	sylib 220	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → ∪ 𝑥 ⊆ 𝑂)
13	12	adantl 484	. . . . . 6 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ∪ 𝑥 ⊆ 𝑂)
14		vuniex 7465	. . . . . . 7 ⊢ ∪ 𝑥 ∈ V
15	14	elpw 4543	. . . . . 6 ⊢ (∪ 𝑥 ∈ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
16	13, 15	sylibr 236	. . . . 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 3182	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂))
19	5, 9, 18	3jca 1124	. 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 31415	. 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 585	1 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138  {crab 3142  Vcvv 3494   ∖ cdif 3933   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  ∪ cuni 4838  Disj wdisj 5031   class class class wbr 5066  ωcom 7580   ≼ cdom 8507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-pw 4541  df-uni 4839

This theorem is referenced by:  ldgenpisyslem1  31422