Theorem pwldsys 30764

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 5077	. . . 4 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V)
2		pwidg 4392	. . . 4 ⊢ (𝒫 𝑂 ∈ V → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
3	1, 2	syl 17	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
4		0elpw 5055	. . . . 5 ⊢ ∅ ∈ 𝒫 𝑂
5	4	a1i 11	. . . 4 ⊢ (𝑂 ∈ 𝑉 → ∅ ∈ 𝒫 𝑂)
6		pwidg 4392	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
7	6	adantr 474	. . . . . 6 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑂) → 𝑂 ∈ 𝒫 𝑂)
8	7	elpwdifcl 29905	. . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝑂) → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
9	8	ralrimiva 3174	. . . 4 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
10		elpwi 4387	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → 𝑥 ⊆ 𝒫 𝑂)
11		pwuniss 29917	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝒫 𝑂 → ∪ 𝑥 ⊆ 𝑂)
12	10, 11	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → ∪ 𝑥 ⊆ 𝑂)
13	12	adantl 475	. . . . . . . 8 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ∪ 𝑥 ⊆ 𝑂)
14		vuniex 7213	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V
15	14	elpw 4383	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
16	13, 15	sylibr 226	. . . . . . 7 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ∪ 𝑥 ∈ 𝒫 𝑂)
17	16	adantr 474	. . . . . 6 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝒫 𝑂)
18	17	ex 403	. . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑥 ∈ 𝒫 𝒫 𝑂) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂))
19	18	ralrimiva 3174	. . . 4 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝒫 𝑂))
20	5, 9, 19	3jca 1164	. . 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 509	. 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 30763	. 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 226	1 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ 𝐿)

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3116  {crab 3120  Vcvv 3413   ∖ cdif 3794   ⊆ wss 3797  ∅c0 4143  𝒫 cpw 4377  ∪ cuni 4657  Disj wdisj 4840   class class class wbr 4872  ωcom 7325   ≼ cdom 8219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-pw 4379  df-sn 4397  df-pr 4399  df-uni 4658

This theorem is referenced by:  ldgenpisyslem1  30770