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Theorem pwldsys 33453
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 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
Assertion
Ref Expression
pwldsys (𝑂𝑉 → 𝒫 𝑂𝐿)
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
StepHypRef Expression
1 pwexg 5375 . . 3 (𝑂𝑉 → 𝒫 𝑂 ∈ V)
2 pwidg 4621 . . 3 (𝒫 𝑂 ∈ V → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
31, 2syl 17 . 2 (𝑂𝑉 → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
4 0elpw 5353 . . . 4 ∅ ∈ 𝒫 𝑂
54a1i 11 . . 3 (𝑂𝑉 → ∅ ∈ 𝒫 𝑂)
6 pwidg 4621 . . . . . 6 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
76adantr 479 . . . . 5 ((𝑂𝑉𝑥 ∈ 𝒫 𝑂) → 𝑂 ∈ 𝒫 𝑂)
87elpwdifcl 32031 . . . 4 ((𝑂𝑉𝑥 ∈ 𝒫 𝑂) → (𝑂𝑥) ∈ 𝒫 𝑂)
98ralrimiva 3144 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂)
10 elpwi 4608 . . . . . . . 8 (𝑥 ∈ 𝒫 𝒫 𝑂𝑥 ⊆ 𝒫 𝑂)
11 sspwuni 5102 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑂 𝑥𝑂)
1210, 11sylib 217 . . . . . . 7 (𝑥 ∈ 𝒫 𝒫 𝑂 𝑥𝑂)
1312adantl 480 . . . . . 6 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → 𝑥𝑂)
14 vuniex 7731 . . . . . . 7 𝑥 ∈ V
1514elpw 4605 . . . . . 6 ( 𝑥 ∈ 𝒫 𝑂 𝑥𝑂)
1613, 15sylibr 233 . . . . 5 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → 𝑥 ∈ 𝒫 𝑂)
1716a1d 25 . . . 4 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂))
1817ralrimiva 3144 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂))
195, 9, 183jca 1126 . 2 (𝑂𝑉 → (∅ ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂)))
20 isldsys.l . . 3 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
2120isldsys 33452 . 2 (𝒫 𝑂𝐿 ↔ (𝒫 𝑂 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂))))
223, 19, 21sylanbrc 581 1 (𝑂𝑉 → 𝒫 𝑂𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1085   = wceq 1539  wcel 2104  wral 3059  {crab 3430  Vcvv 3472  cdif 3944  wss 3947  c0 4321  𝒫 cpw 4601   cuni 4907  Disj wdisj 5112   class class class wbr 5147  ωcom 7857  cdom 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-uni 4908
This theorem is referenced by:  ldgenpisyslem1  33459
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