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Theorem pwldsys 30025
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 4815 . . . 4 (𝑂𝑉 → 𝒫 𝑂 ∈ V)
2 pwidg 4149 . . . 4 (𝒫 𝑂 ∈ V → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
31, 2syl 17 . . 3 (𝑂𝑉 → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
4 0elpw 4799 . . . . 5 ∅ ∈ 𝒫 𝑂
54a1i 11 . . . 4 (𝑂𝑉 → ∅ ∈ 𝒫 𝑂)
6 pwidg 4149 . . . . . . 7 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
76adantr 481 . . . . . 6 ((𝑂𝑉𝑥 ∈ 𝒫 𝑂) → 𝑂 ∈ 𝒫 𝑂)
87elpwdifcl 29228 . . . . 5 ((𝑂𝑉𝑥 ∈ 𝒫 𝑂) → (𝑂𝑥) ∈ 𝒫 𝑂)
98ralrimiva 2961 . . . 4 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂)
10 elpwi 4145 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝒫 𝑂𝑥 ⊆ 𝒫 𝑂)
11 pwuniss 29239 . . . . . . . . . 10 (𝑥 ⊆ 𝒫 𝑂 𝑥𝑂)
1210, 11syl 17 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝒫 𝑂 𝑥𝑂)
1312adantl 482 . . . . . . . 8 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → 𝑥𝑂)
14 vuniex 6914 . . . . . . . . 9 𝑥 ∈ V
1514elpw 4141 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝑂 𝑥𝑂)
1613, 15sylibr 224 . . . . . . 7 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → 𝑥 ∈ 𝒫 𝑂)
1716adantr 481 . . . . . 6 (((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑂)
1817ex 450 . . . . 5 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂))
1918ralrimiva 2961 . . . 4 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂))
205, 9, 193jca 1240 . . 3 (𝑂𝑉 → (∅ ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂)))
213, 20jca 554 . 2 (𝑂𝑉 → (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))))
22 isldsys.l . . 3 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
2322isldsys 30024 . 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))))
2421, 23sylibr 224 1 (𝑂𝑉 → 𝒫 𝑂𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  cdif 3556  wss 3559  c0 3896  𝒫 cpw 4135   cuni 4407  Disj wdisj 4588   class class class wbr 4618  ωcom 7019  cdom 7905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-pw 4137  df-sn 4154  df-pr 4156  df-uni 4408
This theorem is referenced by:  ldgenpisyslem1  30031
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