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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 ∧ ∀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 5077 . . . 4 (𝑂𝑉 → 𝒫 𝑂 ∈ V)
2 pwidg 4392 . . . 4 (𝒫 𝑂 ∈ V → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
31, 2syl 17 . . 3 (𝑂𝑉 → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
4 0elpw 5055 . . . . 5 ∅ ∈ 𝒫 𝑂
54a1i 11 . . . 4 (𝑂𝑉 → ∅ ∈ 𝒫 𝑂)
6 pwidg 4392 . . . . . . 7 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
76adantr 474 . . . . . 6 ((𝑂𝑉𝑥 ∈ 𝒫 𝑂) → 𝑂 ∈ 𝒫 𝑂)
87elpwdifcl 29905 . . . . 5 ((𝑂𝑉𝑥 ∈ 𝒫 𝑂) → (𝑂𝑥) ∈ 𝒫 𝑂)
98ralrimiva 3174 . . . 4 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂)
10 elpwi 4387 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝒫 𝑂𝑥 ⊆ 𝒫 𝑂)
11 pwuniss 29917 . . . . . . . . . 10 (𝑥 ⊆ 𝒫 𝑂 𝑥𝑂)
1210, 11syl 17 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝒫 𝑂 𝑥𝑂)
1312adantl 475 . . . . . . . 8 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → 𝑥𝑂)
14 vuniex 7213 . . . . . . . . 9 𝑥 ∈ V
1514elpw 4383 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝑂 𝑥𝑂)
1613, 15sylibr 226 . . . . . . 7 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → 𝑥 ∈ 𝒫 𝑂)
1716adantr 474 . . . . . 6 (((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑂)
1817ex 403 . . . . 5 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂))
1918ralrimiva 3174 . . . 4 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂))
205, 9, 193jca 1164 . . 3 (𝑂𝑉 → (∅ ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂)))
213, 20jca 509 . 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 30763 . 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 226 1 (𝑂𝑉 → 𝒫 𝑂𝐿)
Colors of variables: wff setvar class
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
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