Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwldsys Structured version   Visualization version   GIF version

Theorem pwldsys 30555
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 5059 . . . 4 (𝑂𝑉 → 𝒫 𝑂 ∈ V)
2 pwidg 4377 . . . 4 (𝒫 𝑂 ∈ V → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
31, 2syl 17 . . 3 (𝑂𝑉 → 𝒫 𝑂 ∈ 𝒫 𝒫 𝑂)
4 0elpw 5037 . . . . 5 ∅ ∈ 𝒫 𝑂
54a1i 11 . . . 4 (𝑂𝑉 → ∅ ∈ 𝒫 𝑂)
6 pwidg 4377 . . . . . . 7 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
76adantr 468 . . . . . 6 ((𝑂𝑉𝑥 ∈ 𝒫 𝑂) → 𝑂 ∈ 𝒫 𝑂)
87elpwdifcl 29693 . . . . 5 ((𝑂𝑉𝑥 ∈ 𝒫 𝑂) → (𝑂𝑥) ∈ 𝒫 𝑂)
98ralrimiva 3165 . . . 4 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂)
10 elpwi 4372 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝒫 𝑂𝑥 ⊆ 𝒫 𝑂)
11 pwuniss 29705 . . . . . . . . . 10 (𝑥 ⊆ 𝒫 𝑂 𝑥𝑂)
1210, 11syl 17 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝒫 𝑂 𝑥𝑂)
1312adantl 469 . . . . . . . 8 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → 𝑥𝑂)
14 vuniex 7191 . . . . . . . . 9 𝑥 ∈ V
1514elpw 4368 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝑂 𝑥𝑂)
1613, 15sylibr 225 . . . . . . 7 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → 𝑥 ∈ 𝒫 𝑂)
1716adantr 468 . . . . . 6 (((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 ∈ 𝒫 𝑂)
1817ex 399 . . . . 5 ((𝑂𝑉𝑥 ∈ 𝒫 𝒫 𝑂) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂))
1918ralrimiva 3165 . . . 4 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂))
205, 9, 193jca 1151 . . 3 (𝑂𝑉 → (∅ ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 ∈ 𝒫 𝑂)))
213, 20jca 503 . 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 30554 . 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 225 1 (𝑂𝑉 → 𝒫 𝑂𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1100   = wceq 1637  wcel 2157  wral 3107  {crab 3111  Vcvv 3402  cdif 3777  wss 3780  c0 4127  𝒫 cpw 4362   cuni 4641  Disj wdisj 4823   class class class wbr 4855  ωcom 7302  cdom 8197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-pw 4364  df-sn 4382  df-pr 4384  df-uni 4642
This theorem is referenced by:  ldgenpisyslem1  30561
  Copyright terms: Public domain W3C validator