Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sorpsscmpl Structured version   Visualization version   GIF version

Theorem sorpsscmpl 6846
 Description: The componentwise complement of a chain of sets is also a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Assertion
Ref Expression
sorpsscmpl ( [] Or 𝑌 → [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌})
Distinct variable groups:   𝑢,𝑌   𝑢,𝐴

Proof of Theorem sorpsscmpl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difeq2 3684 . . . . . . 7 (𝑢 = 𝑥 → (𝐴𝑢) = (𝐴𝑥))
21eleq1d 2672 . . . . . 6 (𝑢 = 𝑥 → ((𝐴𝑢) ∈ 𝑌 ↔ (𝐴𝑥) ∈ 𝑌))
32elrab 3331 . . . . 5 (𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌))
4 difeq2 3684 . . . . . . 7 (𝑢 = 𝑦 → (𝐴𝑢) = (𝐴𝑦))
54eleq1d 2672 . . . . . 6 (𝑢 = 𝑦 → ((𝐴𝑢) ∈ 𝑌 ↔ (𝐴𝑦) ∈ 𝑌))
65elrab 3331 . . . . 5 (𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌))
7 an4 861 . . . . . 6 (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌)) ↔ ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
87biimpi 205 . . . . 5 (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌)) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
93, 6, 8syl2anb 495 . . . 4 ((𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ∧ 𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
10 sorpssi 6841 . . . . . . . 8 (( [] Or 𝑌 ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)) → ((𝐴𝑥) ⊆ (𝐴𝑦) ∨ (𝐴𝑦) ⊆ (𝐴𝑥)))
1110expcom 450 . . . . . . 7 (((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌) → ( [] Or 𝑌 → ((𝐴𝑥) ⊆ (𝐴𝑦) ∨ (𝐴𝑦) ⊆ (𝐴𝑥))))
12 selpw 4115 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
13 dfss4 3820 . . . . . . . . . . 11 (𝑥𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
1412, 13bitri 263 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
15 selpw 4115 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
16 dfss4 3820 . . . . . . . . . . 11 (𝑦𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
1715, 16bitri 263 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
18 sscon 3706 . . . . . . . . . . . 12 ((𝐴𝑦) ⊆ (𝐴𝑥) → (𝐴 ∖ (𝐴𝑥)) ⊆ (𝐴 ∖ (𝐴𝑦)))
19 sseq12 3591 . . . . . . . . . . . 12 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴𝑥)) ⊆ (𝐴 ∖ (𝐴𝑦)) ↔ 𝑥𝑦))
2018, 19syl5ib 233 . . . . . . . . . . 11 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴𝑦) ⊆ (𝐴𝑥) → 𝑥𝑦))
21 sscon 3706 . . . . . . . . . . . 12 ((𝐴𝑥) ⊆ (𝐴𝑦) → (𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)))
22 sseq12 3591 . . . . . . . . . . . . 13 (((𝐴 ∖ (𝐴𝑦)) = 𝑦 ∧ (𝐴 ∖ (𝐴𝑥)) = 𝑥) → ((𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)) ↔ 𝑦𝑥))
2322ancoms 468 . . . . . . . . . . . 12 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)) ↔ 𝑦𝑥))
2421, 23syl5ib 233 . . . . . . . . . . 11 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴𝑥) ⊆ (𝐴𝑦) → 𝑦𝑥))
2520, 24orim12d 879 . . . . . . . . . 10 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → (((𝐴𝑦) ⊆ (𝐴𝑥) ∨ (𝐴𝑥) ⊆ (𝐴𝑦)) → (𝑥𝑦𝑦𝑥)))
2614, 17, 25syl2anb 495 . . . . . . . . 9 ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) → (((𝐴𝑦) ⊆ (𝐴𝑥) ∨ (𝐴𝑥) ⊆ (𝐴𝑦)) → (𝑥𝑦𝑦𝑥)))
2726com12 32 . . . . . . . 8 (((𝐴𝑦) ⊆ (𝐴𝑥) ∨ (𝐴𝑥) ⊆ (𝐴𝑦)) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) → (𝑥𝑦𝑦𝑥)))
2827orcoms 403 . . . . . . 7 (((𝐴𝑥) ⊆ (𝐴𝑦) ∨ (𝐴𝑦) ⊆ (𝐴𝑥)) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) → (𝑥𝑦𝑦𝑥)))
2911, 28syl6 34 . . . . . 6 (((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌) → ( [] Or 𝑌 → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) → (𝑥𝑦𝑦𝑥))))
3029com3l 87 . . . . 5 ( [] Or 𝑌 → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) → (((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌) → (𝑥𝑦𝑦𝑥))))
3130impd 446 . . . 4 ( [] Or 𝑌 → (((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)) → (𝑥𝑦𝑦𝑥)))
329, 31syl5 33 . . 3 ( [] Or 𝑌 → ((𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ∧ 𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}) → (𝑥𝑦𝑦𝑥)))
3332ralrimivv 2953 . 2 ( [] Or 𝑌 → ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} (𝑥𝑦𝑦𝑥))
34 sorpss 6840 . 2 ( [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} (𝑥𝑦𝑦𝑥))
3533, 34sylibr 223 1 ( [] Or 𝑌 → [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  𝒫 cpw 4108   Or wor 4958   [⊊] crpss 6834 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-rpss 6835 This theorem is referenced by:  fin2i2  9023  isfin2-2  9024
 Copyright terms: Public domain W3C validator