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

Theorem sorpsscmpl 7724
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 4117 . . . . . . 7 (𝑢 = 𝑥 → (𝐴𝑢) = (𝐴𝑥))
21eleq1d 2819 . . . . . 6 (𝑢 = 𝑥 → ((𝐴𝑢) ∈ 𝑌 ↔ (𝐴𝑥) ∈ 𝑌))
32elrab 3684 . . . . 5 (𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌))
4 difeq2 4117 . . . . . . 7 (𝑢 = 𝑦 → (𝐴𝑢) = (𝐴𝑦))
54eleq1d 2819 . . . . . 6 (𝑢 = 𝑦 → ((𝐴𝑢) ∈ 𝑌 ↔ (𝐴𝑦) ∈ 𝑌))
65elrab 3684 . . . . 5 (𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌))
7 an4 655 . . . . . 6 (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌)) ↔ ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
87biimpi 215 . . . . 5 (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌)) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
93, 6, 8syl2anb 599 . . . 4 ((𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ∧ 𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
10 sorpssi 7719 . . . . . . . 8 (( [] Or 𝑌 ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)) → ((𝐴𝑥) ⊆ (𝐴𝑦) ∨ (𝐴𝑦) ⊆ (𝐴𝑥)))
1110expcom 415 . . . . . . 7 (((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌) → ( [] Or 𝑌 → ((𝐴𝑥) ⊆ (𝐴𝑦) ∨ (𝐴𝑦) ⊆ (𝐴𝑥))))
12 velpw 4608 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
13 dfss4 4259 . . . . . . . . . . 11 (𝑥𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
1412, 13bitri 275 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
15 velpw 4608 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
16 dfss4 4259 . . . . . . . . . . 11 (𝑦𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
1715, 16bitri 275 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
18 sscon 4139 . . . . . . . . . . . 12 ((𝐴𝑦) ⊆ (𝐴𝑥) → (𝐴 ∖ (𝐴𝑥)) ⊆ (𝐴 ∖ (𝐴𝑦)))
19 sseq12 4010 . . . . . . . . . . . 12 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴𝑥)) ⊆ (𝐴 ∖ (𝐴𝑦)) ↔ 𝑥𝑦))
2018, 19imbitrid 243 . . . . . . . . . . 11 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴𝑦) ⊆ (𝐴𝑥) → 𝑥𝑦))
21 sscon 4139 . . . . . . . . . . . 12 ((𝐴𝑥) ⊆ (𝐴𝑦) → (𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)))
22 sseq12 4010 . . . . . . . . . . . . 13 (((𝐴 ∖ (𝐴𝑦)) = 𝑦 ∧ (𝐴 ∖ (𝐴𝑥)) = 𝑥) → ((𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)) ↔ 𝑦𝑥))
2322ancoms 460 . . . . . . . . . . . 12 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)) ↔ 𝑦𝑥))
2421, 23imbitrid 243 . . . . . . . . . . 11 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴𝑥) ⊆ (𝐴𝑦) → 𝑦𝑥))
2520, 24orim12d 964 . . . . . . . . . 10 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → (((𝐴𝑦) ⊆ (𝐴𝑥) ∨ (𝐴𝑥) ⊆ (𝐴𝑦)) → (𝑥𝑦𝑦𝑥)))
2614, 17, 25syl2anb 599 . . . . . . . . 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 871 . . . . . . 7 (((𝐴𝑥) ⊆ (𝐴𝑦) ∨ (𝐴𝑦) ⊆ (𝐴𝑥)) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) → (𝑥𝑦𝑦𝑥)))
2911, 28syl6 35 . . . . . 6 (((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌) → ( [] Or 𝑌 → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) → (𝑥𝑦𝑦𝑥))))
3029com3l 89 . . . . 5 ( [] Or 𝑌 → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) → (((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌) → (𝑥𝑦𝑦𝑥))))
3130impd 412 . . . 4 ( [] Or 𝑌 → (((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)) → (𝑥𝑦𝑦𝑥)))
329, 31syl5 34 . . 3 ( [] Or 𝑌 → ((𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ∧ 𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}) → (𝑥𝑦𝑦𝑥)))
3332ralrimivv 3199 . 2 ( [] Or 𝑌 → ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} (𝑥𝑦𝑦𝑥))
34 sorpss 7718 . 2 ( [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} (𝑥𝑦𝑦𝑥))
3533, 34sylibr 233 1 ( [] Or 𝑌 → [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3062  {crab 3433  cdif 3946  wss 3949  𝒫 cpw 4603   Or wor 5588   [] crpss 7712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-rpss 7713
This theorem is referenced by:  fin2i2  10313  isfin2-2  10314
  Copyright terms: Public domain W3C validator