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Theorem sorpsscmpl 7449
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 4090 . . . . . . 7 (𝑢 = 𝑥 → (𝐴𝑢) = (𝐴𝑥))
21eleq1d 2894 . . . . . 6 (𝑢 = 𝑥 → ((𝐴𝑢) ∈ 𝑌 ↔ (𝐴𝑥) ∈ 𝑌))
32elrab 3677 . . . . 5 (𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌))
4 difeq2 4090 . . . . . . 7 (𝑢 = 𝑦 → (𝐴𝑢) = (𝐴𝑦))
54eleq1d 2894 . . . . . 6 (𝑢 = 𝑦 → ((𝐴𝑢) ∈ 𝑌 ↔ (𝐴𝑦) ∈ 𝑌))
65elrab 3677 . . . . 5 (𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌))
7 an4 652 . . . . . 6 (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌)) ↔ ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
87biimpi 217 . . . . 5 (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌)) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
93, 6, 8syl2anb 597 . . . 4 ((𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ∧ 𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
10 sorpssi 7444 . . . . . . . 8 (( [] Or 𝑌 ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)) → ((𝐴𝑥) ⊆ (𝐴𝑦) ∨ (𝐴𝑦) ⊆ (𝐴𝑥)))
1110expcom 414 . . . . . . 7 (((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌) → ( [] Or 𝑌 → ((𝐴𝑥) ⊆ (𝐴𝑦) ∨ (𝐴𝑦) ⊆ (𝐴𝑥))))
12 velpw 4543 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
13 dfss4 4232 . . . . . . . . . . 11 (𝑥𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
1412, 13bitri 276 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
15 velpw 4543 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
16 dfss4 4232 . . . . . . . . . . 11 (𝑦𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
1715, 16bitri 276 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
18 sscon 4112 . . . . . . . . . . . 12 ((𝐴𝑦) ⊆ (𝐴𝑥) → (𝐴 ∖ (𝐴𝑥)) ⊆ (𝐴 ∖ (𝐴𝑦)))
19 sseq12 3991 . . . . . . . . . . . 12 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴𝑥)) ⊆ (𝐴 ∖ (𝐴𝑦)) ↔ 𝑥𝑦))
2018, 19syl5ib 245 . . . . . . . . . . 11 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴𝑦) ⊆ (𝐴𝑥) → 𝑥𝑦))
21 sscon 4112 . . . . . . . . . . . 12 ((𝐴𝑥) ⊆ (𝐴𝑦) → (𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)))
22 sseq12 3991 . . . . . . . . . . . . 13 (((𝐴 ∖ (𝐴𝑦)) = 𝑦 ∧ (𝐴 ∖ (𝐴𝑥)) = 𝑥) → ((𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)) ↔ 𝑦𝑥))
2322ancoms 459 . . . . . . . . . . . 12 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)) ↔ 𝑦𝑥))
2421, 23syl5ib 245 . . . . . . . . . . 11 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴𝑥) ⊆ (𝐴𝑦) → 𝑦𝑥))
2520, 24orim12d 958 . . . . . . . . . 10 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → (((𝐴𝑦) ⊆ (𝐴𝑥) ∨ (𝐴𝑥) ⊆ (𝐴𝑦)) → (𝑥𝑦𝑦𝑥)))
2614, 17, 25syl2anb 597 . . . . . . . . 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 868 . . . . . . 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 411 . . . 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 3187 . 2 ( [] Or 𝑌 → ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} (𝑥𝑦𝑦𝑥))
34 sorpss 7443 . 2 ( [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} (𝑥𝑦𝑦𝑥))
3533, 34sylibr 235 1 ( [] Or 𝑌 → [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3135  {crab 3139  cdif 3930  wss 3933  𝒫 cpw 4535   Or wor 5466   [] crpss 7437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-rpss 7438
This theorem is referenced by:  fin2i2  9728  isfin2-2  9729
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