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Theorem sorpsscmpl 6990
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 3755 . . . . . . 7 (𝑢 = 𝑥 → (𝐴𝑢) = (𝐴𝑥))
21eleq1d 2715 . . . . . 6 (𝑢 = 𝑥 → ((𝐴𝑢) ∈ 𝑌 ↔ (𝐴𝑥) ∈ 𝑌))
32elrab 3396 . . . . 5 (𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌))
4 difeq2 3755 . . . . . . 7 (𝑢 = 𝑦 → (𝐴𝑢) = (𝐴𝑦))
54eleq1d 2715 . . . . . 6 (𝑢 = 𝑦 → ((𝐴𝑢) ∈ 𝑌 ↔ (𝐴𝑦) ∈ 𝑌))
65elrab 3396 . . . . 5 (𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌))
7 an4 882 . . . . . 6 (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌)) ↔ ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
87biimpi 206 . . . . 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 6985 . . . . . . . 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 4198 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
13 dfss4 3891 . . . . . . . . . . 11 (𝑥𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
1412, 13bitri 264 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
15 selpw 4198 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
16 dfss4 3891 . . . . . . . . . . 11 (𝑦𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
1715, 16bitri 264 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
18 sscon 3777 . . . . . . . . . . . 12 ((𝐴𝑦) ⊆ (𝐴𝑥) → (𝐴 ∖ (𝐴𝑥)) ⊆ (𝐴 ∖ (𝐴𝑦)))
19 sseq12 3661 . . . . . . . . . . . 12 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴𝑥)) ⊆ (𝐴 ∖ (𝐴𝑦)) ↔ 𝑥𝑦))
2018, 19syl5ib 234 . . . . . . . . . . 11 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴𝑦) ⊆ (𝐴𝑥) → 𝑥𝑦))
21 sscon 3777 . . . . . . . . . . . 12 ((𝐴𝑥) ⊆ (𝐴𝑦) → (𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)))
22 sseq12 3661 . . . . . . . . . . . . 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 234 . . . . . . . . . . 11 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴𝑥) ⊆ (𝐴𝑦) → 𝑦𝑥))
2520, 24orim12d 901 . . . . . . . . . 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 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 446 . . . 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 2999 . 2 ( [] Or 𝑌 → ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} (𝑥𝑦𝑦𝑥))
34 sorpss 6984 . 2 ( [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} (𝑥𝑦𝑦𝑥))
3533, 34sylibr 224 1 ( [] Or 𝑌 → [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cdif 3604  wss 3607  𝒫 cpw 4191   Or wor 5063   [] crpss 6978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-rpss 6979
This theorem is referenced by:  fin2i2  9178  isfin2-2  9179
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