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Theorem sorpsscmpl 7174
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 3921 . . . . . . 7 (𝑢 = 𝑥 → (𝐴𝑢) = (𝐴𝑥))
21eleq1d 2870 . . . . . 6 (𝑢 = 𝑥 → ((𝐴𝑢) ∈ 𝑌 ↔ (𝐴𝑥) ∈ 𝑌))
32elrab 3559 . . . . 5 (𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌))
4 difeq2 3921 . . . . . . 7 (𝑢 = 𝑦 → (𝐴𝑢) = (𝐴𝑦))
54eleq1d 2870 . . . . . 6 (𝑢 = 𝑦 → ((𝐴𝑢) ∈ 𝑌 ↔ (𝐴𝑦) ∈ 𝑌))
65elrab 3559 . . . . 5 (𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌))
7 an4 638 . . . . . 6 (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌)) ↔ ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
87biimpi 207 . . . . 5 (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴𝑦) ∈ 𝑌)) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
93, 6, 8syl2anb 587 . . . 4 ((𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ∧ 𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}) → ((𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)))
10 sorpssi 7169 . . . . . . . 8 (( [] Or 𝑌 ∧ ((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌)) → ((𝐴𝑥) ⊆ (𝐴𝑦) ∨ (𝐴𝑦) ⊆ (𝐴𝑥)))
1110expcom 400 . . . . . . 7 (((𝐴𝑥) ∈ 𝑌 ∧ (𝐴𝑦) ∈ 𝑌) → ( [] Or 𝑌 → ((𝐴𝑥) ⊆ (𝐴𝑦) ∨ (𝐴𝑦) ⊆ (𝐴𝑥))))
12 selpw 4358 . . . . . . . . . . 11 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
13 dfss4 4060 . . . . . . . . . . 11 (𝑥𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
1412, 13bitri 266 . . . . . . . . . 10 (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴𝑥)) = 𝑥)
15 selpw 4358 . . . . . . . . . . 11 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
16 dfss4 4060 . . . . . . . . . . 11 (𝑦𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
1715, 16bitri 266 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴𝑦)) = 𝑦)
18 sscon 3943 . . . . . . . . . . . 12 ((𝐴𝑦) ⊆ (𝐴𝑥) → (𝐴 ∖ (𝐴𝑥)) ⊆ (𝐴 ∖ (𝐴𝑦)))
19 sseq12 3825 . . . . . . . . . . . 12 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴𝑥)) ⊆ (𝐴 ∖ (𝐴𝑦)) ↔ 𝑥𝑦))
2018, 19syl5ib 235 . . . . . . . . . . 11 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴𝑦) ⊆ (𝐴𝑥) → 𝑥𝑦))
21 sscon 3943 . . . . . . . . . . . 12 ((𝐴𝑥) ⊆ (𝐴𝑦) → (𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)))
22 sseq12 3825 . . . . . . . . . . . . 13 (((𝐴 ∖ (𝐴𝑦)) = 𝑦 ∧ (𝐴 ∖ (𝐴𝑥)) = 𝑥) → ((𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)) ↔ 𝑦𝑥))
2322ancoms 448 . . . . . . . . . . . 12 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴𝑦)) ⊆ (𝐴 ∖ (𝐴𝑥)) ↔ 𝑦𝑥))
2421, 23syl5ib 235 . . . . . . . . . . 11 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → ((𝐴𝑥) ⊆ (𝐴𝑦) → 𝑦𝑥))
2520, 24orim12d 978 . . . . . . . . . 10 (((𝐴 ∖ (𝐴𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴𝑦)) = 𝑦) → (((𝐴𝑦) ⊆ (𝐴𝑥) ∨ (𝐴𝑥) ⊆ (𝐴𝑦)) → (𝑥𝑦𝑦𝑥)))
2614, 17, 25syl2anb 587 . . . . . . . . 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 890 . . . . . . 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 398 . . . 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 3158 . 2 ( [] Or 𝑌 → ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} (𝑥𝑦𝑦𝑥))
34 sorpss 7168 . 2 ( [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} ↔ ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌} (𝑥𝑦𝑦𝑥))
3533, 34sylibr 225 1 ( [] Or 𝑌 → [] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴𝑢) ∈ 𝑌})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 865   = wceq 1637  wcel 2156  wral 3096  {crab 3100  cdif 3766  wss 3769  𝒫 cpw 4351   Or wor 5231   [] crpss 7162
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 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-po 5232  df-so 5233  df-xp 5317  df-rel 5318  df-rpss 7163
This theorem is referenced by:  fin2i2  9421  isfin2-2  9422
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