Theorem sorpsscmpl 7208

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 ∣ (𝐴 ∖ 𝑢) ∈ 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.

Step	Hyp	Ref	Expression
1		difeq2 3949	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝐴 ∖ 𝑢) = (𝐴 ∖ 𝑥))
2	1	eleq1d 2891	. . . . . 6 ⊢ (𝑢 = 𝑥 → ((𝐴 ∖ 𝑢) ∈ 𝑌 ↔ (𝐴 ∖ 𝑥) ∈ 𝑌))
3	2	elrab 3585	. . . . 5 ⊢ (𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝑌))
4		difeq2 3949	. . . . . . 7 ⊢ (𝑢 = 𝑦 → (𝐴 ∖ 𝑢) = (𝐴 ∖ 𝑦))
5	4	eleq1d 2891	. . . . . 6 ⊢ (𝑢 = 𝑦 → ((𝐴 ∖ 𝑢) ∈ 𝑌 ↔ (𝐴 ∖ 𝑦) ∈ 𝑌))
6	5	elrab 3585	. . . . 5 ⊢ (𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌))
7		an4 646	. . . . . 6 ⊢ (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌)) ↔ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌)))
8	7	biimpi 208	. . . . 5 ⊢ (((𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝑌) ∧ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌)) → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌)))
9	3, 6, 8	syl2anb 591	. . . 4 ⊢ ((𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} ∧ 𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌}) → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌)))
10		sorpssi 7203	. . . . . . . 8 ⊢ (( [⊊] Or 𝑌 ∧ ((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌)) → ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) ∨ (𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥)))
11	10	expcom 404	. . . . . . 7 ⊢ (((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌) → ( [⊊] Or 𝑌 → ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) ∨ (𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥))))
12		selpw 4385	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13		dfss4 4088	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥)
14	12, 13	bitri 267	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥)
15		selpw 4385	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
16		dfss4 4088	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦)
17	15, 16	bitri 267	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦)
18		sscon 3971	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) → (𝐴 ∖ (𝐴 ∖ 𝑥)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑦)))
19		sseq12 3853	. . . . . . . . . . . 12 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴 ∖ 𝑥)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑦)) ↔ 𝑥 ⊆ 𝑦))
20	18, 19	syl5ib 236	. . . . . . . . . . 11 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ 𝑦))
21		sscon 3971	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) → (𝐴 ∖ (𝐴 ∖ 𝑦)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑥)))
22		sseq12 3853	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦 ∧ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥) → ((𝐴 ∖ (𝐴 ∖ 𝑦)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑥)) ↔ 𝑦 ⊆ 𝑥))
23	22	ancoms 452	. . . . . . . . . . . 12 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴 ∖ 𝑦)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑥)) ↔ 𝑦 ⊆ 𝑥))
24	21, 23	syl5ib 236	. . . . . . . . . . 11 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) → 𝑦 ⊆ 𝑥))
25	20, 24	orim12d 992	. . . . . . . . . 10 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → (((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) ∨ (𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦)) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
26	14, 17, 25	syl2anb 591	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) ∨ (𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦)) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
27	26	com12 32	. . . . . . . 8 ⊢ (((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) ∨ (𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦)) → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
28	27	orcoms 903	. . . . . . 7 ⊢ (((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) ∨ (𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥)) → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
29	11, 28	syl6 35	. . . . . 6 ⊢ (((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌) → ( [⊊] Or 𝑌 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))))
30	29	com3l 89	. . . . 5 ⊢ ( [⊊] Or 𝑌 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))))
31	30	impd 400	. . . 4 ⊢ ( [⊊] Or 𝑌 → (((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌)) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
32	9, 31	syl5 34	. . 3 ⊢ ( [⊊] Or 𝑌 → ((𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} ∧ 𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌}) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
33	32	ralrimivv 3179	. 2 ⊢ ( [⊊] Or 𝑌 → ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
34		sorpss 7202	. 2 ⊢ ( [⊊] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} ↔ ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
35	33, 34	sylibr 226	1 ⊢ ( [⊊] Or 𝑌 → [⊊] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌})

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 878   = wceq 1656   ∈ wcel 2164  ∀wral 3117  {crab 3121   ∖ cdif 3795   ⊆ wss 3798  𝒫 cpw 4378   Or wor 5262   [⊊] crpss 7196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-po 5263  df-so 5264  df-xp 5348  df-rel 5349  df-rpss 7197

This theorem is referenced by:  fin2i2  9455  isfin2-2  9456