Theorem sorpsscmpl 7741

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 4075	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝐴 ∖ 𝑢) = (𝐴 ∖ 𝑥))
2	1	eleq1d 2850	. . . . . 6 ⊢ (𝑢 = 𝑥 → ((𝐴 ∖ 𝑢) ∈ 𝑌 ↔ (𝐴 ∖ 𝑥) ∈ 𝑌))
3	2	elrab 3652	. . . . 5 ⊢ (𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝑌))
4		difeq2 4075	. . . . . . 7 ⊢ (𝑢 = 𝑦 → (𝐴 ∖ 𝑢) = (𝐴 ∖ 𝑦))
5	4	eleq1d 2850	. . . . . 6 ⊢ (𝑢 = 𝑦 → ((𝐴 ∖ 𝑢) ∈ 𝑌 ↔ (𝐴 ∖ 𝑦) ∈ 𝑌))
6	5	elrab 3652	. . . . 5 ⊢ (𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌))
7		an4 669	. . . . . 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 219	. . . . 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 610	. . . 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 7736	. . . . . . . 8 ⊢ (( [⊊] Or 𝑌 ∧ ((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌)) → ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) ∨ (𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥)))
11	10	expcom 419	. . . . . . 7 ⊢ (((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌) → ( [⊊] Or 𝑌 → ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) ∨ (𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥))))
12		velpw 4569	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13		dfss4 4222	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥)
14	12, 13	bitri 278	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥)
15		velpw 4569	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
16		dfss4 4222	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦)
17	15, 16	bitri 278	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦)
18		sscon 4097	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) → (𝐴 ∖ (𝐴 ∖ 𝑥)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑦)))
19		sseq12 3965	. . . . . . . . . . . 12 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴 ∖ 𝑥)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑦)) ↔ 𝑥 ⊆ 𝑦))
20	18, 19	imbitrid 247	. . . . . . . . . . 11 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ 𝑦))
21		sscon 4097	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) → (𝐴 ∖ (𝐴 ∖ 𝑦)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑥)))
22		sseq12 3965	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦 ∧ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥) → ((𝐴 ∖ (𝐴 ∖ 𝑦)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑥)) ↔ 𝑦 ⊆ 𝑥))
23	22	ancoms 464	. . . . . . . . . . . 12 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴 ∖ 𝑦)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑥)) ↔ 𝑦 ⊆ 𝑥))
24	21, 23	imbitrid 247	. . . . . . . . . . 11 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) → 𝑦 ⊆ 𝑥))
25	20, 24	orim12d 979	. . . . . . . . . 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 610	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) ∨ (𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦)) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
27	26	com12 33	. . . . . . . 8 ⊢ (((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) ∨ (𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦)) → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
28	27	orcoms 886	. . . . . . 7 ⊢ (((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) ∨ (𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥)) → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
29	11, 28	syl6 36	. . . . . 6 ⊢ (((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌) → ( [⊊] Or 𝑌 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))))
30	29	com3l 90	. . . . 5 ⊢ ( [⊊] Or 𝑌 → ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) → (((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))))
31	30	impd 416	. . . 4 ⊢ ( [⊊] Or 𝑌 → (((𝑥 ∈ 𝒫 𝐴 ∧ 𝑦 ∈ 𝒫 𝐴) ∧ ((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌)) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
32	9, 31	syl5 35	. . 3 ⊢ ( [⊊] Or 𝑌 → ((𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} ∧ 𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌}) → (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥)))
33	32	ralrimivv 3208	. 2 ⊢ ( [⊊] Or 𝑌 → ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
34		sorpss 7735	. 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 237	1 ⊢ ( [⊊] Or 𝑌 → [⊊] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418   ∖ cdif 3903   ⊆ wss 3906  𝒫 cpw 4564   Or wor 5570   [⊊] crpss 7729

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-rpss 7730

This theorem is used by:  fin2i2  10317  isfin2-2  10318