Theorem sorpsscmpl 6948

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 3722	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝐴 ∖ 𝑢) = (𝐴 ∖ 𝑥))
2	1	eleq1d 2686	. . . . . 6 ⊢ (𝑢 = 𝑥 → ((𝐴 ∖ 𝑢) ∈ 𝑌 ↔ (𝐴 ∖ 𝑥) ∈ 𝑌))
3	2	elrab 3363	. . . . 5 ⊢ (𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑥) ∈ 𝑌))
4		difeq2 3722	. . . . . . 7 ⊢ (𝑢 = 𝑦 → (𝐴 ∖ 𝑢) = (𝐴 ∖ 𝑦))
5	4	eleq1d 2686	. . . . . 6 ⊢ (𝑢 = 𝑦 → ((𝐴 ∖ 𝑢) ∈ 𝑌 ↔ (𝐴 ∖ 𝑦) ∈ 𝑌))
6	5	elrab 3363	. . . . 5 ⊢ (𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌))
7		an4 865	. . . . . 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 206	. . . . 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 496	. . . 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 6943	. . . . . . . 8 ⊢ (( [⊊] Or 𝑌 ∧ ((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌)) → ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) ∨ (𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥)))
11	10	expcom 451	. . . . . . 7 ⊢ (((𝐴 ∖ 𝑥) ∈ 𝑌 ∧ (𝐴 ∖ 𝑦) ∈ 𝑌) → ( [⊊] Or 𝑌 → ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) ∨ (𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥))))
12		selpw 4165	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13		dfss4 3858	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥)
14	12, 13	bitri 264	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥)
15		selpw 4165	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
16		dfss4 3858	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦)
17	15, 16	bitri 264	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦)
18		sscon 3744	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) → (𝐴 ∖ (𝐴 ∖ 𝑥)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑦)))
19		sseq12 3628	. . . . . . . . . . . 12 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴 ∖ 𝑥)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑦)) ↔ 𝑥 ⊆ 𝑦))
20	18, 19	syl5ib 234	. . . . . . . . . . 11 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ 𝑦) ⊆ (𝐴 ∖ 𝑥) → 𝑥 ⊆ 𝑦))
21		sscon 3744	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) → (𝐴 ∖ (𝐴 ∖ 𝑦)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑥)))
22		sseq12 3628	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦 ∧ (𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥) → ((𝐴 ∖ (𝐴 ∖ 𝑦)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑥)) ↔ 𝑦 ⊆ 𝑥))
23	22	ancoms 469	. . . . . . . . . . . 12 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ (𝐴 ∖ 𝑦)) ⊆ (𝐴 ∖ (𝐴 ∖ 𝑥)) ↔ 𝑦 ⊆ 𝑥))
24	21, 23	syl5ib 234	. . . . . . . . . . 11 ⊢ (((𝐴 ∖ (𝐴 ∖ 𝑥)) = 𝑥 ∧ (𝐴 ∖ (𝐴 ∖ 𝑦)) = 𝑦) → ((𝐴 ∖ 𝑥) ⊆ (𝐴 ∖ 𝑦) → 𝑦 ⊆ 𝑥))
25	20, 24	orim12d 883	. . . . . . . . . 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 496	. . . . . . . . 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 404	. . . . . . 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 447	. . . 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 2970	. 2 ⊢ ( [⊊] Or 𝑌 → ∀𝑥 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌}∀𝑦 ∈ {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌} (𝑥 ⊆ 𝑦 ∨ 𝑦 ⊆ 𝑥))
34		sorpss 6942	. 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 224	1 ⊢ ( [⊊] Or 𝑌 → [⊊] Or {𝑢 ∈ 𝒫 𝐴 ∣ (𝐴 ∖ 𝑢) ∈ 𝑌})

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ∖ cdif 3571   ⊆ wss 3574  𝒫 cpw 4158   Or wor 5034   [⊊] crpss 6936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-rpss 6937

This theorem is referenced by:  fin2i2  9140  isfin2-2  9141