Theorem ssunsn2 4673

Description: The property of being sandwiched between two sets naturally splits under union with a singleton. This is the induction hypothesis for the determination of large powersets such as pwtp 4746. (Contributed by Mario Carneiro, 2-Jul-2016.)

Assertion

Ref	Expression
ssunsn2	⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷}))))

Proof of Theorem ssunsn2

Step	Hyp	Ref	Expression
1		snssi 4654	. . . . 5 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴)
2		unss 4087	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴) ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴)
3	2	bicomi 225	. . . . . 6 ⊢ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴))
4	3	rbaibr 538	. . . . 5 ⊢ ({𝐷} ⊆ 𝐴 → (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴))
5	1, 4	syl 17	. . . 4 ⊢ (𝐷 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴))
6	5	anbi1d 629	. . 3 ⊢ (𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷}))))
7	2	biimpi 217	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴) → (𝐵 ∪ {𝐷}) ⊆ 𝐴)
8	7	expcom 414	. . . . . 6 ⊢ ({𝐷} ⊆ 𝐴 → (𝐵 ⊆ 𝐴 → (𝐵 ∪ {𝐷}) ⊆ 𝐴))
9	1, 8	syl 17	. . . . 5 ⊢ (𝐷 ∈ 𝐴 → (𝐵 ⊆ 𝐴 → (𝐵 ∪ {𝐷}) ⊆ 𝐴))
10		ssun3 4077	. . . . . 6 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝐶 ∪ {𝐷}))
11	10	a1i 11	. . . . 5 ⊢ (𝐷 ∈ 𝐴 → (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝐶 ∪ {𝐷})))
12	9, 11	anim12d 608	. . . 4 ⊢ (𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) → ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷}))))
13		pm4.72 944	. . . 4 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) → ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷}))) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))))
14	12, 13	sylib 219	. . 3 ⊢ (𝐷 ∈ 𝐴 → (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))))
15	6, 14	bitrd 280	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))))
16		disjsn 4560	. . . . . . 7 ⊢ ((𝐴 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷 ∈ 𝐴)
17		disj3 4323	. . . . . . 7 ⊢ ((𝐴 ∩ {𝐷}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐷}))
18	16, 17	bitr3i 278	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐴 ↔ 𝐴 = (𝐴 ∖ {𝐷}))
19		sseq1 3919	. . . . . 6 ⊢ (𝐴 = (𝐴 ∖ {𝐷}) → (𝐴 ⊆ 𝐶 ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶))
20	18, 19	sylbi 218	. . . . 5 ⊢ (¬ 𝐷 ∈ 𝐴 → (𝐴 ⊆ 𝐶 ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶))
21		uncom 4056	. . . . . . 7 ⊢ ({𝐷} ∪ 𝐶) = (𝐶 ∪ {𝐷})
22	21	sseq2i 3923	. . . . . 6 ⊢ (𝐴 ⊆ ({𝐷} ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ {𝐷}))
23		ssundif 4353	. . . . . 6 ⊢ (𝐴 ⊆ ({𝐷} ∪ 𝐶) ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶)
24	22, 23	bitr3i 278	. . . . 5 ⊢ (𝐴 ⊆ (𝐶 ∪ {𝐷}) ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶)
25	20, 24	syl6rbbr 291	. . . 4 ⊢ (¬ 𝐷 ∈ 𝐴 → (𝐴 ⊆ (𝐶 ∪ {𝐷}) ↔ 𝐴 ⊆ 𝐶))
26	25	anbi2d 628	. . 3 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶)))
27	3	simplbi 498	. . . . . . 7 ⊢ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 → 𝐵 ⊆ 𝐴)
28	27	a1i 11	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝐵 ∪ {𝐷}) ⊆ 𝐴 → 𝐵 ⊆ 𝐴))
29	25	biimpd 230	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐴 → (𝐴 ⊆ (𝐶 ∪ {𝐷}) → 𝐴 ⊆ 𝐶))
30	28, 29	anim12d 608	. . . . 5 ⊢ (¬ 𝐷 ∈ 𝐴 → (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶)))
31		pm4.72 944	. . . . 5 ⊢ ((((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶)) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶))))
32	30, 31	sylib 219	. . . 4 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶))))
33		orcom 865	. . . 4 ⊢ ((((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶)) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷}))))
34	32, 33	syl6bb 288	. . 3 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))))
35	26, 34	bitrd 280	. 2 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))))
36	15, 35	pm2.61i 183	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   = wceq 1525   ∈ wcel 2083   ∖ cdif 3862   ∪ cun 3863   ∩ cin 3864   ⊆ wss 3865  ∅c0 4217  {csn 4478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-sn 4479

This theorem is referenced by:  ssunsn  4674  ssunpr  4678  sstp  4680