Theorem ssunsn2 4832

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 4907. (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 4813	. . . . 5 ⊢ (𝐷 ∈ 𝐴 → {𝐷} ⊆ 𝐴)
2		unss 4200	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴) ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴)
3	2	bicomi 224	. . . . . 6 ⊢ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴))
4	3	rbaibr 537	. . . . 5 ⊢ ({𝐷} ⊆ 𝐴 → (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴))
5	1, 4	syl 17	. . . 4 ⊢ (𝐷 ∈ 𝐴 → (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴))
6	5	anbi1d 631	. . 3 ⊢ (𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷}))))
7	2	biimpi 216	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐴 ∧ {𝐷} ⊆ 𝐴) → (𝐵 ∪ {𝐷}) ⊆ 𝐴)
8	7	expcom 413	. . . . . 6 ⊢ ({𝐷} ⊆ 𝐴 → (𝐵 ⊆ 𝐴 → (𝐵 ∪ {𝐷}) ⊆ 𝐴))
9	1, 8	syl 17	. . . . 5 ⊢ (𝐷 ∈ 𝐴 → (𝐵 ⊆ 𝐴 → (𝐵 ∪ {𝐷}) ⊆ 𝐴))
10		ssun3 4190	. . . . . 6 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝐶 ∪ {𝐷}))
11	10	a1i 11	. . . . 5 ⊢ (𝐷 ∈ 𝐴 → (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝐶 ∪ {𝐷})))
12	9, 11	anim12d 609	. . . 4 ⊢ (𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) → ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷}))))
13		pm4.72 951	. . . 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 218	. . 3 ⊢ (𝐷 ∈ 𝐴 → (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))))
15	6, 14	bitrd 279	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))))
16		uncom 4168	. . . . . . 7 ⊢ ({𝐷} ∪ 𝐶) = (𝐶 ∪ {𝐷})
17	16	sseq2i 4025	. . . . . 6 ⊢ (𝐴 ⊆ ({𝐷} ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ {𝐷}))
18		ssundif 4494	. . . . . 6 ⊢ (𝐴 ⊆ ({𝐷} ∪ 𝐶) ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶)
19	17, 18	bitr3i 277	. . . . 5 ⊢ (𝐴 ⊆ (𝐶 ∪ {𝐷}) ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶)
20		disjsn 4716	. . . . . . 7 ⊢ ((𝐴 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷 ∈ 𝐴)
21		disj3 4460	. . . . . . 7 ⊢ ((𝐴 ∩ {𝐷}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐷}))
22	20, 21	bitr3i 277	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐴 ↔ 𝐴 = (𝐴 ∖ {𝐷}))
23		sseq1 4021	. . . . . 6 ⊢ (𝐴 = (𝐴 ∖ {𝐷}) → (𝐴 ⊆ 𝐶 ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶))
24	22, 23	sylbi 217	. . . . 5 ⊢ (¬ 𝐷 ∈ 𝐴 → (𝐴 ⊆ 𝐶 ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶))
25	19, 24	bitr4id 290	. . . 4 ⊢ (¬ 𝐷 ∈ 𝐴 → (𝐴 ⊆ (𝐶 ∪ {𝐷}) ↔ 𝐴 ⊆ 𝐶))
26	25	anbi2d 630	. . 3 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶)))
27	3	simplbi 497	. . . . . . 7 ⊢ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 → 𝐵 ⊆ 𝐴)
28	27	a1i 11	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝐵 ∪ {𝐷}) ⊆ 𝐴 → 𝐵 ⊆ 𝐴))
29	25	biimpd 229	. . . . . 6 ⊢ (¬ 𝐷 ∈ 𝐴 → (𝐴 ⊆ (𝐶 ∪ {𝐷}) → 𝐴 ⊆ 𝐶))
30	28, 29	anim12d 609	. . . . 5 ⊢ (¬ 𝐷 ∈ 𝐴 → (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) → (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶)))
31		pm4.72 951	. . . . 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 218	. . . 4 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶))))
33		orcom 870	. . . 4 ⊢ ((((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶)) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷}))))
34	32, 33	bitrdi 287	. . 3 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))))
35	26, 34	bitrd 279	. 2 ⊢ (¬ 𝐷 ∈ 𝐴 → ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})))))
36	15, 35	pm2.61i 182	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐶 ∪ {𝐷}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632

This theorem is referenced by:  ssunsn  4833  ssunpr  4839  sstp  4841