Theorem ssunpr 3869

Description: Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)

Assertion

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

Proof of Theorem ssunpr

Step	Hyp	Ref	Expression
1		df-pr 3743	. . . . . 6 ⊢ {C, D} = ({C} ∪ {D})
2	1	uneq2i 3416	. . . . 5 ⊢ (B ∪ {C, D}) = (B ∪ ({C} ∪ {D}))
3		unass 3421	. . . . 5 ⊢ ((B ∪ {C}) ∪ {D}) = (B ∪ ({C} ∪ {D}))
4	2, 3	eqtr4i 2376	. . . 4 ⊢ (B ∪ {C, D}) = ((B ∪ {C}) ∪ {D})
5	4	sseq2i 3297	. . 3 ⊢ (A ⊆ (B ∪ {C, D}) ↔ A ⊆ ((B ∪ {C}) ∪ {D}))
6	5	anbi2i 675	. 2 ⊢ ((B ⊆ A ∧ A ⊆ (B ∪ {C, D})) ↔ (B ⊆ A ∧ A ⊆ ((B ∪ {C}) ∪ {D})))
7		ssunsn2 3866	. 2 ⊢ ((B ⊆ A ∧ A ⊆ ((B ∪ {C}) ∪ {D})) ↔ ((B ⊆ A ∧ A ⊆ (B ∪ {C})) ∨ ((B ∪ {D}) ⊆ A ∧ A ⊆ ((B ∪ {C}) ∪ {D}))))
8		ssunsn 3867	. . 3 ⊢ ((B ⊆ A ∧ A ⊆ (B ∪ {C})) ↔ (A = B ∨ A = (B ∪ {C})))
9		un23 3423	. . . . . 6 ⊢ ((B ∪ {C}) ∪ {D}) = ((B ∪ {D}) ∪ {C})
10	9	sseq2i 3297	. . . . 5 ⊢ (A ⊆ ((B ∪ {C}) ∪ {D}) ↔ A ⊆ ((B ∪ {D}) ∪ {C}))
11	10	anbi2i 675	. . . 4 ⊢ (((B ∪ {D}) ⊆ A ∧ A ⊆ ((B ∪ {C}) ∪ {D})) ↔ ((B ∪ {D}) ⊆ A ∧ A ⊆ ((B ∪ {D}) ∪ {C})))
12		ssunsn 3867	. . . 4 ⊢ (((B ∪ {D}) ⊆ A ∧ A ⊆ ((B ∪ {D}) ∪ {C})) ↔ (A = (B ∪ {D}) ∨ A = ((B ∪ {D}) ∪ {C})))
13	4, 9	eqtr2i 2374	. . . . . 6 ⊢ ((B ∪ {D}) ∪ {C}) = (B ∪ {C, D})
14	13	eqeq2i 2363	. . . . 5 ⊢ (A = ((B ∪ {D}) ∪ {C}) ↔ A = (B ∪ {C, D}))
15	14	orbi2i 505	. . . 4 ⊢ ((A = (B ∪ {D}) ∨ A = ((B ∪ {D}) ∪ {C})) ↔ (A = (B ∪ {D}) ∨ A = (B ∪ {C, D})))
16	11, 12, 15	3bitri 262	. . 3 ⊢ (((B ∪ {D}) ⊆ A ∧ A ⊆ ((B ∪ {C}) ∪ {D})) ↔ (A = (B ∪ {D}) ∨ A = (B ∪ {C, D})))
17	8, 16	orbi12i 507	. 2 ⊢ (((B ⊆ A ∧ A ⊆ (B ∪ {C})) ∨ ((B ∪ {D}) ⊆ A ∧ A ⊆ ((B ∪ {C}) ∪ {D}))) ↔ ((A = B ∨ A = (B ∪ {C})) ∨ (A = (B ∪ {D}) ∨ A = (B ∪ {C, D}))))
18	6, 7, 17	3bitri 262	1 ⊢ ((B ⊆ A ∧ A ⊆ (B ∪ {C, D})) ↔ ((A = B ∨ A = (B ∪ {C})) ∨ (A = (B ∪ {D}) ∨ A = (B ∪ {C, D}))))

Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∪ cun 3208   ⊆ wss 3258  {csn 3738  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743

This theorem is referenced by:  sspr  3870  sstp  3871