Theorem sstp 3871

Description: The subsets of a triple. (Contributed by Mario Carneiro, 2-Jul-2016.)

Assertion

Ref	Expression
sstp	⊢ (A ⊆ {B, C, D} ↔ (((A = ∅ ∨ A = {B}) ∨ (A = {C} ∨ A = {B, C})) ∨ ((A = {D} ∨ A = {B, D}) ∨ (A = {C, D} ∨ A = {B, C, D}))))

Proof of Theorem sstp

Step	Hyp	Ref	Expression
1		df-tp 3744	. . 3 ⊢ {B, C, D} = ({B, C} ∪ {D})
2	1	sseq2i 3297	. 2 ⊢ (A ⊆ {B, C, D} ↔ A ⊆ ({B, C} ∪ {D}))
3		0ss 3580	. . 3 ⊢ ∅ ⊆ A
4	3	biantrur 492	. 2 ⊢ (A ⊆ ({B, C} ∪ {D}) ↔ (∅ ⊆ A ∧ A ⊆ ({B, C} ∪ {D})))
5		ssunsn2 3866	. . 3 ⊢ ((∅ ⊆ A ∧ A ⊆ ({B, C} ∪ {D})) ↔ ((∅ ⊆ A ∧ A ⊆ {B, C}) ∨ ((∅ ∪ {D}) ⊆ A ∧ A ⊆ ({B, C} ∪ {D}))))
6	3	biantrur 492	. . . . 5 ⊢ (A ⊆ {B, C} ↔ (∅ ⊆ A ∧ A ⊆ {B, C}))
7		sspr 3870	. . . . 5 ⊢ (A ⊆ {B, C} ↔ ((A = ∅ ∨ A = {B}) ∨ (A = {C} ∨ A = {B, C})))
8	6, 7	bitr3i 242	. . . 4 ⊢ ((∅ ⊆ A ∧ A ⊆ {B, C}) ↔ ((A = ∅ ∨ A = {B}) ∨ (A = {C} ∨ A = {B, C})))
9		uncom 3409	. . . . . . . 8 ⊢ (∅ ∪ {D}) = ({D} ∪ ∅)
10		un0 3576	. . . . . . . 8 ⊢ ({D} ∪ ∅) = {D}
11	9, 10	eqtri 2373	. . . . . . 7 ⊢ (∅ ∪ {D}) = {D}
12	11	sseq1i 3296	. . . . . 6 ⊢ ((∅ ∪ {D}) ⊆ A ↔ {D} ⊆ A)
13		uncom 3409	. . . . . . 7 ⊢ ({B, C} ∪ {D}) = ({D} ∪ {B, C})
14	13	sseq2i 3297	. . . . . 6 ⊢ (A ⊆ ({B, C} ∪ {D}) ↔ A ⊆ ({D} ∪ {B, C}))
15	12, 14	anbi12i 678	. . . . 5 ⊢ (((∅ ∪ {D}) ⊆ A ∧ A ⊆ ({B, C} ∪ {D})) ↔ ({D} ⊆ A ∧ A ⊆ ({D} ∪ {B, C})))
16		ssunpr 3869	. . . . 5 ⊢ (({D} ⊆ A ∧ A ⊆ ({D} ∪ {B, C})) ↔ ((A = {D} ∨ A = ({D} ∪ {B})) ∨ (A = ({D} ∪ {C}) ∨ A = ({D} ∪ {B, C}))))
17		uncom 3409	. . . . . . . . 9 ⊢ ({D} ∪ {B}) = ({B} ∪ {D})
18		df-pr 3743	. . . . . . . . 9 ⊢ {B, D} = ({B} ∪ {D})
19	17, 18	eqtr4i 2376	. . . . . . . 8 ⊢ ({D} ∪ {B}) = {B, D}
20	19	eqeq2i 2363	. . . . . . 7 ⊢ (A = ({D} ∪ {B}) ↔ A = {B, D})
21	20	orbi2i 505	. . . . . 6 ⊢ ((A = {D} ∨ A = ({D} ∪ {B})) ↔ (A = {D} ∨ A = {B, D}))
22		uncom 3409	. . . . . . . . 9 ⊢ ({D} ∪ {C}) = ({C} ∪ {D})
23		df-pr 3743	. . . . . . . . 9 ⊢ {C, D} = ({C} ∪ {D})
24	22, 23	eqtr4i 2376	. . . . . . . 8 ⊢ ({D} ∪ {C}) = {C, D}
25	24	eqeq2i 2363	. . . . . . 7 ⊢ (A = ({D} ∪ {C}) ↔ A = {C, D})
26	1, 13	eqtr2i 2374	. . . . . . . 8 ⊢ ({D} ∪ {B, C}) = {B, C, D}
27	26	eqeq2i 2363	. . . . . . 7 ⊢ (A = ({D} ∪ {B, C}) ↔ A = {B, C, D})
28	25, 27	orbi12i 507	. . . . . 6 ⊢ ((A = ({D} ∪ {C}) ∨ A = ({D} ∪ {B, C})) ↔ (A = {C, D} ∨ A = {B, C, D}))
29	21, 28	orbi12i 507	. . . . 5 ⊢ (((A = {D} ∨ A = ({D} ∪ {B})) ∨ (A = ({D} ∪ {C}) ∨ A = ({D} ∪ {B, C}))) ↔ ((A = {D} ∨ A = {B, D}) ∨ (A = {C, D} ∨ A = {B, C, D})))
30	15, 16, 29	3bitri 262	. . . 4 ⊢ (((∅ ∪ {D}) ⊆ A ∧ A ⊆ ({B, C} ∪ {D})) ↔ ((A = {D} ∨ A = {B, D}) ∨ (A = {C, D} ∨ A = {B, C, D})))
31	8, 30	orbi12i 507	. . 3 ⊢ (((∅ ⊆ A ∧ A ⊆ {B, C}) ∨ ((∅ ∪ {D}) ⊆ A ∧ A ⊆ ({B, C} ∪ {D}))) ↔ (((A = ∅ ∨ A = {B}) ∨ (A = {C} ∨ A = {B, C})) ∨ ((A = {D} ∨ A = {B, D}) ∨ (A = {C, D} ∨ A = {B, C, D}))))
32	5, 31	bitri 240	. 2 ⊢ ((∅ ⊆ A ∧ A ⊆ ({B, C} ∪ {D})) ↔ (((A = ∅ ∨ A = {B}) ∨ (A = {C} ∨ A = {B, C})) ∨ ((A = {D} ∨ A = {B, D}) ∨ (A = {C, D} ∨ A = {B, C, D}))))
33	2, 4, 32	3bitri 262	1 ⊢ (A ⊆ {B, C, D} ↔ (((A = ∅ ∨ A = {B}) ∨ (A = {C} ∨ A = {B, C})) ∨ ((A = {D} ∨ A = {B, D}) ∨ (A = {C, D} ∨ A = {B, C, D}))))

Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∪ cun 3208   ⊆ wss 3258  ∅c0 3551  {csn 3738  {cpr 3739  {ctp 3740

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  df-tp 3744

This theorem is referenced by:  pwtp  3885