Theorem sstp 3870

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

Assertion

Ref	Expression
sstp	|- (A C_ {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 3743	. . 3 |- {B, C, D} = ({B, C} u. {D})
2	1	sseq2i 3296	. 2 |- (A C_ {B, C, D} <-> A C_ ({B, C} u. {D}))
3		0ss 3579	. . 3 |- (/) C_ A
4	3	biantrur 492	. 2 |- (A C_ ({B, C} u. {D}) <-> ((/) C_ A /\ A C_ ({B, C} u. {D})))
5		ssunsn2 3865	. . 3 |- (((/) C_ A /\ A C_ ({B, C} u. {D})) <-> (((/) C_ A /\ A C_ {B, C}) \/ (((/) u. {D}) C_ A /\ A C_ ({B, C} u. {D}))))
6	3	biantrur 492	. . . . 5 |- (A C_ {B, C} <-> ((/) C_ A /\ A C_ {B, C}))
7		sspr 3869	. . . . 5 |- (A C_ {B, C} <-> ((A = (/) \/ A = {B}) \/ (A = {C} \/ A = {B, C})))
8	6, 7	bitr3i 242	. . . 4 |- (((/) C_ A /\ A C_ {B, C}) <-> ((A = (/) \/ A = {B}) \/ (A = {C} \/ A = {B, C})))
9		uncom 3408	. . . . . . . 8 |- ((/) u. {D}) = ({D} u. (/))
10		un0 3575	. . . . . . . 8 |- ({D} u. (/)) = {D}
11	9, 10	eqtri 2373	. . . . . . 7 |- ((/) u. {D}) = {D}
12	11	sseq1i 3295	. . . . . 6 |- (((/) u. {D}) C_ A <-> {D} C_ A)
13		uncom 3408	. . . . . . 7 |- ({B, C} u. {D}) = ({D} u. {B, C})
14	13	sseq2i 3296	. . . . . 6 |- (A C_ ({B, C} u. {D}) <-> A C_ ({D} u. {B, C}))
15	12, 14	anbi12i 678	. . . . 5 |- ((((/) u. {D}) C_ A /\ A C_ ({B, C} u. {D})) <-> ({D} C_ A /\ A C_ ({D} u. {B, C})))
16		ssunpr 3868	. . . . 5 |- (({D} C_ A /\ A C_ ({D} u. {B, C})) <-> ((A = {D} \/ A = ({D} u. {B})) \/ (A = ({D} u. {C}) \/ A = ({D} u. {B, C}))))
17		uncom 3408	. . . . . . . . 9 |- ({D} u. {B}) = ({B} u. {D})
18		df-pr 3742	. . . . . . . . 9 |- {B, D} = ({B} u. {D})
19	17, 18	eqtr4i 2376	. . . . . . . 8 |- ({D} u. {B}) = {B, D}
20	19	eqeq2i 2363	. . . . . . 7 |- (A = ({D} u. {B}) <-> A = {B, D})
21	20	orbi2i 505	. . . . . 6 |- ((A = {D} \/ A = ({D} u. {B})) <-> (A = {D} \/ A = {B, D}))
22		uncom 3408	. . . . . . . . 9 |- ({D} u. {C}) = ({C} u. {D})
23		df-pr 3742	. . . . . . . . 9 |- {C, D} = ({C} u. {D})
24	22, 23	eqtr4i 2376	. . . . . . . 8 |- ({D} u. {C}) = {C, D}
25	24	eqeq2i 2363	. . . . . . 7 |- (A = ({D} u. {C}) <-> A = {C, D})
26	1, 13	eqtr2i 2374	. . . . . . . 8 |- ({D} u. {B, C}) = {B, C, D}
27	26	eqeq2i 2363	. . . . . . 7 |- (A = ({D} u. {B, C}) <-> A = {B, C, D})
28	25, 27	orbi12i 507	. . . . . 6 |- ((A = ({D} u. {C}) \/ A = ({D} u. {B, C})) <-> (A = {C, D} \/ A = {B, C, D}))
29	21, 28	orbi12i 507	. . . . 5 |- (((A = {D} \/ A = ({D} u. {B})) \/ (A = ({D} u. {C}) \/ A = ({D} u. {B, C}))) <-> ((A = {D} \/ A = {B, D}) \/ (A = {C, D} \/ A = {B, C, D})))
30	15, 16, 29	3bitri 262	. . . 4 |- ((((/) u. {D}) C_ A /\ A C_ ({B, C} u. {D})) <-> ((A = {D} \/ A = {B, D}) \/ (A = {C, D} \/ A = {B, C, D})))
31	8, 30	orbi12i 507	. . 3 |- ((((/) C_ A /\ A C_ {B, C}) \/ (((/) u. {D}) C_ A /\ A C_ ({B, C} u. {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 |- (((/) C_ A /\ A C_ ({B, C} u. {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 C_ {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   u. cun 3207   C_ wss 3257  (/)c0 3550  {csn 3737  {cpr 3738  {ctp 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 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-tp 3743

This theorem is referenced by:  pwtp  3884