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 C_ A /\ A C_ (B u. {C, D})) <-> ((A = B \/ A = (B u. {C})) \/ (A = (B u. {D}) \/ A = (B u. {C, D}))))

Proof of Theorem ssunpr

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

Syntax hints:   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642   u. cun 3208   C_ 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