Theorem ssunsn 3867

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

Proof of Theorem ssunsn

Step	Hyp	Ref	Expression
1		ssunsn2 3866	. 2 |- ((B C_ A /\ A C_ (B u. {C})) <-> ((B C_ A /\ A C_ B) \/ ((B u. {C}) C_ A /\ A C_ (B u. {C}))))
2		ancom 437	. . . 4 |- ((B C_ A /\ A C_ B) <-> (A C_ B /\ B C_ A))
3		eqss 3288	. . . 4 |- (A = B <-> (A C_ B /\ B C_ A))
4	2, 3	bitr4i 243	. . 3 |- ((B C_ A /\ A C_ B) <-> A = B)
5		ancom 437	. . . 4 |- (((B u. {C}) C_ A /\ A C_ (B u. {C})) <-> (A C_ (B u. {C}) /\ (B u. {C}) C_ A))
6		eqss 3288	. . . 4 |- (A = (B u. {C}) <-> (A C_ (B u. {C}) /\ (B u. {C}) C_ A))
7	5, 6	bitr4i 243	. . 3 |- (((B u. {C}) C_ A /\ A C_ (B u. {C})) <-> A = (B u. {C}))
8	4, 7	orbi12i 507	. 2 |- (((B C_ A /\ A C_ B) \/ ((B u. {C}) C_ A /\ A C_ (B u. {C}))) <-> (A = B \/ A = (B u. {C})))
9	1, 8	bitri 240	1 |- ((B C_ A /\ A C_ (B u. {C})) <-> (A = B \/ A = (B u. {C})))

Syntax hints:   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642   u. cun 3208   C_ wss 3258  {csn 3738

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

This theorem is referenced by:  ssunpr  3869