Theorem ssunsn 4360

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 4359	. 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 466	. . . 4 |- ( ( B C_ A /\ A C_ B ) <-> ( A C_ B /\ B C_ A ) )
3		eqss 3618	. . . 4 |- ( A = B <-> ( A C_ B /\ B C_ A ) )
4	2, 3	bitr4i 267	. . 3 |- ( ( B C_ A /\ A C_ B ) <-> A = B )
5		ancom 466	. . . 4 |- ( ( ( B u. { C } ) C_ A /\ A C_ ( B u. { C } ) ) <-> ( A C_ ( B u. { C } ) /\ ( B u. { C } ) C_ A ) )
6		eqss 3618	. . . 4 |- ( A = ( B u. { C } ) <-> ( A C_ ( B u. { C } ) /\ ( B u. { C } ) C_ A ) )
7	5, 6	bitr4i 267	. . 3 |- ( ( ( B u. { C } ) C_ A /\ A C_ ( B u. { C } ) ) <-> A = ( B u. { C } ) )
8	4, 7	orbi12i 543	. 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 264	1 |- ( ( B C_ A /\ A C_ ( B u. { C } ) ) <-> ( A = B \/ A = ( B u. { C } ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   u. cun 3572   C_ wss 3574   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178

This theorem is referenced by:  ssunpr  4365