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Theorem ssunsn 3919
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
StepHypRef Expression
1 ssunsn2 3918 . 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 438 . . . 4  |-  ( ( B  C_  A  /\  A  C_  B )  <->  ( A  C_  B  /\  B  C_  A ) )
3 eqss 3323 . . . 4  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
42, 3bitr4i 244 . . 3  |-  ( ( B  C_  A  /\  A  C_  B )  <->  A  =  B )
5 ancom 438 . . . 4  |-  ( ( ( B  u.  { C } )  C_  A  /\  A  C_  ( B  u.  { C }
) )  <->  ( A  C_  ( B  u.  { C } )  /\  ( B  u.  { C } )  C_  A
) )
6 eqss 3323 . . . 4  |-  ( A  =  ( B  u.  { C } )  <->  ( A  C_  ( B  u.  { C } )  /\  ( B  u.  { C } )  C_  A
) )
75, 6bitr4i 244 . . 3  |-  ( ( ( B  u.  { C } )  C_  A  /\  A  C_  ( B  u.  { C }
) )  <->  A  =  ( B  u.  { C } ) )
84, 7orbi12i 508 . 2  |-  ( ( ( B  C_  A  /\  A  C_  B )  \/  ( ( B  u.  { C }
)  C_  A  /\  A  C_  ( B  u.  { C } ) ) )  <->  ( A  =  B  \/  A  =  ( B  u.  { C } ) ) )
91, 8bitri 241 1  |-  ( ( B  C_  A  /\  A  C_  ( B  u.  { C } ) )  <-> 
( A  =  B  \/  A  =  ( B  u.  { C } ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    u. cun 3278    C_ wss 3280   {csn 3774
This theorem is referenced by:  ssunpr  3921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-sn 3780
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