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Theorem ssunsn 3959
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 3958 . 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 3363 . . . 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 3363 . . . 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 1652    u. cun 3318    C_ wss 3320   {csn 3814
This theorem is referenced by:  ssunpr  3961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820
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