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Theorem ssunsn 4187
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 4186 . 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 450 . . . 4  |-  ( ( B  C_  A  /\  A  C_  B )  <->  ( A  C_  B  /\  B  C_  A ) )
3 eqss 3519 . . . 4  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
42, 3bitr4i 252 . . 3  |-  ( ( B  C_  A  /\  A  C_  B )  <->  A  =  B )
5 ancom 450 . . . 4  |-  ( ( ( B  u.  { C } )  C_  A  /\  A  C_  ( B  u.  { C }
) )  <->  ( A  C_  ( B  u.  { C } )  /\  ( B  u.  { C } )  C_  A
) )
6 eqss 3519 . . . 4  |-  ( A  =  ( B  u.  { C } )  <->  ( A  C_  ( B  u.  { C } )  /\  ( B  u.  { C } )  C_  A
) )
75, 6bitr4i 252 . . 3  |-  ( ( ( B  u.  { C } )  C_  A  /\  A  C_  ( B  u.  { C }
) )  <->  A  =  ( B  u.  { C } ) )
84, 7orbi12i 521 . 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 249 1  |-  ( ( B  C_  A  /\  A  C_  ( B  u.  { C } ) )  <-> 
( A  =  B  \/  A  =  ( B  u.  { C } ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    u. cun 3474    C_ wss 3476   {csn 4027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028
This theorem is referenced by:  ssunpr  4189
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