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Theorem ssunsn 3774
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 3773 . 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 3194 . . . 4  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
42, 3bitr4i 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 3194 . . . 4  |-  ( A  =  ( B  u.  { C } )  <->  ( A  C_  ( B  u.  { C } )  /\  ( B  u.  { C } )  C_  A
) )
75, 6bitr4i 243 . . 3  |-  ( ( ( B  u.  { C } )  C_  A  /\  A  C_  ( B  u.  { C }
) )  <->  A  =  ( B  u.  { C } ) )
84, 7orbi12i 507 . 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 240 1  |-  ( ( B  C_  A  /\  A  C_  ( B  u.  { C } ) )  <-> 
( A  =  B  \/  A  =  ( B  u.  { C } ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    u. cun 3150    C_ wss 3152   {csn 3640
This theorem is referenced by:  ssunpr  3776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646
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