MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssunsn Structured version   Unicode version

Theorem ssunsn 4154
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 4153 . 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 451 . . . 4  |-  ( ( B  C_  A  /\  A  C_  B )  <->  ( A  C_  B  /\  B  C_  A ) )
3 eqss 3476 . . . 4  |-  ( A  =  B  <->  ( A  C_  B  /\  B  C_  A ) )
42, 3bitr4i 255 . . 3  |-  ( ( B  C_  A  /\  A  C_  B )  <->  A  =  B )
5 ancom 451 . . . 4  |-  ( ( ( B  u.  { C } )  C_  A  /\  A  C_  ( B  u.  { C }
) )  <->  ( A  C_  ( B  u.  { C } )  /\  ( B  u.  { C } )  C_  A
) )
6 eqss 3476 . . . 4  |-  ( A  =  ( B  u.  { C } )  <->  ( A  C_  ( B  u.  { C } )  /\  ( B  u.  { C } )  C_  A
) )
75, 6bitr4i 255 . . 3  |-  ( ( ( B  u.  { C } )  C_  A  /\  A  C_  ( B  u.  { C }
) )  <->  A  =  ( B  u.  { C } ) )
84, 7orbi12i 523 . 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 252 1  |-  ( ( B  C_  A  /\  A  C_  ( B  u.  { C } ) )  <-> 
( A  =  B  \/  A  =  ( B  u.  { C } ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    u. cun 3431    C_ wss 3433   {csn 3993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-sn 3994
This theorem is referenced by:  ssunpr  4156
  Copyright terms: Public domain W3C validator