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Theorem nbrne1 4056
Description: Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012.)
Assertion
Ref Expression
nbrne1  |-  ( ( A R B  /\  -.  A R C )  ->  B  =/=  C
)

Proof of Theorem nbrne1
StepHypRef Expression
1 breq2 4043 . . . 4  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
21biimpcd 215 . . 3  |-  ( A R B  ->  ( B  =  C  ->  A R C ) )
32necon3bd 2496 . 2  |-  ( A R B  ->  ( -.  A R C  ->  B  =/=  C ) )
43imp 418 1  |-  ( ( A R B  /\  -.  A R C )  ->  B  =/=  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    =/= wne 2459   class class class wbr 4039
This theorem is referenced by:  dalem43  30526  cdleme3h  31046  cdleme7ga  31059  cdlemeg46req  31340  cdlemh  31628  cdlemk12  31661  cdlemk12u  31683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040
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