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Theorem nbrne1 4040
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 4027 . . . 4  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
21biimpcd 215 . . 3  |-  ( A R B  ->  ( B  =  C  ->  A R C ) )
32necon3bd 2483 . 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 1623    =/= wne 2446   class class class wbr 4023
This theorem is referenced by:  dalem43  29904  cdleme3h  30424  cdleme7ga  30437  cdlemeg46req  30718  cdlemh  31006  cdlemk12  31039  cdlemk12u  31061
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-3an 936  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-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024
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