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Theorem nbrne1 4464
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 4451 . . . 4  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
21biimpcd 224 . . 3  |-  ( A R B  ->  ( B  =  C  ->  A R C ) )
32necon3bd 2679 . 2  |-  ( A R B  ->  ( -.  A R C  ->  B  =/=  C ) )
43imp 429 1  |-  ( ( A R B  /\  -.  A R C )  ->  B  =/=  C
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    =/= wne 2662   class class class wbr 4447
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-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448
This theorem is referenced by:  dalem43  34511  cdleme3h  35031  cdleme7ga  35044  cdlemeg46req  35325  cdlemh  35613  cdlemk12  35646  cdlemk12u  35668
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