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Theorem nbrne1 4229
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 4216 . . . 4  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
21biimpcd 216 . . 3  |-  ( A R B  ->  ( B  =  C  ->  A R C ) )
32necon3bd 2638 . 2  |-  ( A R B  ->  ( -.  A R C  ->  B  =/=  C ) )
43imp 419 1  |-  ( ( A R B  /\  -.  A R C )  ->  B  =/=  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    =/= wne 2599   class class class wbr 4212
This theorem is referenced by:  dalem43  30512  cdleme3h  31032  cdleme7ga  31045  cdlemeg46req  31326  cdlemh  31614  cdlemk12  31647  cdlemk12u  31669
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213
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