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Theorem nbrne1 4473
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 4460 . . . 4  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
21biimpcd 224 . . 3  |-  ( A R B  ->  ( B  =  C  ->  A R C ) )
32necon3bd 2669 . 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 1395    =/= wne 2652   class class class wbr 4456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457
This theorem is referenced by:  dalem43  35582  cdleme3h  36103  cdleme7ga  36116  cdlemeg46req  36398  cdlemh  36686  cdlemk12  36719  cdlemk12u  36741
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