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Theorem nbrne1 4416
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 4403 . . . 4  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
21biimpcd 224 . . 3  |-  ( A R B  ->  ( B  =  C  ->  A R C ) )
32necon3bd 2663 . 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 1370    =/= wne 2647   class class class wbr 4399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400
This theorem is referenced by:  dalem43  33682  cdleme3h  34202  cdleme7ga  34215  cdlemeg46req  34496  cdlemh  34784  cdlemk12  34817  cdlemk12u  34839
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