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Theorem nbrne1 3781
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 3768 . . . 4  |-  ( B  =  C  ->  ( A R B  <->  A R C ) )
21biimpcd 148 . . 3  |-  ( A R B  ->  ( B  =  C  ->  A R C ) )
32necon3bd 2248 . 2  |-  ( A R B  ->  ( -.  A R C  ->  B  =/=  C ) )
43imp 115 1  |-  ( ( A R B  /\  -.  A R C )  ->  B  =/=  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    = wceq 1243    =/= wne 2204   class class class wbr 3764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765
This theorem is referenced by: (None)
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