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Theorem nbrne2 4458
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
nbrne2  |-  ( ( A R C  /\  -.  B R C )  ->  A  =/=  B
)

Proof of Theorem nbrne2
StepHypRef Expression
1 breq1 4443 . . . 4  |-  ( A  =  B  ->  ( A R C  <->  B R C ) )
21biimpcd 224 . . 3  |-  ( A R C  ->  ( A  =  B  ->  B R C ) )
32necon3bd 2672 . 2  |-  ( A R C  ->  ( -.  B R C  ->  A  =/=  B ) )
43imp 429 1  |-  ( ( A R C  /\  -.  B R C )  ->  A  =/=  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    =/= wne 2655   class class class wbr 4440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441
This theorem is referenced by:  frfi  7754  hl2at  34076  2atjm  34116  atbtwn  34117  atbtwnexOLDN  34118  atbtwnex  34119  dalem21  34365  dalem23  34367  dalem27  34370  dalem54  34397  2llnma1b  34457  lhpexle1lem  34678  lhpexle3lem  34682  lhp2at0nle  34706  4atexlemunv  34737  4atexlemnclw  34741  4atexlemcnd  34743  cdlemc5  34866  cdleme0b  34883  cdleme0c  34884  cdleme0fN  34889  cdleme01N  34892  cdleme0ex2N  34895  cdleme3b  34900  cdleme3c  34901  cdleme3g  34905  cdleme3h  34906  cdleme7aa  34913  cdleme7b  34915  cdleme7c  34916  cdleme7d  34917  cdleme7e  34918  cdleme7ga  34919  cdleme11fN  34935  cdlemesner  34967  cdlemednpq  34970  cdleme19a  34974  cdleme19c  34976  cdleme21c  34998  cdleme21ct  35000  cdleme22cN  35013  cdleme22f2  35018  cdleme22g  35019  cdleme41sn3aw  35145  cdlemeg46rgv  35199  cdlemeg46req  35200  cdlemf1  35232  cdlemg27b  35367  cdlemg33b0  35372  cdlemg33c0  35373  cdlemh  35488  cdlemk14  35525  dia2dimlem1  35736
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