Theorem nbrne2 3863

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

Step	Hyp	Ref	Expression
1		breq1 3848	. . . 4 |- ( A = B -> ( A R C <-> B R C ) )
2	1	biimpcd 157	. . 3 |- ( A R C -> ( A = B -> B R C ) )
3	2	necon3bd 2298	. 2 |- ( A R C -> ( -. B R C -> A =/= B ) )
4	3	imp 122	1 |- ( ( A R C /\ -. B R C ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1289   =/= wne 2255   class class class wbr 3845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846

This theorem is referenced by: (None)