Theorem nbrne2 3803

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 3788	. . . 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 2288	. 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 1284   =/= wne 2245   class class class wbr 3785

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786

This theorem is referenced by: (None)