Theorem nbrne1 3828

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

Step	Hyp	Ref	Expression
1		breq2 3815	. . . 4 |- ( B = C -> ( A R B <-> A R C ) )
2	1	biimpcd 157	. . 3 |- ( A R B -> ( B = C -> A R C ) )
3	2	necon3bd 2292	. 2 |- ( A R B -> ( -. A R C -> B =/= C ) )
4	3	imp 122	1 |- ( ( A R B /\ -. A R C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1285   =/= wne 2249   class class class wbr 3811

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812

This theorem is referenced by:  zeneo  10651