Theorem nbrne1 4008

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 3993	. . . 4 |- ( B = C -> ( A R B <-> A R C ) )
2	1	biimpcd 158	. . 3 |- ( A R B -> ( B = C -> A R C ) )
3	2	necon3bd 2383	. 2 |- ( A R B -> ( -. A R C -> B =/= C ) )
4	3	imp 123	1 |- ( ( A R B /\ -. A R C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1348   =/= wne 2340   class class class wbr 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990

This theorem is referenced by:  zeneo  11830