Theorem nbrne1 4672

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 4657	. . . 4 |- ( B = C -> ( A R B <-> A R C ) )
2	1	biimpcd 239	. . 3 |- ( A R B -> ( B = C -> A R C ) )
3	2	necon3bd 2808	. 2 |- ( A R B -> ( -. A R C -> B =/= C ) )
4	3	imp 445	1 |- ( ( A R B /\ -. A R C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  zeneo  15063  dalem43  35001  cdleme3h  35522  cdleme7ga  35535  cdlemeg46req  35817  cdlemh  36105  cdlemk12  36138  cdlemk12u  36160  lighneallem1  41522