Theorem nbrne1 4112

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 4097	. . . 4 |- ( B = C -> ( A R B <-> A R C ) )
2	1	biimpcd 159	. . 3 |- ( A R B -> ( B = C -> A R C ) )
3	2	necon3bd 2446	. 2 |- ( A R B -> ( -. A R C -> B =/= C ) )
4	3	imp 124	1 |- ( ( A R B /\ -. A R C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1398   =/= wne 2403   class class class wbr 4093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094

This theorem is referenced by:  zeneo  12512