Theorem nbrne1 4052

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 4037	. . . 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 2410	. 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 1364   =/= wne 2367   class class class wbr 4033

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034

This theorem is referenced by:  zeneo  12036