Theorem nbrne2 3782

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
nbrne2	|- ( ( A R C /\ -. B R C ) -> A =/= B )

Proof of Theorem nbrne2

Step	Hyp	Ref	Expression
1		breq1 3767	. . . 4 |- ( A = B -> ( A R C <-> B R C ) )
2	1	biimpcd 148	. . 3 |- ( A R C -> ( A = B -> B R C ) )
3	2	necon3bd 2248	. 2 |- ( A R C -> ( -. B R C -> A =/= B ) )
4	3	imp 115	1 |- ( ( A R C /\ -. B R C ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   = wceq 1243   =/= wne 2204   class class class wbr 3764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765

This theorem is referenced by: (None)