Theorem nbrne2 4410

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 4395	. . . 4 |- ( A = B -> ( A R C <-> B R C ) )
2	1	biimpcd 224	. . 3 |- ( A R C -> ( A = B -> B R C ) )
3	2	necon3bd 2660	. 2 |- ( A R C -> ( -. B R C -> A =/= B ) )
4	3	imp 429	1 |- ( ( A R C /\ -. B R C ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1370   =/= wne 2644   class class class wbr 4392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393

This theorem is referenced by:  frfi  7660  hl2at  33357  2atjm  33397  atbtwn  33398  atbtwnexOLDN  33399  atbtwnex  33400  dalem21  33646  dalem23  33648  dalem27  33651  dalem54  33678  2llnma1b  33738  lhpexle1lem  33959  lhpexle3lem  33963  lhp2at0nle  33987  4atexlemunv  34018  4atexlemnclw  34022  4atexlemcnd  34024  cdlemc5  34147  cdleme0b  34164  cdleme0c  34165  cdleme0fN  34170  cdleme01N  34173  cdleme0ex2N  34176  cdleme3b  34181  cdleme3c  34182  cdleme3g  34186  cdleme3h  34187  cdleme7aa  34194  cdleme7b  34196  cdleme7c  34197  cdleme7d  34198  cdleme7e  34199  cdleme7ga  34200  cdleme11fN  34216  cdlemesner  34248  cdlemednpq  34251  cdleme19a  34255  cdleme19c  34257  cdleme21c  34279  cdleme21ct  34281  cdleme22cN  34294  cdleme22f2  34299  cdleme22g  34300  cdleme41sn3aw  34426  cdlemeg46rgv  34480  cdlemeg46req  34481  cdlemf1  34513  cdlemg27b  34648  cdlemg33b0  34653  cdlemg33c0  34654  cdlemh  34769  cdlemk14  34806  dia2dimlem1  35017