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Theorem nbrne2 4603
 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 ((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴𝐵)

Proof of Theorem nbrne2
StepHypRef Expression
1 breq1 4586 . . . 4 (𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
21biimpcd 238 . . 3 (𝐴𝑅𝐶 → (𝐴 = 𝐵𝐵𝑅𝐶))
32necon3bd 2796 . 2 (𝐴𝑅𝐶 → (¬ 𝐵𝑅𝐶𝐴𝐵))
43imp 444 1 ((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ≠ wne 2780   class class class wbr 4583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584 This theorem is referenced by:  frfi  8090  hl2at  33709  2atjm  33749  atbtwn  33750  atbtwnexOLDN  33751  atbtwnex  33752  dalem21  33998  dalem23  34000  dalem27  34003  dalem54  34030  2llnma1b  34090  lhpexle1lem  34311  lhpexle3lem  34315  lhp2at0nle  34339  4atexlemunv  34370  4atexlemnclw  34374  4atexlemcnd  34376  cdlemc5  34500  cdleme0b  34517  cdleme0c  34518  cdleme0fN  34523  cdleme01N  34526  cdleme0ex2N  34529  cdleme3b  34534  cdleme3c  34535  cdleme3g  34539  cdleme3h  34540  cdleme7aa  34547  cdleme7b  34549  cdleme7c  34550  cdleme7d  34551  cdleme7e  34552  cdleme7ga  34553  cdleme11fN  34569  cdlemesner  34601  cdlemednpq  34604  cdleme19a  34609  cdleme19c  34611  cdleme21c  34633  cdleme21ct  34635  cdleme22cN  34648  cdleme22f2  34653  cdleme22g  34654  cdleme41sn3aw  34780  cdlemeg46rgv  34834  cdlemeg46req  34835  cdlemf1  34867  cdlemg27b  35002  cdlemg33b0  35007  cdlemg33c0  35008  cdlemh  35123  cdlemk14  35160  dia2dimlem1  35371
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