MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbrne1 Structured version   Visualization version   GIF version

Theorem nbrne1 4823
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 ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵𝐶)

Proof of Theorem nbrne1
StepHypRef Expression
1 breq2 4808 . . . 4 (𝐵 = 𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
21biimpcd 239 . . 3 (𝐴𝑅𝐵 → (𝐵 = 𝐶𝐴𝑅𝐶))
32necon3bd 2946 . 2 (𝐴𝑅𝐵 → (¬ 𝐴𝑅𝐶𝐵𝐶))
43imp 444 1 ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wne 2932   class class class wbr 4804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805
This theorem is referenced by:  zeneo  15265  dalem43  35504  cdleme3h  36025  cdleme7ga  36038  cdlemeg46req  36319  cdlemh  36607  cdlemk12  36640  cdlemk12u  36662  lighneallem1  42032
  Copyright terms: Public domain W3C validator