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

Theorem nbrne1 5076
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 5061 . . . 4 (𝐵 = 𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
21biimpcd 251 . . 3 (𝐴𝑅𝐵 → (𝐵 = 𝐶𝐴𝑅𝐶))
32necon3bd 3028 . 2 (𝐴𝑅𝐵 → (¬ 𝐴𝑅𝐶𝐵𝐶))
43imp 409 1 ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1531  wne 3014   class class class wbr 5057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058
This theorem is referenced by:  zeneo  15680  dalem43  36843  cdleme3h  37363  cdleme7ga  37376  cdlemeg46req  37657  cdlemh  37945  cdlemk12  37978  cdlemk12u  38000  lighneallem1  43761
  Copyright terms: Public domain W3C validator