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Theorem nbrne2 3918
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 3902 . . . 4 (𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
21biimpcd 158 . . 3 (𝐴𝑅𝐶 → (𝐴 = 𝐵𝐵𝑅𝐶))
32necon3bd 2328 . 2 (𝐴𝑅𝐶 → (¬ 𝐵𝑅𝐶𝐴𝐵))
43imp 123 1 ((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1316  wne 2285   class class class wbr 3899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900
This theorem is referenced by: (None)
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