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Theorem nbrne2 3985
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 3969 . . . 4 (𝐴 = 𝐵 → (𝐴𝑅𝐶𝐵𝑅𝐶))
21biimpcd 158 . . 3 (𝐴𝑅𝐶 → (𝐴 = 𝐵𝐵𝑅𝐶))
32necon3bd 2370 . 2 (𝐴𝑅𝐶 → (¬ 𝐵𝑅𝐶𝐴𝐵))
43imp 123 1 ((𝐴𝑅𝐶 ∧ ¬ 𝐵𝑅𝐶) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1335  wne 2327   class class class wbr 3966
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967
This theorem is referenced by: (None)
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