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Theorem nbrne1 4037
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 4022 . . . 4 (𝐵 = 𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
21biimpcd 159 . . 3 (𝐴𝑅𝐵 → (𝐵 = 𝐶𝐴𝑅𝐶))
32necon3bd 2403 . 2 (𝐴𝑅𝐵 → (¬ 𝐴𝑅𝐶𝐵𝐶))
43imp 124 1 ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1364  wne 2360   class class class wbr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019
This theorem is referenced by:  zeneo  11907
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