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Theorem nbrne1 5128
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 5113 . . . 4 (𝐵 = 𝐶 → (𝐴𝑅𝐵𝐴𝑅𝐶))
21biimpcd 249 . . 3 (𝐴𝑅𝐵 → (𝐵 = 𝐶𝐴𝑅𝐶))
32necon3bd 2954 . 2 (𝐴𝑅𝐵 → (¬ 𝐴𝑅𝐶𝐵𝐶))
43imp 408 1 ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wne 2940   class class class wbr 5109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110
This theorem is referenced by:  zeneo  16229  dalem43  38228  cdleme3h  38748  cdleme7ga  38761  cdlemeg46req  39042  cdlemh  39330  cdlemk12  39363  cdlemk12u  39385  lighneallem1  45887
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