Theorem nbrne1 5108

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) → 𝐵 ≠ 𝐶)

Proof of Theorem nbrne1

Step	Hyp	Ref	Expression
1		breq2 5093	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶))
2	1	biimpcd 249	. . 3 ⊢ (𝐴𝑅𝐵 → (𝐵 = 𝐶 → 𝐴𝑅𝐶))
3	2	necon3bd 2942	. 2 ⊢ (𝐴𝑅𝐵 → (¬ 𝐴𝑅𝐶 → 𝐵 ≠ 𝐶))
4	3	imp 406	1 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ≠ wne 2928   class class class wbr 5089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090

This theorem is referenced by:  zeneo  16250  dalem43  39824  cdleme3h  40344  cdleme7ga  40357  cdlemeg46req  40638  cdlemh  40926  cdlemk12  40959  cdlemk12u  40981  lighneallem1  47715