Theorem nbrne1 5094

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 5079	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶))
2	1	biimpcd 251	. . 3 ⊢ (𝐴𝑅𝐵 → (𝐵 = 𝐶 → 𝐴𝑅𝐶))
3	2	necon3bd 2950	. 2 ⊢ (𝐴𝑅𝐵 → (¬ 𝐴𝑅𝐶 → 𝐵 ≠ 𝐶))
4	3	imp 408	1 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1548   ≠ wne 2936   class class class wbr 5075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076

This theorem is referenced by:  zeneo  16303  dalem43  40222  cdleme3h  40742  cdleme7ga  40755  cdlemeg46req  41036  cdlemh  41324  cdlemk12  41357  cdlemk12u  41379  lighneallem1  48097