Theorem nbrne1 3947

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 3933	. . . 4 ⊢ (𝐵 = 𝐶 → (𝐴𝑅𝐵 ↔ 𝐴𝑅𝐶))
2	1	biimpcd 158	. . 3 ⊢ (𝐴𝑅𝐵 → (𝐵 = 𝐶 → 𝐴𝑅𝐶))
3	2	necon3bd 2351	. 2 ⊢ (𝐴𝑅𝐵 → (¬ 𝐴𝑅𝐶 → 𝐵 ≠ 𝐶))
4	3	imp 123	1 ⊢ ((𝐴𝑅𝐵 ∧ ¬ 𝐴𝑅𝐶) → 𝐵 ≠ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1331   ≠ wne 2308   class class class wbr 3929

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930

This theorem is referenced by:  zeneo  11568