Theorem nbrne1

Description: Two classes are different if they don't have the same relationship to a third class. (Contributed by NM, 3-Jun-2012)

	Ref	Expression
Assertion	nbrne1	⊢ ( ( 𝐴 𝑅 𝐵 ∧ ¬ 𝐴 𝑅 𝐶 ) → 𝐵 ≠ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		breq2	⊢ ( 𝐵 = 𝐶 → ( 𝐴 𝑅 𝐵 ↔ 𝐴 𝑅 𝐶 ) )
2	1	biimpcd	⊢ ( 𝐴 𝑅 𝐵 → ( 𝐵 = 𝐶 → 𝐴 𝑅 𝐶 ) )
3	2	necon3bd	⊢ ( 𝐴 𝑅 𝐵 → ( ¬ 𝐴 𝑅 𝐶 → 𝐵 ≠ 𝐶 ) )
4	3	imp	⊢ ( ( 𝐴 𝑅 𝐵 ∧ ¬ 𝐴 𝑅 𝐶 ) → 𝐵 ≠ 𝐶 )