Theorem cvrne

Description: The covers relation implies inequality. (Contributed by NM, 13-Oct-2011)

	Ref	Expression
Hypotheses	cvrne.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cvrne.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
Assertion	cvrne	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → 𝑋 ≠ 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		cvrne.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cvrne.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
3		eqid	⊢ ( lt ‘ 𝐾 ) = ( lt ‘ 𝐾 )
4	1 3 2	cvrlt	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → 𝑋 ( lt ‘ 𝐾 ) 𝑌 )
5		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6	5 3	pltval	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( lt ‘ 𝐾 ) 𝑌 ↔ ( 𝑋 ( le ‘ 𝐾 ) 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
7	6	simplbda	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ( lt ‘ 𝐾 ) 𝑌 ) → 𝑋 ≠ 𝑌 )
8	4 7	syldan	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → 𝑋 ≠ 𝑌 )