Theorem cvrntr

Description: The covers relation is not transitive. ( cvntr analog.) (Contributed by NM, 18-Jun-2012)

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

Proof

Step	Hyp	Ref	Expression
1		cvrntr.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cvrntr.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	4	ex	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 → 𝑋 ( lt ‘ 𝐾 ) 𝑌 ) )
6	5	3adant3r3	⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 𝐶 𝑌 → 𝑋 ( lt ‘ 𝐾 ) 𝑌 ) )
7	1 3 2	ltcvrntr	⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ( lt ‘ 𝐾 ) 𝑌 ∧ 𝑌 𝐶 𝑍 ) → ¬ 𝑋 𝐶 𝑍 ) )
8	6 7	syland	⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 𝐶 𝑌 ∧ 𝑌 𝐶 𝑍 ) → ¬ 𝑋 𝐶 𝑍 ) )