Theorem ltcvrntr

Description: Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012)

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

Proof

Step	Hyp	Ref	Expression
1		ltltncvr.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		ltltncvr.s	⊢ < = ( lt ‘ 𝐾 )
3		ltltncvr.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
4	1 2 3	cvrlt	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ∧ 𝑌 𝐶 𝑍 ) → 𝑌 < 𝑍 )
5	4	ex	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) → ( 𝑌 𝐶 𝑍 → 𝑌 < 𝑍 ) )
6	5	3adant3r1	⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑌 𝐶 𝑍 → 𝑌 < 𝑍 ) )
7	1 2 3	ltltncvr	⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑍 ) → ¬ 𝑋 𝐶 𝑍 ) )
8	6 7	sylan2d	⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 𝐶 𝑍 ) → ¬ 𝑋 𝐶 𝑍 ) )