Theorem ltltncvr

Description: A chained strong ordering is not a 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	ltltncvr	⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 < 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		simpll	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → 𝐾 ∈ 𝐴 )
5		simplr1	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → 𝑋 ∈ 𝐵 )
6		simplr3	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → 𝑍 ∈ 𝐵 )
7		simplr2	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → 𝑌 ∈ 𝐵 )
8		simpr	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → 𝑋 𝐶 𝑍 )
9	1 2 3	cvrnbtwn	⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑍 ) → ¬ ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑍 ) )
10	4 5 6 7 8 9	syl131anc	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) ∧ 𝑋 𝐶 𝑍 ) → ¬ ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑍 ) )
11	10	ex	⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( 𝑋 𝐶 𝑍 → ¬ ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑍 ) ) )
12	11	con2d	⊢ ( ( 𝐾 ∈ 𝐴 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 < 𝑌 ∧ 𝑌 < 𝑍 ) → ¬ 𝑋 𝐶 𝑍 ) )