Theorem cvrlt

Description: The covers relation implies the less-than relation. ( cvpss analog.) (Contributed by NM, 8-Oct-2011)

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

Proof

Step	Hyp	Ref	Expression
1		cvrfval.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cvrfval.s	⊢ < = ( lt ‘ 𝐾 )
3		cvrfval.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
4	1 2 3	cvrval	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝐶 𝑌 ↔ ( 𝑋 < 𝑌 ∧ ¬ ∃ 𝑧 ∈ 𝐵 ( 𝑋 < 𝑧 ∧ 𝑧 < 𝑌 ) ) ) )
5	4	simprbda	⊢ ( ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 𝐶 𝑌 ) → 𝑋 < 𝑌 )