Theorem cvrnle

Description: The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011)

	Ref	Expression
Hypotheses	cvrle.b	$$⊢B={\mathrm{Base}}_K$$
	cvrle.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cvrle.c	$$⊢C={⋖}_K$$
Assertion	cvrnle	$$⊢\left(\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\wedge XCY\right)\to \neg Y\underset{\dot{}}{\le }X$$

Proof

Step	Hyp	Ref	Expression
1		cvrle.b	$$⊢B={\mathrm{Base}}_K$$
2		cvrle.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cvrle.c	$$⊢C={⋖}_K$$
4		eqid	$$⊢{<}_K={<}_K$$
5	1 4 3	cvrlt	$$⊢\left(\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\wedge XCY\right)\to X{<}_{K}Y$$
6	1 2 4	pltnle	$$⊢\left(\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\wedge X{<}_{K}Y\right)\to \neg Y\underset{\dot{}}{\le }X$$
7	5 6	syldan	$$⊢\left(\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\wedge XCY\right)\to \neg Y\underset{\dot{}}{\le }X$$