Theorem cvrle

Description: The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011)

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

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	$$⊢\left(\left(K\in A\wedge X\in B\wedge Y\in B\right)\wedge XCY\right)\to X{<}_{K}Y$$
6	2 4	pltval	$$⊢\left(K\in A\wedge X\in B\wedge Y\in B\right)\to \left(X{<}_{K}Y\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)$$
7	6	simprbda	$$⊢\left(\left(K\in A\wedge X\in B\wedge Y\in B\right)\wedge X{<}_{K}Y\right)\to X\underset{\dot{}}{\le }Y$$
8	5 7	syldan	$$⊢\left(\left(K\in A\wedge X\in B\wedge Y\in B\right)\wedge XCY\right)\to X\underset{\dot{}}{\le }Y$$