Theorem pltnle

Description: "Less than" implies not converse "less than or equal to". (Contributed by NM, 18-Oct-2011)

	Ref	Expression
Hypotheses	pleval2.b	$$⊢B={\mathrm{Base}}_K$$
	pleval2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	pleval2.s	$$⊢\underset{\dot{}}{<}={<}_K$$
Assertion	pltnle	$$⊢\left(\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{<}Y\right)\to \neg Y\underset{\dot{}}{\le }X$$

Proof

Step	Hyp	Ref	Expression
1		pleval2.b	$$⊢B={\mathrm{Base}}_K$$
2		pleval2.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		pleval2.s	$$⊢\underset{\dot{}}{<}={<}_K$$
4	2 3	pltval	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{<}Y\leftrightarrow \left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\right)$$
5	1 2	posasymb	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }X\right)\leftrightarrow X=Y\right)$$
6	5	biimpd	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }X\right)\to X=Y\right)$$
7	6	expdimp	$$⊢\left(\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(Y\underset{\dot{}}{\le }X\to X=Y\right)$$
8	7	necon3ad	$$⊢\left(\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(X\ne Y\to \neg Y\underset{\dot{}}{\le }X\right)$$
9	8	expimpd	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge X\ne Y\right)\to \neg Y\underset{\dot{}}{\le }X\right)$$
10	4 9	sylbid	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(X\underset{\dot{}}{<}Y\to \neg Y\underset{\dot{}}{\le }X\right)$$
11	10	imp	$$⊢\left(\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{<}Y\right)\to \neg Y\underset{\dot{}}{\le }X$$