Theorem latasymb

Description: A lattice ordering is asymmetric. ( eqss analog.) (Contributed by NM, 22-Oct-2011)

	Ref	Expression
Hypotheses	latref.b	$$⊢B={\mathrm{Base}}_K$$
	latref.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
Assertion	latasymb	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }X\right)\leftrightarrow X=Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		latref.b	$$⊢B={\mathrm{Base}}_K$$
2		latref.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		latpos	$$⊢K\in \mathrm{Lat}\to K\in \mathrm{Poset}$$
4	1 2	posasymb	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\wedge Y\in B\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }X\right)\leftrightarrow X=Y\right)$$
5	3 4	syl3an1	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\in B\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }X\right)\leftrightarrow X=Y\right)$$