Theorem postr

Description: A poset ordering is transitive. (Contributed by NM, 11-Sep-2011)

	Ref	Expression
Hypotheses	posi.b	$$⊢B={\mathrm{Base}}_K$$
	posi.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
Assertion	postr	$$⊢\left(K\in \mathrm{Poset}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\to X\underset{\dot{}}{\le }Z\right)$$

Proof

Step	Hyp	Ref	Expression
1		posi.b	$$⊢B={\mathrm{Base}}_K$$
2		posi.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3	1 2	posi	$$⊢\left(K\in \mathrm{Poset}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(X\underset{\dot{}}{\le }X\wedge \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }X\right)\to X=Y\right)\wedge \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\to X\underset{\dot{}}{\le }Z\right)\right)$$
4	3	simp3d	$$⊢\left(K\in \mathrm{Poset}\wedge \left(X\in B\wedge Y\in B\wedge Z\in B\right)\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge Y\underset{\dot{}}{\le }Z\right)\to X\underset{\dot{}}{\le }Z\right)$$