Theorem posref

Description: A poset ordering is reflexive. (Contributed by NM, 11-Sep-2011) (Proof shortened by OpenAI, 25-Mar-2020)

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

Proof

Step	Hyp	Ref	Expression
1		posi.b	$$⊢B={\mathrm{Base}}_K$$
2		posi.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		posprs	$$⊢K\in \mathrm{Poset}\to K\in \mathrm{Proset}$$
4	1 2	prsref	$$⊢\left(K\in \mathrm{Proset}\wedge X\in B\right)\to X\underset{\dot{}}{\le }X$$
5	3 4	sylan	$$⊢\left(K\in \mathrm{Poset}\wedge X\in B\right)\to X\underset{\dot{}}{\le }X$$