Theorem drsbn0

Description: The base of a directed set is not empty. (Contributed by Stefan O'Rear, 1-Feb-2015)

	Ref	Expression
Hypothesis	drsbn0.b	$$⊢B={\mathrm{Base}}_K$$
Assertion	drsbn0	$$⊢K\in \mathrm{Dirset}\to B\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		drsbn0.b	$$⊢B={\mathrm{Base}}_K$$
2		eqid	$$⊢{\le }_K={\le }_K$$
3	1 2	isdrs	$$⊢K\in \mathrm{Dirset}\leftrightarrow \left(K\in \mathrm{Proset}\wedge B\ne \varnothing \wedge \forall x\in B\forall y\in B\exists z\in B\left(x{\le }_{K}z\wedge y{\le }_{K}z\right)\right)$$
4	3	simp2bi	$$⊢K\in \mathrm{Dirset}\to B\ne \varnothing$$