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 = ( Base ‘ 𝐾 )
Assertion	drsbn0	⊢ ( 𝐾 ∈ Dirset → 𝐵 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		drsbn0.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
3	1 2	isdrs	⊢ ( 𝐾 ∈ Dirset ↔ ( 𝐾 ∈ Proset ∧ 𝐵 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∃ 𝑧 ∈ 𝐵 ( 𝑥 ( le ‘ 𝐾 ) 𝑧 ∧ 𝑦 ( le ‘ 𝐾 ) 𝑧 ) ) )
4	3	simp2bi	⊢ ( 𝐾 ∈ Dirset → 𝐵 ≠ ∅ )