Theorem drsprs

Description: A directed set is a proset. (Contributed by Stefan O'Rear, 1-Feb-2015)

	Ref	Expression
Assertion	drsprs	⊢ ( 𝐾 ∈ Dirset → 𝐾 ∈ Proset )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
2		eqid	⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
3	1 2	isdrs	⊢ ( 𝐾 ∈ Dirset ↔ ( 𝐾 ∈ Proset ∧ ( Base ‘ 𝐾 ) ≠ ∅ ∧ ∀ 𝑥 ∈ ( Base ‘ 𝐾 ) ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∃ 𝑧 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( le ‘ 𝐾 ) 𝑧 ∧ 𝑦 ( le ‘ 𝐾 ) 𝑧 ) ) )
4	3	simp1bi	⊢ ( 𝐾 ∈ Dirset → 𝐾 ∈ Proset )