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Metamath Proof Explorer

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 ≠ ∅ )