Metamath Proof Explorer


Theorem po0

Description: Any relation is a partial order on the empty set. (Contributed by NM, 28-Mar-1997) (Proof shortened by Andrew Salmon, 25-Jul-2011)

Ref Expression
Assertion po0 𝑅 Po ∅

Proof

Step Hyp Ref Expression
1 ral0 𝑥 ∈ ∅ ∀ 𝑦 ∈ ∅ ∀ 𝑧 ∈ ∅ ( ¬ 𝑥 𝑅 𝑥 ∧ ( ( 𝑥 𝑅 𝑦𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) )
2 df-po ( 𝑅 Po ∅ ↔ ∀ 𝑥 ∈ ∅ ∀ 𝑦 ∈ ∅ ∀ 𝑧 ∈ ∅ ( ¬ 𝑥 𝑅 𝑥 ∧ ( ( 𝑥 𝑅 𝑦𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) )
3 1 2 mpbir 𝑅 Po ∅