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 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) |
2 |
|
df-po |
⊢ ( 𝑅 Po ∅ ↔ ∀ 𝑥 ∈ ∅ ∀ 𝑦 ∈ ∅ ∀ 𝑧 ∈ ∅ ( ¬ 𝑥 𝑅 𝑥 ∧ ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) ) |
3 |
1 2
|
mpbir |
⊢ 𝑅 Po ∅ |