Theorem po0 5595

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.)

Assertion

Ref	Expression
po0	⊢ 𝑅 Po ∅

Proof of Theorem po0

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4504	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2		df-po 5578	. 2 ⊢ (𝑅 Po ∅ ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
3	1, 2	mpbir 230	1 ⊢ 𝑅 Po ∅

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wral 3053  ∅c0 4314   class class class wbr 5138   Po wpo 5576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-ral 3054  df-dif 3943  df-nul 4315  df-po 5578

This theorem is referenced by:  so0  5614  posn  5751  dfpo2  6285  ipo0  43697