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Theorem po0 5460
 Description: Any relation is a partial ordering of 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.
StepHypRef Expression
1 ral0 4406 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2 df-po 5444 . 2 (𝑅 Po ∅ ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
31, 2mpbir 234 1 𝑅 Po ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wral 3071  ∅c0 4226   class class class wbr 5033   Po wpo 5442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-ral 3076  df-dif 3862  df-nul 4227  df-po 5444 This theorem is referenced by:  so0  5479  posn  5607  dfpo2  33239  ipo0  41527
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