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Theorem po0 5194
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 4212 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2 df-po 5179 . 2 (𝑅 Po ∅ ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ ∀𝑧 ∈ ∅ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
31, 2mpbir 221 1 𝑅 Po ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wral 3042  c0 4050   class class class wbr 4796   Po wpo 5177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-v 3334  df-dif 3710  df-nul 4051  df-po 5179
This theorem is referenced by:  so0  5212  posn  5336  dfpo2  31944  ipo0  39147
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