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Theorem so0 5634
Description: Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
so0 𝑅 Or ∅

Proof of Theorem so0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 po0 5614 . 2 𝑅 Po ∅
2 ral0 4519 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
3 df-so 5598 . 2 (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
41, 2, 3mpbir2an 711 1 𝑅 Or ∅
Colors of variables: wff setvar class
Syntax hints:  w3o 1085  wral 3059  c0 4339   class class class wbr 5148   Po wpo 5595   Or wor 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-ral 3060  df-dif 3966  df-nul 4340  df-po 5597  df-so 5598
This theorem is referenced by:  we0  5684  wemapso2  9591
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