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Theorem so0 5478
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 5459 . 2 𝑅 Po ∅
2 ral0 4399 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
3 df-so 5443 . 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 1087  wral 3053  c0 4211   class class class wbr 5030   Po wpo 5440   Or wor 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-ral 3058  df-dif 3846  df-nul 4212  df-po 5442  df-so 5443
This theorem is referenced by:  we0  5520  wemapso2  9090
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