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Theorem so0 5309
 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 5290 . 2 𝑅 Po ∅
2 ral0 4299 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
3 df-so 5275 . 2 (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
41, 2, 3mpbir2an 701 1 𝑅 Or ∅
 Colors of variables: wff setvar class Syntax hints:   ∨ w3o 1070  ∀wral 3090  ∅c0 4141   class class class wbr 4886   Po wpo 5272   Or wor 5273 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-ral 3095  df-dif 3795  df-nul 4142  df-po 5274  df-so 5275 This theorem is referenced by:  we0  5350  wemapso2  8747
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