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Theorem so0 5539
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 5520 . 2 𝑅 Po ∅
2 ral0 4443 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
3 df-so 5504 . 2 (𝑅 Or ∅ ↔ (𝑅 Po ∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
41, 2, 3mpbir2an 708 1 𝑅 Or ∅
Colors of variables: wff setvar class
Syntax hints:  w3o 1085  wral 3064  c0 4256   class class class wbr 5074   Po wpo 5501   Or wor 5502
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 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-ral 3069  df-dif 3890  df-nul 4257  df-po 5503  df-so 5504
This theorem is referenced by:  we0  5584  wemapso2  9312
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