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Theorem so0 5630
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 5609 . 2 𝑅 Po ∅
2 ral0 4513 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
3 df-so 5593 . 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 1086  wral 3061  c0 4333   class class class wbr 5143   Po wpo 5590   Or wor 5591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-ral 3062  df-dif 3954  df-nul 4334  df-po 5592  df-so 5593
This theorem is referenced by:  we0  5680  wemapso2  9593
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