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Theorem so0 4109
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  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 po0 4094 . 2  |-  R  Po  (/)
2 ral0 3359 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  A. z  e.  (/)  ( x R y  ->  ( x R z  \/  z R y ) )
3 df-iso 4080 . 2  |-  ( R  Or  (/)  <->  ( R  Po  (/) 
/\  A. x  e.  (/)  A. y  e.  (/)  A. z  e.  (/)  ( x R y  ->  ( x R z  \/  z R y ) ) ) )
41, 2, 3mpbir2an 884 1  |-  R  Or  (/)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662   A.wral 2353   (/)c0 3267   class class class wbr 3805    Po wpo 4077    Or wor 4078
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-dif 2984  df-nul 3268  df-po 4079  df-iso 4080
This theorem is referenced by: (None)
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