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Theorem so0 4063
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 4048 . 2  |-  R  Po  (/)
2 ral0 3322 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  A. z  e.  (/)  ( x R y  ->  ( x R z  \/  z R y ) )
3 df-iso 4034 . 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 849 1  |-  R  Or  (/)
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 629   A.wral 2306   (/)c0 3224   class class class wbr 3764    Po wpo 4031    Or wor 4032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-nul 3225  df-po 4033  df-iso 4034
This theorem is referenced by: (None)
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