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Theorem po0 4074
Description: Any relation is a partial ordering of the empty set. (Contributed by NM, 28-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
po0  |-  R  Po  (/)

Proof of Theorem po0
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 3350 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  A. z  e.  (/)  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) )
2 df-po 4059 . 2  |-  ( R  Po  (/)  <->  A. x  e.  (/)  A. y  e.  (/)  A. z  e.  (/)  ( -.  x R x  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
31, 2mpbir 144 1  |-  R  Po  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wral 2349   (/)c0 3258   class class class wbr 3793    Po wpo 4057
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 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-dif 2976  df-nul 3259  df-po 4059
This theorem is referenced by:  so0  4089
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