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Theorem po0 4329
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 3558 . 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 4314 . 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 200 1  |-  R  Po  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wral 2543   (/)c0 3455   class class class wbr 4023    Po wpo 4312
This theorem is referenced by:  so0  4347  posn  4758  dfpo2  24112  ipo0  27652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-dif 3155  df-nul 3456  df-po 4314
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