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Theorem po0 4769
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 3873 . 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 4754 . 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 213 1  |-  R  Po  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371   A.wral 2736   (/)c0 3730   class class class wbr 4401    Po wpo 4752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ral 2741  df-v 3046  df-dif 3406  df-nul 3731  df-po 4754
This theorem is referenced by:  so0  4787  posn  4902  dfpo2  30388  ipo0  36796
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