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Theorem po0 4785
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 3902 . 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 4770 . 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 212 1  |-  R  Po  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370   A.wral 2775   (/)c0 3761   class class class wbr 4420    Po wpo 4768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-v 3083  df-dif 3439  df-nul 3762  df-po 4770
This theorem is referenced by:  so0  4803  posn  4918  dfpo2  30389  ipo0  36659
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