MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  po0 Structured version   Unicode version

Theorem po0 4520
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 3734 . 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 4505 . 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 202 1  |-  R  Po  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360   A.wral 2707   (/)c0 3630   class class class wbr 4214    Po wpo 4503
This theorem is referenced by:  so0  4538  posn  4948  dfpo2  25380  ipo0  27630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-dif 3325  df-nul 3631  df-po 4505
  Copyright terms: Public domain W3C validator