MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  so0 Unicode version

Theorem so0 4384
Description: Any relation is a strict ordering of the empty set. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
so0  |-  R  Or  (/)

Proof of Theorem so0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 po0 4366 . 2  |-  R  Po  (/)
2 ral0 3592 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  ( x R y  \/  x  =  y  \/  y R x )
3 df-so 4352 . 2  |-  ( R  Or  (/)  <->  ( R  Po  (/) 
/\  A. x  e.  (/)  A. y  e.  (/)  ( x R y  \/  x  =  y  \/  y R x ) ) )
41, 2, 3mpbir2an 886 1  |-  R  Or  (/)
Colors of variables: wff set class
Syntax hints:    \/ w3o 933    = wceq 1633   A.wral 2577   (/)c0 3489   class class class wbr 4060    Po wpo 4349    Or wor 4350
This theorem is referenced by:  we0  4425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-v 2824  df-dif 3189  df-nul 3490  df-po 4351  df-so 4352
  Copyright terms: Public domain W3C validator