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Theorem so0 4347
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 4329 . 2  |-  R  Po  (/)
2 ral0 3558 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  ( x R y  \/  x  =  y  \/  y R x )
3 df-so 4315 . 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 1623   A.wral 2543   (/)c0 3455   class class class wbr 4023    Po wpo 4312    Or wor 4313
This theorem is referenced by:  we0  4388
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  df-so 4315
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