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

Theorem issod 4820
Description: An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
issod.1  |-  ( ph  ->  R  Po  A )
issod.2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x R y  \/  x  =  y  \/  y R x ) )
Assertion
Ref Expression
issod  |-  ( ph  ->  R  Or  A )
Distinct variable groups:    x, y, R    x, A, y    ph, x, y

Proof of Theorem issod
StepHypRef Expression
1 issod.1 . 2  |-  ( ph  ->  R  Po  A )
2 issod.2 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x R y  \/  x  =  y  \/  y R x ) )
32ralrimivva 2864 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )
4 df-so 4791 . 2  |-  ( R  Or  A  <->  ( R  Po  A  /\  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) ) )
51, 3, 4sylanbrc 664 1  |-  ( ph  ->  R  Or  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    \/ w3o 973    e. wcel 1804   A.wral 2793   class class class wbr 4437    Po wpo 4788    Or wor 4789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691
This theorem depends on definitions:  df-bi 185  df-an 371  df-ral 2798  df-so 4791
This theorem is referenced by:  issoi  4821  swoso  7344  wemapsolem  7978  legso  23857  socnv  29169  fin2so  30015
  Copyright terms: Public domain W3C validator