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Theorem issod 4344
Description: A 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 2635 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )
4 df-so 4315 . 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 645 1  |-  ( ph  ->  R  Or  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023    Po wpo 4312    Or wor 4313
This theorem is referenced by:  issoi  4345  swoso  6691  wemapso2lem  7265
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-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-nf 1532  df-ral 2548  df-so 4315
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