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Theorem issod 4536
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 2800 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  A  ( x R y  \/  x  =  y  \/  y R x ) )
4 df-so 4507 . 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 647 1  |-  ( ph  ->  R  Or  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    \/ w3o 936    e. wcel 1726   A.wral 2707   class class class wbr 4215    Po wpo 4504    Or wor 4505
This theorem is referenced by:  issoi  4537  swoso  6939  wemapso2lem  7522  socnv  25393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-ral 2712  df-so 4507
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