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Theorem issoi 4345
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
issoi.1  |-  ( x  e.  A  ->  -.  x R x )
issoi.2  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( ( x R y  /\  y R z )  ->  x R z ) )
issoi.3  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  \/  x  =  y  \/  y R x ) )
Assertion
Ref Expression
issoi  |-  R  Or  A
Distinct variable groups:    x, y,
z, R    x, A, y, z

Proof of Theorem issoi
StepHypRef Expression
1 issoi.1 . . . . 5  |-  ( x  e.  A  ->  -.  x R x )
21adantl 452 . . . 4  |-  ( (  T.  /\  x  e.  A )  ->  -.  x R x )
3 issoi.2 . . . . 5  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  ->  ( ( x R y  /\  y R z )  ->  x R z ) )
43adantl 452 . . . 4  |-  ( (  T.  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  y R z )  ->  x R z ) )
52, 4ispod 4322 . . 3  |-  (  T. 
->  R  Po  A
)
6 issoi.3 . . . 4  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x R y  \/  x  =  y  \/  y R x ) )
76adantl 452 . . 3  |-  ( (  T.  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x R y  \/  x  =  y  \/  y R x ) )
85, 7issod 4344 . 2  |-  (  T. 
->  R  Or  A
)
98trud 1314 1  |-  R  Or  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    T. wtru 1307    = wceq 1623    e. wcel 1684   class class class wbr 4023    Or wor 4313
This theorem is referenced by:  isso2i  4346  ltsopr  8656  sltsolem1  24322
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-3an 936  df-tru 1310  df-nf 1532  df-ral 2548  df-po 4314  df-so 4315
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