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Theorem issoi 4424
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
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 4401 . . 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 4423 . 2  |-  (  T. 
->  R  Or  A
)
98trud 1323 1  |-  R  Or  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    T. wtru 1316    = wceq 1642    e. wcel 1710   class class class wbr 4102    Or wor 4392
This theorem is referenced by:  isso2i  4425  ltsopr  8743  sltsolem1  24880
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2624  df-po 4393  df-so 4394
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