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Theorem ispod 4514
Description: Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
Hypotheses
Ref Expression
ispod.1  |-  ( (
ph  /\  x  e.  A )  ->  -.  x R x )
ispod.2  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  y R z )  ->  x R z ) )
Assertion
Ref Expression
ispod  |-  ( ph  ->  R  Po  A )
Distinct variable groups:    x, y,
z, A    x, R, y, z    ph, x, y, z

Proof of Theorem ispod
StepHypRef Expression
1 ispod.1 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  -.  x R x )
213ad2antr1 1123 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  ->  -.  x R x )
3 ispod.2 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( ( x R y  /\  y R z )  ->  x R z ) )
42, 3jca 520 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
54ralrimivvva 2801 . 2  |-  ( ph  ->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
6 df-po 4506 . 2  |-  ( R  Po  A  <->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( -.  x R x  /\  (
( x R y  /\  y R z )  ->  x R
z ) ) )
75, 6sylibr 205 1  |-  ( ph  ->  R  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    e. wcel 1726   A.wral 2707   class class class wbr 4215    Po wpo 4504
This theorem is referenced by:  swopo  4516  pofun  4522  issoi  4537  wemappo  7521  pospo  14435  pocnv  25392
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-3an 939  df-ex 1552  df-nf 1555  df-ral 2712  df-po 4506
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