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Theorem ispod 4471
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 1122 . . . 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 519 . . 3  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A ) )  -> 
( -.  x R x  /\  ( ( x R y  /\  y R z )  ->  x R z ) ) )
54ralrimivvva 2759 . 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 4463 . 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 204 1  |-  ( ph  ->  R  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721   A.wral 2666   class class class wbr 4172    Po wpo 4461
This theorem is referenced by:  swopo  4473  pofun  4479  issoi  4494  wemappo  7474  pospo  14385
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-ex 1548  df-nf 1551  df-ral 2671  df-po 4463
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